year stringdate 1961-01-01 00:00:00 2025-01-01 00:00:00 β | tier stringclasses 5
values | problem_label stringclasses 119
values | problem_type stringclasses 13
values | exam stringclasses 28
values | problem stringlengths 87 2.77k | solution stringlengths 834 13k | metadata dict | problem_tokens int64 50 903 | solution_tokens int64 500 3.93k |
|---|---|---|---|---|---|---|---|---|---|
2019 | T0 | 5 | null | USA_TSTST | Let $A B C$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. A line through $H$ intersects segments $A B$ and $A C$ at $E$ and $F$, respectively. Let $K$ be the circumcenter of $\triangle A E F$, and suppose line $A K$ intersects $\Gamma$ again at a point $D$. Prove that line $H K$ and the line thro... | γ Seventh solution using moving points (Zack Chroman). We state the converse of the problem as follows: Take a point $D$ on $\Gamma$, and let $G \in \Gamma$ such that $\overline{D G} \perp \overline{B C}$. Then define $K$ to lie on $\overline{G H}, \overline{A D}$, and take $L \in \overline{A D}$ such that $K$ is the m... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2019.jsonl",
"solution_match": null
} | 112 | 869 |
2019 | T0 | 7 | null | USA_TSTST | Let $f: \mathbb{Z} \rightarrow\left\{1,2, \ldots, 10^{100}\right\}$ be a function satisfying $$ \operatorname{gcd}(f(x), f(y))=\operatorname{gcd}(f(x), x-y) $$ for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x)=\operatorname{gcd}(m+x, n)$ for all integers $x$. | Let $\mathcal{P}$ be the set of primes not exceeding $10^{100}$. For each $p \in \mathcal{P}$, let $e_{p}=\max _{x} \nu_{p}(f(x))$ and let $c_{p} \in \operatorname{argmax}_{x} \nu_{p}(f(x))$. We show that this is good enough to compute all values of $x$, by looking at the exponent at each individual prime. Claim - For ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2019.jsonl",
"solution_match": null
} | 113 | 593 |
2019 | T0 | 8 | null | USA_TSTST | Let $\mathcal{S}$ be a set of 16 points in the plane, no three collinear. Let $\chi(\mathcal{S})$ denote the number of ways to draw 8 line segments with endpoints in $\mathcal{S}$, such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of $\chi(\mathcal{S})$ across all such $\mat... | The answer is 1430 . In general, we prove that with $2 n$ points the answer is the $n^{\text {th }}$ Catalan number $C_{n}=\frac{1}{n+1}\binom{2 n}{n}$. First of all, it is well-known that if $\mathcal{S}$ is a convex $2 n$-gon, then $\chi(\mathcal{S})=C_{n}$. It remains to prove the lower bound. We proceed by (strong)... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2019.jsonl",
"solution_match": null
} | 92 | 626 |
2019 | T0 | 9 | null | USA_TSTST | Let $A B C$ be a triangle with incenter $I$. Points $K$ and $L$ are chosen on segment $B C$ such that the incircles of $\triangle A B K$ and $\triangle A B L$ are tangent at $P$, and the incircles of $\triangle A C K$ and $\triangle A C L$ are tangent at $Q$. Prove that $I P=I Q$. | γ First solution, mostly elementary (original). Let $I_{B}, J_{B}, I_{C}, J_{C}$ be the incenters of $\triangle A B K, \triangle A B L, \triangle A C K, \triangle A C L$ respectively. We begin with the following claim which does not depend on the existence of tangency points $P$ and $Q$. Claim - Lines $B C, I_{B} J_{C}... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2019.jsonl",
"solution_match": null
} | 90 | 663 |
2019 | T0 | 9 | null | USA_TSTST | Let $A B C$ be a triangle with incenter $I$. Points $K$ and $L$ are chosen on segment $B C$ such that the incircles of $\triangle A B K$ and $\triangle A B L$ are tangent at $P$, and the incircles of $\triangle A C K$ and $\triangle A C L$ are tangent at $Q$. Prove that $I P=I Q$. | γ Second solution, inversion (Nikolai Beluhov). As above, the lines $B C, I_{B} J_{C}, J_{B} I_{C}$ meet at some point $R$ (possibly at infinity). Let $\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}$ be the incircles of $\triangle A B K$, $\triangle A C L, \triangle A B L$, and $\triangle A C K$. Claim - There exists a... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2019.jsonl",
"solution_match": null
} | 90 | 647 |
2020 | T0 | 3 | null | USA_TSTST | We say a nondegenerate triangle whose angles have measures $\theta_{1}, \theta_{2}, \theta_{3}$ is quirky if there exists integers $r_{1}, r_{2}, r_{3}$, not all zero, such that $$ r_{1} \theta_{1}+r_{2} \theta_{2}+r_{3} \theta_{3}=0 $$ Find all integers $n \geq 3$ for which a triangle with side lengths $n-1, n, n+1$... | The answer is $n=3,4,5,7$. We first introduce a variant of the $k$ th Chebyshev polynomials in the following lemma (which is standard, and easily shown by induction). ## Lemma For each $k \geq 0$ there exists $P_{k}(X) \in \mathbb{Z}[X]$, monic for $k \geq 1$ and with degree $k$, such that $$ P_{k}\left(X+X^{-1}\right)... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2020.jsonl",
"solution_match": null
} | 118 | 1,257 |
2020 | T0 | 4 | null | USA_TSTST | Find all pairs of positive integers $(a, b)$ satisfying the following conditions: (i) $a$ divides $b^{4}+1$, (ii) $b$ divides $a^{4}+1$, (iii) $\lfloor\sqrt{a}\rfloor=\lfloor\sqrt{b}\rfloor$. | Obviously, $\operatorname{gcd}(a, b)=1$, so the problem conditions imply $$ a b \mid(a-b)^{4}+1 $$ since each of $a$ and $b$ divide the right-hand side. We define $$ k \stackrel{\text { def }}{=} \frac{(b-a)^{4}+1}{a b} . $$ Claim (Size estimate) β We must have $k \leq 16$. $$ \begin{aligned} a b & \geq n^{2}\left(n^{2... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2020.jsonl",
"solution_match": null
} | 70 | 603 |
2020 | T0 | 8 | null | USA_TSTST | For every positive integer $N$, let $\sigma(N)$ denote the sum of the positive integer divisors of $N$. Find all integers $m \geq n \geq 2$ satisfying $$ \frac{\sigma(m)-1}{m-1}=\frac{\sigma(n)-1}{n-1}=\frac{\sigma(m n)-1}{m n-1} . $$ | The answer is that $m$ and $n$ should be powers of the same prime number. These all work because for a prime power we have $$ \frac{\sigma\left(p^{e}\right)-1}{p^{e}-1}=\frac{\left(1+p+\cdots+p^{e}\right)-1}{p^{e}-1}=\frac{p\left(1+\cdots+p^{e-1}\right)}{p^{e}-1}=\frac{p}{p-1} $$ So we now prove these are the only ones... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2020.jsonl",
