problem_id stringlengths 16 19 | problem stringlengths 69 2.04k | solution stringlengths 60 23.9k |
|---|---|---|
sols-TSTST-2017_1 | Let $ABC$ be a triangle with circumcircle $\Gamma$,
circumcenter $O$, and orthocenter $H$.
Assume that $AB \neq AC$ and $\angle A \neq 90\dg$.
Let $M$ and $N$ be the midpoints of $\ol{AB}$ and $\ol{AC}$,
respectively, and let $E$ and $F$ be the feet of the altitudes
from $B$ and $C$ in $\triangle ABC$, respectively.
Le... | \paragraph{First solution (power of a point).}
Let $\gamma$ denote the nine-point circle of $ABC$.
\begin{center}
\begin{asy}
pair A = dir(125);
pair B = dir(210);
pair C = dir(330);
pair M = midpoint(A--B);
pair N = midpoint(A--C);
pair O = origin;
pair H = A+B+C;
draw(A--B--C--cycle, blue);
pair E = foot(B, A, C);
p... |
sols-TSTST-2017_2 | Ana and Banana are playing a game. First Ana picks a word,
which is defined to be a nonempty sequence of capital English letters.
Then Banana picks a nonnegative integer $k$ and challenges Ana
to supply a word with exactly $k$ subsequences which are equal to Ana's word.
Ana wins if she is able to supply such a word, ot... | First we introduce some notation.
Define a \emph{block} of letters to be a maximal
contiguous subsequence of consecutive letters.
For example, the word $AABBBCAAA$ has four blocks, namely $AA$, $BBB$, $C$, $AAA$.
Throughout the solution, we fix the word $A$ that Ana picks,
and introduce the following notation for its $... |
sols-TSTST-2017_3 | Consider solutions to the equation
\[ x^2 - cx + 1 = \frac{f(x)}{g(x)} \]
where $f$ and $g$ are nonzero polynomials with nonnegative real coefficients.
For each $c > 0$, determine the minimum possible degree of $f$, or show
that no such $f$, $g$ exist. | First, if $c \ge 2$ then we claim no such $f$ and $g$ exist.
Indeed, one simply takes $x=1$ to get $f(1)/g(1) \le 0$, impossible.
For $c < 2$, let $c = 2 \cos \theta$, where $0 < \theta < \pi$.
We claim that $f$ exists and has minimum degree equal to $n$,
where $n$ is defined as the smallest integer
satisfying $\sin n... |
sols-TSTST-2017_4 | Find all nonnegative integer solutions to \[ 2^a + 3^b + 5^c = n!. \] | For $n \le 4$, one can check the only solutions are:
\begin{align*}
2^2 + 3^0 + 5^0 &= 3! \\
2^1 + 3^1 + 5^0 &= 3! \\
2^4 + 3^1 + 5^1 &= 4!.
\end{align*}
Now we prove there are no solutions for $n \ge 5$.
A tricky way to do this is to take modulo $120$, since
\begin{align*}
2^a \pmod{120} &\in \{ 1, 2, 4, 8, 1... |
sols-TSTST-2017_5 | Let $ABC$ be a triangle with incenter $I$.
Let $D$ be a point on side $BC$ and let $\omega_B$ and $\omega_C$
be the incircles of $\triangle ABD$ and $\triangle ACD$, respectively.
Suppose that $\omega_B$ and $\omega_C$ are tangent to segment $BC$
at points $E$ and $F$, respectively.
Let $P$ be the intersection of segme... | \paragraph{First solution (homothety).}
Let $Z$ be the diametrically opposite point on the incircle.
We claim this is the desired intersection.
\begin{center}
\begin{asy}
pair A = dir(110);
pair B = dir(210);
pair C = dir(330);
filldraw(A--B--C--cycle, opacity(0.1)+lightred, red);
filldraw(incircle(A, B, C), opacity(... |
sols-TSTST-2017_6 | A sequence of positive integers $(a_{n})_{n \geq 1}$ is of
\emph{Fibonacci type} if it satisfies the recursive relation
$a_{n+2}=a_{n+1}+a_{n}$ for all $n \geq 1$.