"solution_match": null
} | 83 | 790 |
2020 | T0 | 9 | null | USA_TSTST | Ten million fireflies are glowing in $\mathbb{R}^{3}$ at midnight. Some of the fireflies are friends, and friendship is always mutual. Every second, one firefly moves to a new position so that its distance from each one of its friends is the same as it was before moving. This is the only way that the fireflies ever cha... | In general, we show that when $n \geq 70$, the answer is $f(n)=\left\lfloor\frac{n^{2}}{3}\right\rfloor$. Construction: Choose three pairwise parallel lines $\ell_{A}, \ell_{B}, \ell_{C}$ forming an infinite equilateral triangle prism (with side larger than 1). Split the $n$ fireflies among the lines as equally as poss... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2020.jsonl",
"solution_match": null
} | 145 | 1,592 |
2021 | T0 | 2 | null | USA_TSTST | Let $a_{1}<a_{2}<a_{3}<a_{4}<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence $$ \frac{a_{1}}{1}, \frac{a_{2}}{2}, \frac{a_{3}}{3}, \frac{a_{4}}{4}, \ldots $$ | γ Solution 1 (Merlijn Staps). We argue by contradiction, so suppose that for each $\lambda$ for which the set $S_{\lambda}=\left\{k: a_{k} / k=\lambda\right\}$ is non-empty, it contains at least two elements. Note that $S_{\lambda}$ is always a finite set because $a_{k}=k \lambda$ implies $k<1 / \lambda$. Write $m_{\la... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 97 | 893 |
2021 | T0 | 2 | null | USA_TSTST | Let $a_{1}<a_{2}<a_{3}<a_{4}<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence $$ \frac{a_{1}}{1}, \frac{a_{2}}{2}, \frac{a_{3}}{3}, \frac{a_{4}}{4}, \ldots $$ | γ Solution 2 (Sanjana Das). Assume for the sake of contradiction that no number appears exactly once in the sequence. For every $i<j$ with $a_{i} / i=a_{j} / j$, draw an edge between $i$ and $j$, so every $i$ has an edge (and being connected by an edge is a transitive property). Call $i$ good if it has an edge with som... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 97 | 1,311 |
2021 | T0 | 2 | null | USA_TSTST | Let $a_{1}<a_{2}<a_{3}<a_{4}<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence $$ \frac{a_{1}}{1}, \frac{a_{2}}{2}, \frac{a_{3}}{3}, \frac{a_{4}}{4}, \ldots $$ | γ Solution 3 (Gopal Goel). Suppose for sake of contradiction that the problem is false. Call an index $i$ a pin if $$ \frac{a_{j}}{j}=\frac{a_{i}}{i} \Longrightarrow j \geq i $$ ## Lemma There exists $k$ such that if we have $\frac{a_{i}}{i}=\frac{a_{j}}{j}$ with $j>i \geq k$, then $j \leq 1.1 i$. Suppose no such $k$ e... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 97 | 1,515 |
2021 | T0 | 3 | null | USA_TSTST | Find all positive integers $k>1$ for which there exists a positive integer $n$ such that $\binom{n}{k}$ is divisible by $n$, and $\binom{n}{m}$ is not divisible by $n$ for $2 \leq m<k$. | Such an $n$ exists for any $k$. First, suppose $k$ is prime. We choose $n=(k-1)$ !. For $m<k$, it follows from $m!\mid n$ that $$ \begin{aligned} (n-1)(n-2) \cdots(n-m+1) & \equiv(-1)(-2) \cdots(-m+1) \\ & \equiv(-1)^{m-1}(m-1)! \\ & \not \equiv 0 \bmod m! \end{aligned} $$ We see that $\binom{n}{m}$ is not a multiple o... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 58 | 1,729 |
2021 | T0 | 4 | null | USA_TSTST | Let $a$ and $b$ be positive integers. Suppose that there are infinitely many pairs of positive integers $(m, n)$ for which $m^{2}+a n+b$ and $n^{2}+a m+b$ are both perfect squares. Prove that $a$ divides $2 b$. | Treating $a$ and $b$ as fixed, we are given that there are infinitely many quadrpules $(m, n, r, s)$ which satisfy the system $$ \begin{aligned} m^{2}+a n+b & =(m+r)^{2} \\ n^{2}+a m+b & =(n+s)^{2} \end{aligned} $$ We say that $(r, s)$ is exceptional if there exists infinitely many $(m, n)$ that satisfy. Claim - If $(r... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 66 | 683 |
2021 | T0 | 6 | null | USA_TSTST | Triangles $A B C$ and $D E F$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A, F$, $B, D, C$, and $E$ occur in this order along $\Omega$. Let $\Delta_{A}$ be the triangle formed by lines $A B, A C$, and $E F$, and define triangles $\Delta_{B}, \Delta_{C}, \ldots, \Delta_{F}$ similarly. Furthermore, ... | Let $I$ and $r$ be the center and radius of $\omega$, and let $O$ and $R$ be the center and radius of $\Omega$. Let $O_{A}$ and $I_{A}$ be the circumcenter and incenter of triangle $\Delta_{A}$, and define $O_{B}$, $I_{B}, \ldots, I_{F}$ similarly. Let $\omega$ touch $E F$ at $A_{1}$, and define $B_{1}, C_{1}, \ldots, ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 268 | 982 |
2021 | T0 | 6 | null | USA_TSTST | Triangles $A B C$ and $D E F$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A, F$, $B, D, C$, and $E$ occur in this order along $\Omega$. Let $\Delta_{A}$ be the triangle formed by lines $A B, A C$, and $E F$, and define triangles $\Delta_{B}, \Delta_{C}, \ldots, \Delta_{F}$ similarly. Furthermore, ... | Let $I$ and $r$ be the center and radius of $\omega$, and let $O$ and $R$ be the center and radius of $\Omega$. Let $O_{A}$ and $I_{A}$ be the circumcenter and incenter of triangle $\Delta_{A}$, and define $O_{B}$, $I_{B}, \ldots, I_{F}$ similarly. Let $\omega$ touch $E F$ at $A_{1}$, and define $B_{1}, C_{1}, \ldots, ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 268 | 789 |
2021 | T0 | 6 | null | USA_TSTST | Triangles $A B C$ and $D E F$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A, F$, $B, D, C$, and $E$ occur in this order along $\Omega$. Let $\Delta_{A}$ be the triangle formed by lines $A B, A C$, and $E F$, and define triangles $\Delta_{B}, \Delta_{C}, \ldots, \Delta_{F}$ similarly. Furthermore, ... | Let $I$ and $r$ be the center and radius of $\omega$, and let $O$ and $R$ be the center and radius of $\Omega$. Let $O_{A}$ and $I_{A}$ be the circumcenter and incenter of triangle $\Delta_{A}$, and define $O_{B}$, $I_{B}, \ldots, I_{F}$ similarly. Let $\omega$ touch $E F$ at $A_{1}$, and define $B_{1}, C_{1}, \ldots, ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 268 | 547 |
2021 | T0 | 6 | null | USA_TSTST | Triangles $A B C$ and $D E F$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A, F$, $B, D, C$, and $E$ occur in this order along $\Omega$. Let $\Delta_{A}$ be the triangle formed by lines $A B, A C$, and $E F$, and define triangles $\Delta_{B}, \Delta_{C}, \ldots, \Delta_{F}$ similarly. Furthermore, ... | Let $I$ and $r$ be the center and radius of $\omega$, and let $O$ and $R$ be the center and radius of $\Omega$. Let $O_{A}$ and $I_{A}$ be the circumcenter and incenter of triangle $\Delta_{A}$, and define $O_{B}$, $I_{B}, \ldots, I_{F}$ similarly. Let $\omega$ touch $E F$ at $A_{1}$, and define $B_{1}, C_{1}, \ldots, ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 268 | 935 |
2021 | T0 | 6 | null | USA_TSTST | Triangles $A B C$ and $D E F$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A, F$, $B, D, C$, and $E$ occur in this order along $\Omega$. Let $\Delta_{A}$ be the triangle formed by lines $A B, A C$, and $E F$, and define triangles $\Delta_{B}, \Delta_{C}, \ldots, \Delta_{F}$ similarly. Furthermore, ... | Let $I$ and $r$ be the center and radius of $\omega$, and let $O$ and $R$ be the center and radius of $\Omega$. Let $O_{A}$ and $I_{A}$ be the circumcenter and incenter of triangle $\Delta_{A}$, and define $O_{B}$, $I_{B}, \ldots, I_{F}$ similarly. Let $\omega$ touch $E F$ at $A_{1}$, and define $B_{1}, C_{1}, \ldots, ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 268 | 668 |
2021 | T0 | 6 | null | USA_TSTST | Triangles $A B C$ and $D E F$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A, F$, $B, D, C$, and $E$ occur in this order along $\Omega$. Let $\Delta_{A}$ be the triangle formed by lines $A B, A C$, and $E F$, and define triangles $\Delta_{B}, \Delta_{C}, \ldots, \Delta_{F}$ similarly. Furthermore, ... | Let $I$ and $r$ be the center and radius of $\omega$, and let $O$ and $R$ be the center and radius of $\Omega$. Let $O_{A}$ and $I_{A}$ be the circumcenter and incenter of triangle $\Delta_{A}$, and define $O_{B}$, $I_{B}, \ldots, I_{F}$ similarly. Let $\omega$ touch $E F$ at $A_{1}$, and define $B_{1}, C_{1}, \ldots, ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 268 | 543 |
2021 | T0 | 6 | null | USA_TSTST | Triangles $A B C$ and $D E F$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A, F$, $B, D, C$, and $E$ occur in this order along $\Omega$. Let $\Delta_{A}$ be the triangle formed by lines $A B, A C$, and $E F$, and define triangles $\Delta_{B}, \Delta_{C}, \ldots, \Delta_{F}$ similarly. Furthermore, ... | Let $I$ and $r$ be the center and radius of $\omega$, and let $O$ and $R$ be the center and radius of $\Omega$. Let $O_{A}$ and $I_{A}$ be the circumcenter and incenter of triangle $\Delta_{A}$, and define $O_{B}$, $I_{B}, \ldots, I_{F}$ similarly. Let $\omega$ touch $E F$ at $A_{1}$, and define $B_{1}, C_{1}, \ldots, ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 268 | 798 |
2021 | T0 | 6 | null | USA_TSTST | Triangles $A B C$ and $D E F$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A, F$, $B, D, C$, and $E$ occur in this order along $\Omega$. Let $\Delta_{A}$ be the triangle formed by lines $A B, A C$, and $E F$, and define triangles $\Delta_{B}, \Delta_{C}, \ldots, \Delta_{F}$ similarly. Furthermore, ... | Let $I$ and $r$ be the center and radius of $\omega$, and let $O$ and $R$ be the center and radius of $\Omega$. Let $O_{A}$ and $I_{A}$ be the circumcenter and incenter of triangle $\Delta_{A}$, and define $O_{B}$, $I_{B}, \ldots, I_{F}$ similarly. Let $\omega$ touch $E F$ at $A_{1}$, and define $B_{1}, C_{1}, \ldots, ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 268 | 1,706 |
2021 | T0 | 7 | null | USA_TSTST | Let $M$ be a finite set of lattice points and $n$ be a positive integer. A mine-avoiding path is a path of lattice points with length $n$, beginning at $(0,0)$ and ending at a point on the line $x+y=n$, that does not contain any point in $M$. Prove that if there exists a mine-avoiding path, then there exist at least $2... | ## γ Solution 2. ## Lemma If $|M|<n$, there is more than one mine-avoiding path. - Make the first $i+1$ points $P_{0}, P_{1}, \ldots, P_{i}$. - If $P_{i} \rightarrow P_{i+1}$ is one unit up, go right until $\left(n-y_{i}, y_{i}\right)$. - If $P_{i} \rightarrow P_{i+1}$ is one unit right, go up until $\left(x_{i}, n-x_{... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 100 | 546 |
2021 | T0 | 8 | null | USA_TSTST | Let $A B C$ be a scalene triangle. Points $A_{1}, B_{1}$ and $C_{1}$ are chosen on segments $B C$, $C A$, and $A B$, respectively, such that $\triangle A_{1} B_{1} C_{1}$ and $\triangle A B C$ are similar. Let $A_{2}$ be the unique point on line $B_{1} C_{1}$ such that $A A_{2}=A_{1} A_{2}$. Points $B_{2}$ and $C_{2}$ ... | ΰ€¬ Solution 1 (author). We'll use the following lemma. ## Lemma Suppose that $P Q R S$ is a convex quadrilateral with $\angle P=\angle R$. Then there is a point $T$ on $Q S$ such that $\angle Q P T=\angle S R P, \angle T R Q=\angle R P S$, and $P T=R T$. Before proving the lemma, we will show how it solves the problem. ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 154 | 660 |
2021 | T0 | 8 | null | USA_TSTST | Let $A B C$ be a scalene triangle. Points $A_{1}, B_{1}$ and $C_{1}$ are chosen on segments $B C$, $C A$, and $A B$, respectively, such that $\triangle A_{1} B_{1} C_{1}$ and $\triangle A B C$ are similar. Let $A_{2}$ be the unique point on line $B_{1} C_{1}$ such that $A A_{2}=A_{1} A_{2}$. Points $B_{2}$ and $C_{2}$ ... | Solution 2 (Ankan Bhattacharya). We prove the main claim $\frac{B_{1} A_{2}}{A_{2} C_{1}}=\frac{B A_{1}}{A_{1} C}$. Let $\triangle A_{0} B_{0} C_{0}$ be the medial triangle of $\triangle A B C$. In addition, let $A_{1}^{\prime}$ be the reflection of $A_{1}$ over $\overline{B_{1} C_{1}}$, and let $X$ be the point satisf... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 154 | 668 |