Is it possible to partition the set of positive integers
into an infinite number of Fibonacci type sequences?
\end{enumerate} | Yes, it is possible.
The following solutions were written for me by Kevin Sun and Mark Sellke.
We let $F_1 = F_2 = 1$, $F_3 = 2$, $F_4 = 3$, $F_5 = 5$, \dots
denote the Fibonacci numbers.
\paragraph{First solution (Kevin Sun).}
We are going to appeal to the so-called Zeckendorf theorem:
\begin{theorem*}
[Zeckendorf]... |
sols-TSTST-2018_1 | As usual, let $\ZZ[x]$ denote the set of single-variable
polynomials in $x$ with integer coefficients.
Find all functions $\theta \colon \ZZ[x] \to \ZZ$
such that for any polynomials $p,q \in \ZZ[x]$,
\begin{itemize}
\ii $\theta(p+1) = \theta(p)+1$, and
\ii if $\theta(p) \neq 0$
then $\theta(p)$ divides $\theta(p... | The answer is $\theta : p \mapsto p(c)$, for each choice of $c \in \ZZ$.
Obviously these work, so we prove these are the only ones.
In what follows, $x \in \ZZ[x]$ is the identity polynomial,
and $c = \theta(x)$.
\paragraph{First solution (Merlijn Staps).}
Consider an integer $n \neq c$.
Because $x-n \mid p(x)-p(n)$, ... |
sols-TSTST-2018_2 | In the nation of Onewaynia,
certain pairs of cities are connected by one-way roads.
Every road connects exactly two cities
(roads are allowed to cross each other, e.g., via bridges),
and each pair of cities has at most one road between them.
Moreover, every city has exactly two roads leaving it and
exactly two roads en... | In the language of graph theory, we have a simple digraph $G$
which is 2-regular and we seek the
number of sub-digraphs which are $1$-regular.
We now present two solution paths.
\paragraph{First solution, combinatorial.}
We construct a simple undirected bipartite graph $\Gamma$ as follows:
\begin{itemize}
\ii the ve... |
sols-TSTST-2018_3 | Let $ABC$ be an acute triangle with incenter $I$, circumcenter $O$, and circumcircle $\Gamma$.
Let $M$ be the midpoint of $\ol{AB}$.
Ray $AI$ meets $\ol{BC}$ at $D$.
Denote by $\omega$ and $\gamma$ the circumcircles of $\triangle BIC$ and $\triangle BAD$, respectively.
Line $MO$ meets $\omega$ at $X$ and $Y$, while lin... | Henceforth assume $\angle A \neq 60\dg$; we prove the concurrence.
Let $L$ denote the center of $\omega$, which is the midpoint of minor arc $BC$.
\begin{claim*}
Let $K$ be the point on $\omega$ such that
$\ol{KL} \parallel \ol{AB}$ and $\ol{KC} \parallel \ol{AL}$.
Then $\ol{KA}$ is tangent to $\gamma$, and we m... |
sols-TSTST-2018_4 | For an integer $n > 0$,
denote by $\mathcal F(n)$ the set of integers $m > 0$ for which
the polynomial $p(x) = x^2 + mx + n$ has an integer root.
\begin{enumerate}
\item [(a)] Let $S$ denote the set of integers $n > 0$
for which $\mathcal F(n)$ contains two consecutive integers.
Show that $S$ is infinite but
\[ \... | We prove the following.
\begin{claim*}
The set $S$ is given explicitly by
$S = \{ x(x+1)y(y+1) \mid x,y > 0 \}$.
\end{claim*}
\begin{proof}
Note that $m, m+1 \in \mathcal F(n)$ if and only if
there exist integers $q > p \ge 0$ such that
\begin{align*}
m^2 - 4n &= p^2 \\
(m+1)^2 - 4n &= q^2.