2021 | T0 | 9 | null | USA_TSTST | Let $q=p^{r}$ for a prime number $p$ and positive integer $r$. Let $\zeta=e^{\frac{2 \pi i}{q}}$. Find the least positive integer $n$ such that $$ \sum_{\substack{1 \leq k \leq q \\ \operatorname{gcd}(k, p)=1}} \frac{1}{\left(1-\zeta^{k}\right)^{n}} $$ is not an integer. (The sum is over all $1 \leq k \leq q$ with $p... | Let $S_{q}$ denote the set of primitive $q$ th roots of unity (thus, the sum in question is a sum over $S_{q}$ ). γ Solution 1 (author). Let $\zeta_{p}=e^{2 \pi i / p}$ be a fixed primitive $p$ th root of unity. Observe that the given sum is an integer for all $n \leq 0$ (e.g. because the sum is an integer symmetric po... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 132 | 2,260 |
2021 | T0 | 9 | null | USA_TSTST | Let $q=p^{r}$ for a prime number $p$ and positive integer $r$. Let $\zeta=e^{\frac{2 \pi i}{q}}$. Find the least positive integer $n$ such that $$ \sum_{\substack{1 \leq k \leq q \\ \operatorname{gcd}(k, p)=1}} \frac{1}{\left(1-\zeta^{k}\right)^{n}} $$ is not an integer. (The sum is over all $1 \leq k \leq q$ with $p... | Let $S_{q}$ denote the set of primitive $q$ th roots of unity (thus, the sum in question is a sum over $S_{q}$ ). Solution 2 (Nikolai Beluhov). Suppose that the complex numbers $\frac{1}{1-\omega}$ for $\omega \in S_{q}$ are the roots of $$ P(x)=x^{d}-c_{1} x^{d-1}+c_{2} x^{d-2}-\cdots \pm c_{d} $$ so that $c_{k}$ is t... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2021.jsonl",
"solution_match": null
} | 132 | 2,870 |
2022 | T0 | 1 | null | USA_TSTST | Let $n$ be a positive integer. Find the smallest positive integer $k$ such that for any set $S$ of $n$ points in the interior of the unit square, there exists a set of $k$ rectangles such that the following hold: - The sides of each rectangle are parallel to the sides of the unit square. - Each point in $S$ is not in ... | We claim the answer is $k=2 n+2$. The lower bound is given by picking $$ S=\left\{\left(s_{1}, s_{1}\right),\left(s_{2}, s_{2}\right), \ldots,\left(s_{n}, s_{n}\right)\right\} $$ for some real numbers $0<s_{1}<s_{2}<\cdots<s_{n}<1$. Consider the $4 n$ points $$ S^{\prime}=S+\{(\varepsilon, 0),(0, \varepsilon),(-\vareps... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2022.jsonl",
"solution_match": null
} | 129 | 669 |
2022 | T0 | 3 | null | USA_TSTST | Determine all positive integers $N$ for which there exists a strictly increasing sequence of positive integers $s_{0}<s_{1}<s_{2}<\cdots$ satisfying the following properties: - the sequence $s_{1}-s_{0}, s_{2}-s_{1}, s_{3}-s_{2}, \ldots$ is periodic; and - $s_{s_{n}}-s_{s_{n-1}} \leq N<s_{1+s_{n}}-s_{s_{n-1}}$ for all... | γ Answer. All $N$ such that $t^{2} \leq N<t^{2}+t$ for some positive integer $t$. Solution 1 (local). If $t^{2} \leq N<t^{2}+t$ then the sequence $s_{n}=t n+1$ satisfies both conditions. It remains to show that no other values of $N$ work. Define $a_{n}:=s_{n}-s_{n-1}$, and let $p$ be the minimal period of $\left\{a_{n... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2022.jsonl",
"solution_match": null
} | 123 | 950 |
2022 | T0 | 3 | null | USA_TSTST | Determine all positive integers $N$ for which there exists a strictly increasing sequence of positive integers $s_{0}<s_{1}<s_{2}<\cdots$ satisfying the following properties: - the sequence $s_{1}-s_{0}, s_{2}-s_{1}, s_{3}-s_{2}, \ldots$ is periodic; and - $s_{s_{n}}-s_{s_{n-1}} \leq N<s_{1+s_{n}}-s_{s_{n-1}}$ for all... | γ Answer. All $N$ such that $t^{2} \leq N<t^{2}+t$ for some positive integer $t$. γ Solution 2 (global). Define $\left\{a_{n}\right\}$ and $f$ as in the previous solution. We first show that $s_{i} \not \equiv s_{j}(\bmod p)$ for all $i<j<i+p$. Suppose the contrary, i.e. that $s_{i} \equiv s_{j}$ $(\bmod p)$ for some $... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2022.jsonl",
"solution_match": null
} | 123 | 1,180 |
2022 | T0 | 4 | null | USA_TSTST | A function $f: \mathbb{N} \rightarrow \mathbb{N}$ has the property that for all positive integers $m$ and $n$, exactly one of the $f(n)$ numbers $$ f(m+1), f(m+2), \ldots, f(m+f(n)) $$ is divisible by $n$. Prove that $f(n)=n$ for infinitely many positive integers $n$. | We start with the following claim: $$ \text { Claim - If } a \mid b \text { then } f(a) \mid f(b) $$ In what follows, let $a \geq 2$ be any positive integer. Because $f(a)$ and $f(2 a)$ are both divisible by $f(a)$, there are $a+1$ consecutive values of $f$ of which at least two divisible by $f(a)$. It follows that $f(... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2022.jsonl",
"solution_match": null
} | 89 | 608 |
2022 | T0 | 5 | null | USA_TSTST | Let $A_{1}, \ldots, A_{2022}$ be the vertices of a regular 2022 -gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the plane ... | The answer is 22 . To prove the lower bound, note that there are $2022 \cdot 2021+2>2^{21}$ possible colorings. If Bob makes less than 22 queries, then he can only output $2^{21}$ possible colorings, which means he is wrong on some coloring. Now we show Bob can always win in 22 queries. A key observation is that the se... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2022.jsonl",
"solution_match": null
} | 193 | 627 |
2022 | T0 | 8 | null | USA_TSTST | Find all functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that $$ \left\lfloor\frac{f(m n)}{n}\right\rfloor=f(m) $$ for all positive integers $m, n$. | There are two families of functions that work: for each $\alpha \in \mathbb{R}$ the function $f(n)=\lfloor\alpha n\rfloor$, and for each $\alpha \in \mathbb{R}$ the function $f(n)=\lceil\alpha n\rceil-1$. (For irrational $\alpha$ these two functions coincide.) It is straightforward to check that these functions indeed ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2022.jsonl",
"solution_match": null