\end{ali... |
sols-TSTST-2018_5 | Let $ABC$ be an acute triangle with circumcircle $\omega$,
and let $H$ be the foot of the altitude from $A$ to $\ol{BC}$.
Let $P$ and $Q$ be the points on $\omega$ with $PA = PH$ and $QA = QH$.
The tangent to $\omega$ at $P$ intersects lines $AC$ and $AB$
at $E_1$ and $F_1$ respectively; the tangent to $\omega$ at $Q$
... | Let $O$ be the center of $\omega$,
and let $M = \ol{PQ} \cap \ol{AB}$ and $N = \ol{PQ} \cap \ol{AC}$
be the midpoints of $\ol{AB}$ and $\ol{AC}$ respectively.
Refer to the diagram below.
\begin{center}
\begin{asy}
size(9cm);
pair A, B, C, O, M, N, P, Q, E1, F1, E2, F2, O1, O2;
A = dir(105); B = dir(190); C = dir(350);... |
sols-TSTST-2018_6 | Let $S = \left\{ 1, \dots, 100 \right\}$,
and for every positive integer $n$ define
\[
T_n = \left\{ (a_1, \dots, a_n) \in S^n
\mid a_1 + \dots + a_n \equiv 0 \pmod{100} \right\}.
\]
Determine which $n$ have the following property:
if we color any $75$ elements of $S$ red,
then at least half of the $n$-tuples in ... | We claim this holds exactly for $n$ even.
\paragraph{First solution by generating functions.}
Define
\[
R(x) = \sum_{s \text{ red}} x^s, \qquad
B(x) = \sum_{s \text{ blue}} x^s.
\]
(Here ``blue'' means ``not-red'', as always.)
Then, the number of tuples in $T_n$ with exactly
$k$ red coordinates is exactly equal to... |
sols-TSTST-2018_7 | Let $n$ be a positive integer.
A frog starts on the number line at $0$.
Suppose it makes a finite sequence of hops, subject to two conditions:
\begin{itemize}
\ii The frog visits only points in $\{1, 2, \dots, 2^n-1\}$,
each at most once.
\ii The length of each hop is in $\{2^0, 2^1, 2^2, \dots\}$.
(The hops ma... | We claim the answer is $\frac{4^n - 1}{3}$.
We first prove the bound.
First notice that the hop sizes
are in $\{2^0, 2^1, \dots, 2^{n - 1}\}$,
since the frog must stay within bounds the whole time.
Let $a_i$ be the number of hops of size $2^i$ the frog makes, for $0\le i\le n - 1$.
\begin{claim*}
For any $k = 1, \do... |
sols-TSTST-2018_8 | For which positive integers $b > 2$ do there exist
infinitely many positive integers $n$ such that $n^2$ divides $b^n+1$? | This problem is sort of the union of IMO 1990/3 and IMO 2000/5.
The answer is any $b$ such that $b+1$ is not a power of $2$.
In the forwards direction, we first prove more carefully the
following claim.
\begin{claim*}
If $b+1$ is a power of $2$,
then the only $n$ which is valid is $n = 1$.
\end{claim*}
\begin{pro... |
sols-TSTST-2018_9 | Show that there is an absolute constant $c < 1$
with the following property:
whenever $\mathcal P$ is a polygon with area $1$ in the plane,
one can translate it by a distance of $\frac{1}{100}$ in some direction
to obtain a polygon $\mathcal Q$, for which
the intersection of the interiors of $\mathcal P$ and $\mathcal ... | The following solution is due to Brian Lawrence.
We will prove the result with the generality
of any measurable set $\mathcal{P}$ (rather than a polygon).
For a vector $v$ in the plane,
write $\mathcal{P} + v$ for the translate of $\mathcal{P}$ by $v$.
Suppose $\mathcal{P}$ is a polygon of area $1$,
and $\eps > 0$ is ... |
sols-TSTST-2019_1 | \def\di{\mathbin{\diamondsuit}}
Find all binary operations $\di \colon \RR_{>0} \times \RR_{>0} \to \RR_{>0}$
(meaning $\di$ takes pairs of
positive real numbers to positive real numbers)
such that for any real numbers $a,b,c > 0$,
\begin{itemize}
\ii the equation $a \di (b \di c) = (a \di b) \cdot c$ holds; and
\i... | \def\di{\mathbin{\diamondsuit}}
The answer is only multiplication and division,
which both obviously work.
We present two approaches,
one appealing to theorems on Cauchy's functional equation,
and one which avoids it.
\paragraph{First solution using Cauchy FE.}
We prove:
\begin{claim*}
We have $a \di b = a f(b)$ wh... |
sols-TSTST-2019_2 | Let $ABC$ be an acute triangle with circumcircle $\Omega$
and orthocenter $H$.
Points $D$ and $E$ lie on segments $AB$ and $AC$
respectively, such that $AD = AE$.
The lines through $B$ and $C$ parallel to $\ol{DE}$
intersect $\Omega$ again at $P$ and $Q$, respectively.
Denote by $\omega$ the circumcircle of $\triangle ... | We will give one solution to (a),
then several solutions to (b).
\paragraph{Solution to (a).}
Note that $\dang AQP = \dang ABP = \dang ADE$
and $\dang APQ = \dang ACQ = \dang AED$,
so we have a spiral similarity $\triangle ADE \sim \triangle AQP$.
Therefore, lines $PE$ and $QD$ meet at the second
intersection of $\ome... |
sols-TSTST-2019_3 | On an infinite square grid we place finitely many \emph{cars},
which each occupy a single cell and face in one of the four cardinal directions.
Cars may never occupy the same cell.
It is given that the cell immediately in front of each car is empty,
and moreover no two cars face towards each other
(no right-facing car ... | Let $S$ be any rectangle containing all the cars.
Partition $S$ into horizontal strips of height $1$,
and color them red and green in an alternating fashion.
It is enough to prove all the cars may exit $S$.
\begin{center}
\begin{asy}
size(10cm);
defaultpen(fontsize(12pt));
usepackage("amssymb");
picture base;
draw(bas... |
sols-TSTST-2019_4 | Consider coins with positive real denominations not exceeding $1$.
Find the smallest $C>0$ such that the following holds:
if we are given any $100$ such coins
with total value $50$, then we can
always split them into two stacks
of $50$ coins each such that the absolute difference
between the total values of the two sta... | The answer is $C = \frac{50}{51}$.
The lower bound is obtained
if we have $51$ coins of value $\frac{1}{51}$
and $49$ coins of value $1$.
(Alternatively, $51$ coins of value $1-\frac{\eps}{51}$
and $49$ coins of value $\frac{\eps}{49}$ works fine for $\eps > 0$.)
We now present two (similar)
proofs that this $C = \frac... |
sols-TSTST-2019_5 | Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$.
A line through $H$ intersects segments $AB$ and $AC$ at $E$ and $F$, respectively.
Let $K$ be the circumcenter of $\triangle AEF$,
and suppose line $AK$ intersects $\Gamma$ again at a point $D$.
Prove that line $HK$ and the line through $D$
... | We present several solutions.
(There are more in the official packet; some are omitted here,
which explains the numbering.)
\paragraph{First solution (Andrew Gu).}
We begin with the following two observations.
\begin{claim*}
Point $K$ lies on the radical axis of $(BEH)$ and $(CFH)$.
\end{claim*}
\begin{proof}
Actu... |
sols-TSTST-2019_6 | Suppose $P$ is a polynomial with integer coefficients
such that for every positive integer $n$,
the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number.
Must $P$ be constant? | The answer is yes, $P$ must be constant.
By $S(n)$ we mean the sum of the decimal digits of $|n|$.
We need two claims.
\begin{claim*}
If $P(x) \in \ZZ[x]$ is nonconstant with positive leading coefficient,
then there exists an integer polynomial $F(x)$
such that all coefficients of $P \circ F$
are positive exce... |
sols-TSTST-2019_7 | Let $f \colon \ZZ \to \{1, 2, \dots, 10^{100}\}$
be a function satisfying
\[ \gcd(f(x), f(y)) = \gcd(f(x), x-y) \]
for all integers $x$ and $y$.