} | 57 | 673 |
2022 | T0 | 9 | null | USA_TSTST | Let $k>1$ be a fixed positive integer. Prove that if $n$ is a sufficiently large positive integer, there exists a sequence of integers with the following properties: - Each element of the sequence is between 1 and $n$, inclusive. - For any two different contiguous subsequences of the sequence with length between 2 and... | For any positive integer $n$, define an $(n, k)$-good sequence to be a finite sequence of integers each between 1 and $n$ inclusive satisfying the second property in the problem statement. The problems asks to show that, for all sufficiently large integers $n$, there is an $(n, k)$-good sequence of length at least $0.4... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2022.jsonl",
"solution_match": null
} | 109 | 3,677 |
2023 | T0 | 1 | null | USA_TSTST | Let $A B C$ be a triangle with centroid $G$. Points $R$ and $S$ are chosen on rays $G B$ and $G C$, respectively, such that $$ \angle A B S=\angle A C R=180^{\circ}-\angle B G C . $$ Prove that $\angle R A S+\angle B A C=\angle B G C$. | γ Solution 5 using complex numbers, by Milan Haiman. Note that $\angle R A S+\angle B A C=$ $\angle B A S+\angle R A C$. We compute $\angle B A S$ in complex numbers; then $\angle R A C$ will then be known by symmetry. Let $a, b, c$ be points on the unit circle representing $A, B, C$ respectively. Let $g=\frac{1}{3}(a+... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 86 | 1,141 |
2023 | T0 | 2 | null | USA_TSTST | Let $n \geq m \geq 1$ be integers. Prove that $$ \sum_{k=m}^{n}\left(\frac{1}{k^{2}}+\frac{1}{k^{3}}\right) \geq m \cdot\left(\sum_{k=m}^{n} \frac{1}{k^{2}}\right)^{2} $$ | γ First solution (authors). By Cauchy-Schwarz, we have $$ \begin{aligned} \sum_{k=m}^{n} \frac{k+1}{k^{3}} & =\sum_{k=m}^{n} \frac{\left(\frac{1}{k^{2}}\right)^{2}}{\frac{1}{k(k+1)}} \\ & \geq \frac{\left(\frac{1}{m^{2}}+\frac{1}{(m+1)^{2}}+\cdots+\frac{1}{n^{2}}\right)^{2}}{\frac{1}{m(m+1)}+\frac{1}{(m+1)(m+2)}+\cdots... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 85 | 888 |
2023 | T0 | 2 | null | USA_TSTST | Let $n \geq m \geq 1$ be integers. Prove that $$ \sum_{k=m}^{n}\left(\frac{1}{k^{2}}+\frac{1}{k^{3}}\right) \geq m \cdot\left(\sum_{k=m}^{n} \frac{1}{k^{2}}\right)^{2} $$ | I Third approach by reducing $n \rightarrow \infty$, Michael Ren and Carl Schildkraut. First, we give: Claim (Reduction to $n \rightarrow \infty$ ) - If the problem is true when $n \rightarrow \infty$, it is true for all $n$. However, the region is bounded by a convex curve, and the sequence of points $(0,0)$, $\left(\... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 85 | 967 |
2023 | T0 | 2 | null | USA_TSTST | Let $n \geq m \geq 1$ be integers. Prove that $$ \sum_{k=m}^{n}\left(\frac{1}{k^{2}}+\frac{1}{k^{3}}\right) \geq m \cdot\left(\sum_{k=m}^{n} \frac{1}{k^{2}}\right)^{2} $$ | γ Fourth approach by bashing, Carl Schildkraut. With a bit more work, the third approach can be adapted to avoid the $n \rightarrow \infty$ reduction. Similarly to before, define $$ A=\sum_{k=m}^{n} \frac{1}{k^{2}} \text { and } B=\sum_{k=m}^{n} \frac{1}{k^{3}} $$ we want to show $1+4 m B \geq(2 m A-1)^{2}$. Writing $$... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 85 | 624 |
2023 | T0 | 3 | null | USA_TSTST | Find all positive integers $n$ for which it is possible to color some cells of an infinite grid of unit squares red, such that each rectangle consisting of exactly $n$ cells (and whose edges lie along the lines of the grid) contains an odd number of red cells. | We claim that this is possible for all positive integers $n$. Call a positive integer for which such a coloring is possible good. To show that all positive integers $n$ are good we prove the following: (i) If $n$ is good and $p$ is an odd prime, then $p n$ is good; (ii) For every $k \geq 0$, the number $n=2^{k}$ is goo... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 56 | 1,237 |
2023 | T0 | 5 | null | USA_TSTST | Suppose $a, b$, and $c$ are three complex numbers with product 1 . Assume that none of $a, b$, and $c$ are real or have absolute value 1 . Define $$ p=(a+b+c)+\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \quad \text { and } \quad q=\frac{a}{b}+\frac{b}{c}+\frac{c}{a} . $$ Given that both $p$ and $q$ are real numb... | We show $(p, q)=(-3,3)$ is the only possible ordered pair. ## γT First solution. $$ \begin{aligned} p+3 & =3+\sum_{\text {cyc }}\left(\frac{x}{y}+\frac{y}{x}\right)=3+\frac{x^{2}(y+z)+y^{2}(z+x)+z^{2}(x+y)}{x y z} \\ & =\frac{(x+y+z)(x y+y z+z x)}{x y z} \\ q-3 & =-3+\sum_{\text {cyc }} \frac{y^{2}}{z x}=\frac{x^{3}+y^... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 138 | 631 |
2023 | T0 | 5 | null | USA_TSTST | Suppose $a, b$, and $c$ are three complex numbers with product 1 . Assume that none of $a, b$, and $c$ are real or have absolute value 1 . Define $$ p=(a+b+c)+\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \quad \text { and } \quad q=\frac{a}{b}+\frac{b}{c}+\frac{c}{a} . $$ Given that both $p$ and $q$ are real numb... | We show $(p, q)=(-3,3)$ is the only possible ordered pair. γ Third solution by Luke Robitaille and Daniel Zhu. The answer is $p=-3$ and $q=3$. Let's first prove that no other $(p, q)$ work. Let $e_{1}=a+b+c$ and $e_{2}=a^{-1}+b^{-1}+c^{-1}=a b+a c+b c$. Also, let $f=e_{1} e_{2}$. Note that $p=e_{1}+e_{2}$. Our main ins... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 138 | 1,371 |
2023 | T0 | 6 | null | USA_TSTST | Let $A B C$ be a scalene triangle and let $P$ and $Q$ be two distinct points in its interior. Suppose that the angle bisectors of $\angle P A Q, \angle P B Q$, and $\angle P C Q$ are the altitudes of triangle $A B C$. Prove that the midpoint of $\overline{P Q}$ lies on the Euler line of $A B C$. | γ Solution 1 (Ankit Bisain). Let $H$ be the orthocenter of $A B C$, and construct $P^{\prime}$ using the following claim. Claim - There is a point $P^{\prime}$ for which $$ \measuredangle A P H+\measuredangle A P^{\prime} H=\measuredangle B P H+\measuredangle B P^{\prime} H=\measuredangle C P H+\measuredangle C P^{\pri... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 89 | 618 |
2023 | T0 | 6 | null | USA_TSTST | Let $A B C$ be a scalene triangle and let $P$ and $Q$ be two distinct points in its interior. Suppose that the angle bisectors of $\angle P A Q, \angle P B Q$, and $\angle P C Q$ are the altitudes of triangle $A B C$. Prove that the midpoint of $\overline{P Q}$ lies on the Euler line of $A B C$. | Solution 2 using complex numbers (Carl Schildkraut and Milan Haiman). Let $(A B C)$ be the unit circle in the complex plane, and let $A=a, B=b, C=c$ such that $|a|=|b|=|c|=1$. Let $P=p$ and $Q=q$, and $O=0$ and $H=h=a+b+c$ be the circumcenter and orthocenter of $A B C$ respectively. The first step is to translate the g... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 89 | 1,524 |
2023 | T0 | 6 | null | USA_TSTST | Let $A B C$ be a scalene triangle and let $P$ and $Q$ be two distinct points in its interior. Suppose that the angle bisectors of $\angle P A Q, \angle P B Q$, and $\angle P C Q$ are the altitudes of triangle $A B C$. Prove that the midpoint of $\overline{P Q}$ lies on the Euler line of $A B C$. | γ Solution 3 also using complex numbers (Michael Ren). We use complex numbers as in the previous solution. The angle conditions imply that $\frac{(a-p)(a-q)}{(b-c)^{2}}, \frac{(b-p)(b-q)}{(c-a)^{2}}$, and $\frac{(c-p)(c-q)}{(a-b)^{2}}$ are real numbers. Take a linear combination of these with real coefficients $X$, $Y$... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 89 | 881 |
2023 | T0 | 8 | null | USA_TSTST | Let $A B C$ be an equilateral triangle with side length 1. Points $A_{1}$ and $A_{2}$ are chosen on side $B C$, points $B_{1}$ and $B_{2}$ are chosen on side $C A$, and points $C_{1}$ and $C_{2}$ are chosen on side $A B$ such that $B A_{1}<B A_{2}, C B_{1}<C B_{2}$, and $A C_{1}<A C_{2}$. Suppose that the three line se... | The only possible value of the common perimeter, denoted $p$, is 1 . γ Synthetic approach (from author). We prove the converse of the problem first: Claim ( $p=1$ implies concurrence) - Suppose the six points are chosen so that triangles $A B_{2} C_{1}, B C_{2} A_{1}, C A_{2} B_{1}$ all have perimeter 1. Then lines $\o... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 202 | 511 |
2023 | T0 | 8 | null | USA_TSTST | Let $A B C$ be an equilateral triangle with side length 1. Points $A_{1}$ and $A_{2}$ are chosen on side $B C$, points $B_{1}$ and $B_{2}$ are chosen on side $C A$, and points $C_{1}$ and $C_{2}$ are chosen on side $A B$ such that $B A_{1}<B A_{2}, C B_{1}<C B_{2}$, and $A C_{1}<A C_{2}$. Suppose that the three line se... | The only possible value of the common perimeter, denoted $p$, is 1 . γ Barycentric solution (by Carl, Krit, Milan). We show that, if the common perimeter is 1 , then the lines concur. To do this, we use barycentric coordinates. Let $A=(1: 0: 0)$, $B=(0: 1: 0)$, and $C=(0: 0: 1)$. Let $A_{1}=\left(0: 1-a_{1}: a_{1}\righ... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 202 | 2,100 |
2023 | T0 | 9 | null | USA_TSTST | Let $p$ be a fixed prime and let $a \geq 2$ and $e \geq 1$ be fixed integers. Given a function $f: \mathbb{Z} / a \mathbb{Z} \rightarrow \mathbb{Z} / p^{e} \mathbb{Z}$ and an integer $k \geq 0$, the $k$ th finite difference, denoted $\Delta^{k} f$, is the function from $\mathbb{Z} / a \mathbb{Z}$ to $\mathbb{Z} / p^{e}... | Let $p$ be a fixed prime and let $a \geq 2$ and $e \geq 1$ be fixed integers. Given a function $f: \mathbb{Z} / a \mathbb{Z} \rightarrow \mathbb{Z} / p^{e} \mathbb{Z}$ and an integer $k \geq 0$, the $k$ th finite difference, denoted $\Delta^{k} f$, is the function from $\mathbb{Z} / a \mathbb{Z}$ to $\mathbb{Z} / p^{e}... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 243 | 1,316 |
2023 | T0 | 9 | null | USA_TSTST | Let $p$ be a fixed prime and let $a \geq 2$ and $e \geq 1$ be fixed integers. Given a function $f: \mathbb{Z} / a \mathbb{Z} \rightarrow \mathbb{Z} / p^{e} \mathbb{Z}$ and an integer $k \geq 0$, the $k$ th finite difference, denoted $\Delta^{k} f$, is the function from $\mathbb{Z} / a \mathbb{Z}$ to $\mathbb{Z} / p^{e}... | Let $p$ be a fixed prime and let $a \geq 2$ and $e \geq 1$ be fixed integers. Given a function $f: \mathbb{Z} / a \mathbb{Z} \rightarrow \mathbb{Z} / p^{e} \mathbb{Z}$ and an integer $k \geq 0$, the $k$ th finite difference, denoted $\Delta^{k} f$, is the function from $\mathbb{Z} / a \mathbb{Z}$ to $\mathbb{Z} / p^{e}... | {
"problem_match": null,
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2023.jsonl",
"solution_match": null
} | 243 | 1,704 |
2024 | T0 | 1 | null | USA_TSTST | For every ordered pair of integers \((i,j)\) , not necessarily positive, we wish to select a point \(P_{i,j}\) in the Cartesian plane whose coordinates lie inside the unit square defined by
\[i< x< i + 1,\qquad j< y< j + 1.\]
Find all real numbers \(c > 0\) for which it's possible to choose these points such that... | Answer. \(c\geq 4\)
Proof \(c< 4\) is not possible. Let \(n\) be an arbitrary positive integer. We take an \(n\times n\) subgrid of unit squares (i.e. \(P_{i,j}\) for \(1\leq i,j\leq n\) ), and compute a lower bound on the average of all possible quadrilaterals from this subgrid.
Consider the average length of the ... | {
"problem_match": "1. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2024.jsonl",
"solution_match": "## \\(\\S 1.1\\) TSTST 2024/1, proposed by Karthik Vedula \n"
} | 153 | 782 |
2024 | T0 | 2 | null | USA_TSTST | Let \(p\) be an odd prime number. Suppose \(P\) and \(Q\) are polynomials with integer coefficients such that \(P(0) = Q(0) = 1\) , there is no nonconstant polynomial dividing both \(P\) and \(Q\) , and
\[1 + \frac{x}{1 + \frac{2x}{1 + \frac{\cdots}{1 + (p - 1)x}}} = \frac{P(x)}{Q(x)}.\]
Show that all coefficient... | \(\P\) Solution 1. We first make some general observations about rational functions represented through continued fractions.