Show that there exist positive integers $m$ and $n$ such that
$f(x) = \gcd(m + x, n)$ 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 $c_p \in \opname{argmax}_x \nu_p(f(x))$.
We show that this is good enough to compute all values of $x$,
by looking at the exponent at each individual prime.
\begin{claim*}
For any $p ... |
sols-TSTST-2019_8 | 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 $... | The answer is $1430$.
In general, we prove that with $2n$ points
the answer is the $n$\ts{th} Catalan number
$C_n = \tfrac{1}{n+1}\tbinom{2n}{n}$.
First of all, it is well-known that if $\mathcal{S}$ is a convex $2n$-gon,
then $\chi(\mathcal{S}) = C_n$.
It remains to prove the lower bound.
We proceed by (strong) indu... |
sols-TSTST-2019_9 | Let $ABC$ be a triangle with incenter $I$.
Points $K$ and $L$ are chosen on segment $BC$
such that the incircles of $\triangle ABK$ and $\triangle ABL$ are tangent at $P$,
and the incircles of $\triangle ACK$ and $\triangle ACL$ are tangent at $Q$.
Prove that $IP = IQ$.
\end{enumerate} | We present two solutions.
\paragraph{First solution, mostly elementary (original).}
Let $I_B$, $J_B$, $I_C$, $J_C$ be the incenters
of $\triangle ABK$, $\triangle ABL$,
$\triangle ACK$, $\triangle ACL$ respectively.
\begin{center}
\begin{asy}
unitsize(15);
real r1, r2;
path w1, w2;
pair J1 = (-0.5, 2.7);
pair J2 = (... |
sols-TSTST-2020_1 | Let $a$, $b$, $c$ be fixed positive integers.
There are $a+b+c$ ducks sitting in a circle, one behind the other.
Each duck picks either \emph{rock}, \emph{paper}, or \emph{scissors},
with $a$ ducks picking rock, $b$ ducks picking paper,
and $c$ ducks picking scissors.
A \emph{move} consists of an operation of one of t... | The maximum possible number of moves is $\max(ab, ac, bc)$.
First, we prove this is best possible.
We define a \emph{feisty triplet} to be an unordered triple of ducks,
one of each of rock, paper, scissors,
such that the paper duck is between the rock and scissors duck
and facing the rock duck, as shown.
(There may be... |
sols-TSTST-2020_2 | Let $ABC$ be a scalene triangle with incenter $I$.
The incircle of $ABC$ touches $\ol{BC}$, $\ol{CA}$, $\ol{AB}$
at points $D$, $E$, $F$, respectively.
Let $P$ be the foot of the altitude from $D$ to $\ol{EF}$,
and let $M$ be the midpoint of $\ol{BC}$.
The rays $AP$ and $IP$ intersect the circumcircle
of triangle $ABC$... | Refer to the figure below.
\begin{center}
\begin{asy}
pair A = dir(130);
pair B = dir(210);
pair C = dir(330);
pair I = incenter(A, B, C);
pair D = foot(I, B, C);
pair E = foot(I, C, A);
pair F = foot(I, A, B);
pair P = foot(D, E, F);
pair Y = dir(270);
pair Q = extension(Y, D, I, P);
draw(A--B--C--cycle, lightblue);
p... |
sols-TSTST-2020_3 | We say a nondegenerate triangle whose angles have
measures $\theta_1$, $\theta_2$, $\theta_3$ is \emph{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 $n \ge 3$ 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).
\begin{lemma*}
For each $k \ge 0$ there exists $P_k(X) \in \ZZ[X]$,
monic for $k \ge 1$ and with degree $k$, such that
\[ P_k( X + X^{-1} )... |
sols-TSTST-2020_4 | Find all pairs of positive integers $(a, b)$ satisfying
the following conditions:
\begin{enumerate}[(i)]
\ii $a$ divides $b^4+1$,
\ii $b$ divides $a^4+1$,
\ii $\lfloor \sqrt{a} \rfloor = \lfloor \sqrt{b} \rfloor$.
\end{enumerate} | The only solutions are $(1, 1)$, $(1, 2)$, and $(2, 1)$,
which clearly work. Now we show there are no others.
Obviously, $\gcd(a,b) = 1$, so the problem conditions imply
\[ ab \mid (a-b)^4+1 \]
since each of $a$ and $b$ divide the right-hand side.
We define
\[ k \stackrel{\text{def}}{=} \frac{(b-a)^4 + 1}{ab}. \]
\be... |
sols-TSTST-2020_5 | Let $\NN^2$ denote the set of ordered pairs of positive integers.
A finite subset $S$ of $\NN^2$ is \emph{stable} if whenever $(x,y)$ is in $S$,
then so are all points $(x',y')$ of $\NN^2$ with both $x' \leq x$ and $y' \leq y$.
Prove that if $S$ is a stable set, then among all stable subsets of $S$
(including the empt... | The following inductive solution was given by Nikolai Beluhov.
We proceed by induction on $|S|$, with $|S| \le 1$ clear.
Suppose $|S| \ge 2$.
For any $p \in S$, let $R(p)$ denote
the stable rectangle with upper-right corner $p$.
We say such $p$ is \emph{pivotal} if $p + (1, 1) \notin S$
and $|R(p)|$ is even.
\begin{c... |
sols-TSTST-2020_6 | Let $A$, $B$, $C$, $D$ be four points
such that no three are collinear
and $D$ is not the orthocenter of triangle $ABC$.
Let $P$, $Q$, $R$ be the orthocenters of
$\triangle BCD$, $\triangle CAD$, $\triangle ABD$, respectively.
Suppose that lines $AP$, $BQ$, $CR$ are pairwise distinct
and are concurrent.
Show that the f... | Let $T$ be the concurrency point,
and let $H$ be the orthocenter of $\triangle ABC$.
\begin{center}
\begin{asy}
pair A = dir(130);
pair B = dir(210);
pair C = dir(330);
pair D = dir(97);
pair P = B+C+D;
pair Q = C+A+D;
pair R = A+B+D;
filldraw(R--B--D--cycle, opacity(0.1)+lightcyan, palecyan);
filldraw(Q--A--C--cycle,... |
sols-TSTST-2020_7 | Find all nonconstant polynomials $P(z)$ with complex coefficients
for which all complex roots of the polynomials
$P(z)$ and $P(z)-1$ have absolute value $1$. | The answer is $P(x)$ should be a polynomial
of the form $P(x) = \lambda x^n - \mu$
where $|\lambda| = |\mu|$ and $\opname{Re} \mu = -\half$.
One may check these all work;
let's prove they are the only solutions.
\paragraph{First approach (Evan Chen).}
We introduce the following notations:
\begin{align*}
P(x) &= c_n ... |
sols-TSTST-2020_8 | For every positive integer $N$,
let $\sigma(N)$ denote the sum of the positive integer divisors of $N$.
Find all integers $m \ge n \ge 2$ satisfying
\[ \frac{\sigma(m)-1}{m-1}
= \frac{\sigma(n)-1}{n-1} = \frac{\sigma(mn)-1}{mn-1}. \] | 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
\[
\frac{\sigma(p^e) - 1}{p^e - 1}
= \frac{(1 + p + \dots + p^e) - 1}{p^e - 1}
= \frac{p(1 + \dots + p^{e-1})}{p^e - 1}
= \frac{p}{p-1}.
\]
So we now prove these are the only ones.
Let $\la... |
sols-TSTST-2020_9 | Ten million fireflies are glowing in $\RR^3$ 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 change their... | In general, we show that when $n \ge 70$,
the answer is $f(n) = \lfloor\tfrac{n^2}3\rfloor$.
\bigskip
\textbf{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 ... |
sols-TSTST-2021_1 | Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$.