Claim β Let \(a_{1}, a_{2}, \ldots\) , be a sequence of nonzero integers. Define the sequence of polynomials \(P_{1}(x) = 1\) , \(P_{2}(x) = 1 + a_{1}x\) , and
\[P_{k + 1}(x) = P_{k}(x) + a_{... | {
"problem_match": "2. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2024.jsonl",
"solution_match": "## \\(\\S 1.2\\) TSTST 2024/2, proposed by Andrew Gu \n"
} | 140 | 3,639 |
2024 | T0 | 3 | null | USA_TSTST | Let \(A = \{a_{1},\ldots ,a_{2024}\}\) be a set of 2024 pairwise distinct real numbers. Assume that there exist positive integers \(b_{1},b_{2},\ldots ,b_{2024}\) such that
\[a_{1}b_{1} + a_{2}b_{2} + \cdot \cdot \cdot +a_{2024}b_{2024} = 0.\]
Prove that one can choose \(a_{2025},a_{2026},a_{2027},\ldots\) such t... | It will be convenient to use 0- based indexing here, i.e. \(A = \{a_{0},\ldots ,a_{2023}\}\) and so on. Let \(m = \sum_{i = 0}^{2023}b_{i}\) . By appending \(b_{i} - 1\) copies of \(a_{i}\) for each \(i\) , we may extend the sequence to \(a_{0},\ldots ,a_{m - 1}\) such that \(a_{0} + \cdot \cdot \cdot +a_{m - 1} = 0\) ... | {
"problem_match": "3. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2024.jsonl",
"solution_match": "## \\(\\S 1.3\\) TSTST 2024/3, proposed by Daniel Zhu \n"
} | 214 | 2,958 |
2024 | T0 | 4 | null | USA_TSTST | Let \(ABCD\) be a quadrilateral inscribed in a circle with center \(O\) and \(E\) be the intersection of segments \(AC\) and \(BD\) . Let \(\omega_{1}\) be the circumcircle of \(ADE\) and \(\omega_{2}\) be the circumcircle of \(BCE\) . The tangent to \(\omega_{1}\) at \(A\) and the tangent to \(\omega_{2}\) at \(C\) me... | \(\P\) Solution 1. Let \(R = \overline{AD} \cap \overline{BC}\) (possibly at infinity, but we'll see it's an Euclidean point later).
Claim β \(ACRP\) is an isosceles trapezoid with \(\overline{AC} \parallel \overline{PR}\) . Consequently,... | {
"problem_match": "4. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2024.jsonl",
"solution_match": "## \\(\\S 2.1\\) TSTST 2024/4, proposed by Merlijn Staps \n"
} | 149 | 2,051 |
2024 | T0 | 5 | null | USA_TSTST | For a positive integer \(k\) , let \(s(k)\) denote the number of 1s in the binary representation of \(k\) . Prove that for any positive integer \(n\) ,
\[\sum_{i = 1}^{n}(-1)^{s(3i)} > 0.\] | \(\P\) Solution 1. Given a set of positive integers \(S\) , define
\[f(S) = \sum_{k\in S}(-1)^{s(k)}.\]
We also define
\[S_{\mathrm{even}} = \{k\in S\mid k\mathrm{~is~even}\}\] \[S_{\mathrm{odd}} = \{k\in S\mid k\mathrm{~is~odd}\}\]
and apply functions on sets pointwise, e.g.
\[\frac{S - 1}{2} = \left\{\frac{... | {
"problem_match": "5. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2024.jsonl",
"solution_match": "## \\(\\S 2.2\\) TSTST 2024/5, proposed by Holden Mui \n"
} | 68 | 3,606 |
2024 | T0 | 6 | null | USA_TSTST | Determine whether there exists a function \(f\colon \mathbb{Z}_{>0}\to \mathbb{Z}_{>0}\) such that for all positive integers \(m\) and \(n\) ,
\[f(m + n f(m)) = f(n)^{m} + 2024!\cdot m.\] | The answer is no. Let \(P(m,n)\) denote the given FE.
\(\P\) Solution 1 (Gopal Goel). Suppose there was a function \(f\) , and let \(r = f(1)\) . Note that \(P(1,n)\) gives
\[f(1 + r n) = f(n) + 2024!.\]
Iterating this result gives
\[f(1 + r + \dots +r^{k}) = r + k\cdot 2024!\]
for all \(k\in \mathbb{Z}_{\geq... | {
"problem_match": "6. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2024.jsonl",
"solution_match": "## \\(\\S 2.3\\) TSTST 2024/6, proposed by Jaedon Whyte \n"
} | 72 | 1,089 |
2024 | T0 | 7 | null | USA_TSTST | An infinite sequence \(a_{1},a_{2},a_{3},\ldots\) of real numbers satisfies
\[a_{2n - 1} + a_{2n} > a_{2n + 1} + a_{2n + 2}\qquad \mathrm{and}\qquad a_{2n} + a_{2n + 1}< a_{2n + 2} + a_{2n + 3}\]
for every positive integer \(n\) . Prove that there exists a real number \(C\) such that \(a_{n}a_{n + 1}< C\) for eve... | Aer \(n\) .
It suffices to solve the problem for sufficiently large \(n\) . Let \(d_{n} = (- 1)^{n - 1}(a_{n + 2} - a_{n})\) . The assertion simply says that \(d_{1}, d_{2}, \ldots\) is strictly increasing.
We consider the following cases.
- Suppose that \(d_{k} > 0\) for some \(k\) . Then,
\[a_{2n + 1} = a_{1}... | {
"problem_match": "7. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2024.jsonl",
"solution_match": "## \\(\\S 3.1\\) TSTST 2024/7, proposed by Merlijn Staps \n"
} | 143 | 651 |
2024 | T0 | 8 | null | USA_TSTST | Let \(ABC\) be a scalene triangle, and let \(D\) be a point on side \(BC\) satisfying \(\angle BAD = \angle DAC\) . Suppose that \(X\) and \(Y\) are points inside \(ABC\) such that triangles \(ABX\) and \(ACY\) are similar and quadrilaterals \(ACDX\) and \(ABDY\) are cyclic. Let lines \(BX\) and \(CY\) meet at \(S\) an... | \(\P\) Solution by characterizing \(X\) and \(Y\) . We first state an important property of \(X\) and \(Y\) .
Claim β Points \(X\) and \(Y\) are isogonal conjugates with respect to \(\triangle ABC\) .
Here are two proofs of the claim.
... | {
"problem_match": "8. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2024.jsonl",
"solution_match": "## \\(\\S 3.2\\) TSTST 2024/8, proposed by Michael Ren \n"
} | 121 | 1,968 |
2024 | T0 | 9 | null | USA_TSTST | Let \(n \geq 2\) be a fixed integer. The cells of an \(n \times n\) table are filled with the integers from 1 to \(n^2\) with each number appearing exactly once. Let \(N\) be the number of unordered quadruples of cells on this board which form an axis-aligned rectangle, with the two smaller integers being on opposite v... | L
The largest possible value of \(N\) is \(\frac{1}{12} n^2 (n^2 - 1)\) . Call these rectangles wobbly. We defer the construction until the proof is complete, since the proof suggests the construction.