Points $X$ and $Y$ lie on sides $AB$ and $CD$, respectively.
Suppose the circumcircles of $ADX$ and $BCY$
meet line $XY$ again at $P$ and $Q$, respectively.
Show that $OP=OQ$. | We present many solutions.
\paragraph{First solution, angle chasing only (Ankit Bisain).}
Let lines $BQ$ and $DP$ meet $(ABCD)$ again at $D'$ and $B'$, respectively.
\begin{center}
\begin{asy}[width = 0.5\textwidth]
path omega = circle((0, 0), 1);
pair A = dir(70);
pair B = dir(180);
pair C = dir(200);
pair D = dir(34... |
sols-TSTST-2021_2 | Let $a_1 < a_2 < a_3 < a_4 < \dotsb$ 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
\[ \frac{a_1}{1}, \; \frac{a_2}{2}, \; \frac{a_3}{3}, \; \frac{a_4}{4}, \; \dots. \] | We present three solutions.
\paragraph{Solution 1 (Merlijn Staps).}
We argue by contradiction, so suppose that for each $\lambda$ for which the set
$S_\lambda = \{k : a_k/k = \lambda\}$ 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 <... |
sols-TSTST-2021_3 | 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{align*}
(n-1)(n-2) \dotsm (n-m+1) &\equiv (-1)(-2) \dotsm (-m+1) \\
&\equiv (-1)^{m-1} (m-1)! \\
&\not\equiv 0 \mod m!.
\end{align*}
We see that $\binom{n}{m}$ is not a multipl... |
sols-TSTST-2021_4 | Let $a$ and $b$ be positive integers.
Suppose that there are infinitely many pairs of positive integers $(m, n)$
for which $m^2+an+b$ and $n^2+am+b$ are both perfect squares.
Prove that $a$ divides $2b$. | Treating $a$ and $b$ as fixed,
we are given that there are infinitely many quadrpules $(m,n,r,s)$
which satisfy the system
\begin{gather*}
m^2+an+b=(m+r)^2 \\
n^2+am+b=(n+s)^2
\end{gather*}
We say that $(r,s)$ is \emph{exceptional}
if there exists infinitely many $(m,n)$ that satisfy.
\begin{claim*}
If $(r,s)$ i... |
sols-TSTST-2021_5 | Let $T$ be a tree on $n$ vertices with exactly $k$ leaves.
Suppose that there exists a subset of
at least $\frac{n+k-1}{2}$ vertices of $T$, no two of which are adjacent.
Show that the longest path in $T$ contains an even number of edges. | The longest path in $T$ must go between two leaves. The solutions presented here
will solve the problem by showing that in the unique $2$-coloring of $T$, all
leaves are the same color.
\paragraph{Solution 1 (Ankan Bhattacharya, Jeffery Li).}
\begin{lemma*}
If $S$ is an independent set of $T$, then
\[\sum_{v\in S}... |
sols-TSTST-2021_6 | Triangles $ABC$ and $DEF$ 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 $AB$, $AC$, and $EF$,
and define triangles $\Delta_B$, $\Delta_C, \dots, \Delta_F$ similarly.
Furthermore, let $\... | \begin{center}
\begin{asy}
import geometry;
linemargin = 0;
real R = 120;
pair O = (0, 0);
pair A = rotate(35, O)*(R, 0);
pair B = rotate(210, O)*(R, 0);
pair C = rotate(330, O)*(R, 0);
circle omega = incircle(A, B, C);
circle k = circle((point) O, R);
pair D = rotate(255, O)*(R, 0);
line[] ts = tangents(omega, (p... |
sols-TSTST-2021_7 | Let $M$ be a finite set of lattice points
and $n$ be a positive integer.
A \emph{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 l... | We present two approaches.
\paragraph{Solution 1.}
We prove the statement by induction on $n$. We use $n=0$ as a base case, where
the statement follows from $1\geq 2^{-|M|}$. For the inductive step, let $n >
0$. There exists at least one mine-avoiding path, which must pass through either
$(0, 1)$ or $(1, 0)$. We consi... |
sols-TSTST-2021_8 | Let $ABC$ be a scalene triangle.