Proof of bound. Call a triple of integers \((a, b, c)\) an elbow if \(a\) and \(b\) are in the same row, \(b\) and... | {
"problem_match": "9. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2024.jsonl",
"solution_match": "## Β§3.3 TSTST 2024/9 \n"
} | 92 | 1,407 |
2025 | T0 | 2 | null | IMO | Find all sets \(S \subseteq \mathbb{Z}\) for which there exists a function \(f: \mathbb{R} \to \mathbb{Z}\) such that
- \(f(x - y) - 2f(x) + f(x + y) \geq -1\) for all \(x, y \in \mathbb{R}\) , and
- \(S = \{f(z) \mid z \in \mathbb{R}\}\) . | The answer is \(\{a\}\) , \(\{a,a + 1\}\) , \(\{a,a + 1,a + 2,\ldots \}\) , and \(\mathbb{Z}\) , for arbitrary \(a\in \mathbb{Z}\) . For constructions, it is not hard to show that if \(g\colon \mathbb{R}\to \mathbb{R}\) is a convex function, then \(\lfloor g\rfloor\) satisfies the functional equation. Thus \(f(x) = a\)... | {
"problem_match": "Problem 2. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2025.jsonl",
"solution_match": "## \\(\\S 1.2\\) Solution to TSTST 2, by Daniel Zhu \n"
} | 107 | 989 |
2025 | T0 | 3 | null | IMO | Let \(a_{1}, a_{2}, r\) , and \(s\) be positive integers with \(r\) and \(s\) odd. The sequence \(a_{1}, a_{2}, a_{3}, \ldots\) is defined by
\[a_{n + 2} = r a_{n + 1} + s a_{n}\]
for all \(n \geq 1\) . Determine the maximum possible number of integers \(1 \leq \ell \leq 2025\) such that \(a_{\ell}\) divides \(a_... | Answer 1350.
\(\P\) Solution We first provide the upper bound. We start by dividing out any common factors of \(a_{1}\) and \(a_{2}\) from the whole sequence. Note that since \(r\) and \(s\) are odd, and \(a_{1}\) and \(a_{2}\) cannot both be divisible by 2, the sequence \(a_{1}, a_{2}, a_{3}, \ldots\) (mod 2) must b... | {
"problem_match": "Problem 3. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2025.jsonl",
"solution_match": "## Β§1.3 Solution to TSTST 3, by Carlos Rodriguez, Albert Wang, Kevin Wu, Isaac Zhu, Nathan Cho \n"
} | 154 | 581 |
2025 | T0 | 5 | null | IMO | A tetrahedron \(ABCD\) is said to be angelic if it has nonzero volume and satisfies
\[\angle BAC + \angle CAD + \angle DAB = \angle ABC + \angle CBD + \angle DBA,\] \[\angle ACB + \angle BCD + \angle DCA = \angle ADB + \angle BDC + \angle CDA.\]
Across all angelic tetrahedrons, what is the maximum number of disti... | 
We claim the maximum cardinality is \(\boxed{4}\) . This is attained by taking a non- square rectangle (or parallelogram) \(A C B D\) and folding it along diagonal \(A B\) , creating edges \(A B\) and \(C D\) in the process. Here, \(A C = B... | {
"problem_match": "Problem 5. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2025.jsonl",
"solution_match": "## \\(\\S 2.2\\) Solution to TSTST 5, by Karthik Vedula \n"
} | 126 | 768 |
2025 | T0 | 6 | null | IMO | Alice and Bob play a game on \(n\) vertices labelled \(1, 2, \ldots , n\) . They take turns adding edges \(\{i, j\}\) , with Alice going first. Neither player is allowed to make a move that creates a cycle, and the game ends after \(n - 1\) total turns.
Let the weight of the edge \(\{i, j\}\) be \(|i - j|\) , and le... | \(\P\) Solution Let \(k = \lceil \frac{n - 1}{2}\rceil\) . The answer is
\[\frac{1}{2} (k + 1)(2n - k - 2) = (n - k - 1) + (n - k) + \dots +(n - 1).\]
When \(n = 1\) , this is clear.
Consider now when \(n\geq 2\) . Note that the game consists of \(n - 1\) moves, with Alice making \(k\) moves and Bob making \(n - ... | {
"problem_match": "Problem 6. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2025.jsonl",
"solution_match": "## \\(\\S 2.3\\) Solution to TSTST 6, by Max Lu, Kevin Wu \n"
} | 145 | 1,271 |
2025 | T0 | 7 | null | IMO | For a positive real number \(c\) , the sequence \(a_{1}, a_{2}, \ldots\) of real numbers is defined as follows. Let \(a_{1} = c\) , and for \(n \geq 2\) , let
\[a_{n} = \sum_{i = 1}^{n - 1}(a_{i})^{n - i + 1}.\]
Find all positive real numbers \(c\) such that \(a_{i} > a_{i + 1}\) for all positive integers \(i\) . | \(\P\) Solution (author) The answer is \(c< \frac{\sqrt{5} - 1}{2}\)
To show this is necessary, note that \(a_{2} = c^{2}\) and \(a_{3} = c^{3} + c^{4}\) , so if the sequence is decreasing, we have \(c^{2} > c^{3} + c^{4}\) , implying \(c< \frac{\sqrt{5} - 1}{2}\)
In the other direction, suppose \(c\) is a positive... | {
"problem_match": "Problem 7. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2025.jsonl",
"solution_match": "## \\(\\S 3.1\\) Solution to TSTST 7, by Luke Robitaille \n"
} | 122 | 1,191 |
2025 | T0 | 8 | null | IMO | Find all polynomials \(f\) with integer coefficients such that for all positive integers \(n\) ,
\[n \text{ divides } \frac{f(f(\ldots(f(0)) \ldots) - 1}{n + 1 f^{\mathrm{s}}}\] | There are three families.
- \(f(x) = x + 1\) .
- \(f(x) = x(x - 1)g(x) + 1\) for any polynomial \(g(x)\) (i.e., any \(f(x)\) such that \(f(0) = f(1) = 1\) ).
- \(f(x) = x(x - 1)(x + 1)g(x) + (2x^2 - 1)\) for any polynomial \(g(x)\) (i.e., any \(f(x)\) such that \(f(0) = -1\) , \(f(-1) = f(1) = 1\) ).
These all ... | {
"problem_match": "Problem 8. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2025.jsonl",
"solution_match": "## \\(\\S 3.2\\) Solution to TSTST 8, by Pitchayut Saengrungkonga \n"
} | 60 | 959 |
2025 | T0 | 9 | null | IMO | Let acute triangle \(ABC\) have orthocenter \(H\) . Let \(B_{1}\) , \(C_{1}\) , \(B_{2}\) , and \(C_{2}\) be collinear points which lie on lines \(AB\) , \(AC\) , \(BH\) , and \(CH\) , respectively. Let \(\omega_{B}\) and \(\omega_{C}\) be the circumcircles of triangles \(BB_{1}B_{2}\) and \(CC_{1}C_{2}\) , respectivel... | \(\P\) Solution (author) The first important step is to introduce \(N\) , the circumcenter of \(H B_{2}C_{2}\) .
Claim β Lines \(N B_{2}\) and \(N C_{2}\) are tangent to \((B B_{1}B_{2})\) and \((C C_{1}C_{2})\) , respectively.
Proof. This follows from chasing
\[\angle N B_{2}B = \angle N B_{2}H = 90^{\circ} - \a... | {
"problem_match": "Problem 9. ",
"resource_path": "USA_TSTST/segmented/en-sols-TSTST-2025.jsonl",
"solution_match": "## \\(\\S 3.3\\) Solution to TSTST 9, by Ruben Carpenter \n"
} | 157 | 3,871 |
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