Points $A_1$, $B_1$ and $C_1$ are chosen
on segments $BC$, $CA$, and $AB$, respectively,
such that $\triangle A_1B_1C_1$ and $\triangle ABC$
are similar.
Let $A_2$ be the unique point on line $B_1C_1$
such that $AA_2 = A_1A_2$.
Points $B_2$ and $C_2$ are defined similarly.
Prove that $\... | We give three solutions.
\paragraph{Solution 1 (author).}
We'll use the following lemma.
\begin{lemma*}
Suppose that $PQRS$ is a convex quadrilateral with $\angle P = \angle R$. Then
there is a point $T$ on $QS$ such that $\angle QPT = \angle SRP$, $\angle
TRQ = \angle RPS$, and $PT=RT$.
\end{lemma*}
Before pro... |
sols-TSTST-2021_9 | 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
\[
\sum_{\substack{1 \le k \le q \\ \gcd(k,p) = 1}}
\frac{1}{(1 - \zeta^k)^n}
\]
is not an integer.
(The sum is over all $1\leq k\leq q$ with $p$ not dividing $k$.)
\end{en... | Let $S_q$ denote the set of primitive $q$th roots of unity (thus,
the sum in question is a sum over $S_q$).
\paragraph{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 poly... |
sols-TSTST-2022_1 | 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:
\begin{itemize}
\item The sides of each rectangle are parallel to the sides of the unit square.
\item E... | We give the author's solution.
In terms of $n$, we wish find the smallest integer $k$ for which
$(0, 1)^2 \setminus S$ is always a union of $k$ open rectangles
for every set $S \subset (0, 1)^2$ of size $n$.
We claim the answer is $k = \boxed{2n+2}$.
The lower bound is given by picking
\[ S = \{(s_1, s_1), (s_2, s_2)... |
sols-TSTST-2022_2 | Let $ABC$ be a triangle.
Let $\theta$ be a fixed angle for which
\[ \theta < \frac12 \min(\angle A, \angle B, \angle C). \]
Points $S_A$ and $T_A$ lie on segment $BC$
such that $\angle BAS_A = \angle T_AAC = \theta$.
Let $P_A$ and $Q_A$ be the feet from $B$ and $C$
to $\ol{AS_A}$ and $\ol{AT_A}$ respectively.
Then $\el... | We discard the points $S_A$ and $T_A$ since they are only there
to direct the angles correctly in the problem statement.
\paragraph{First solution, by author.}
Let $X$ be the projection from $C$ to $AP_A$,
$Y$ be the projection from $B$ to $AQ_A$.
\begin{center}
\begin{asy}
pair A = dir(110);
pair B = dir(200);
pair C... |
sols-TSTST-2022_3 | Determine all positive integers $N$ for which there exists a strictly
increasing sequence of positive integers $s_0 < s_1 < s_2 < \dotsb$
satisfying the following properties:
\begin{itemize}
\item the sequence $s_1-s_0$, $s_2-s_1$, $s_3-s_2$, \dots\ is periodic; and
\item $s_{s_n} - s_{s_{n-1}} \le N < s_{1+s_n} - ... | \paragraph{Answer.}
All $N$ such that $t^2 \le N < t^2+t$ for some positive integer $t$.
\paragraph{Solution 1 (local).}
If $t^2\le N < t^2+t$ then the sequence $s_n = tn+1$
satisfies both conditions.
It remains to show that no other values of $N$ work.
Define $a_n \coloneq s_n - s_{n-1}$,
and let $p$ be the minimal ... |
sols-TSTST-2022_4 | A function $f \colon \NN \to \NN$ has the property that
for all positive integers $m$ and $n$, exactly one of the $f(n)$ numbers
\[ f(m+1), f(m+2), \dots, f(m+f(n)) \]
is divisible by $n$.
Prove that $f(n)=n$ for infinitely many positive integers $n$. | We start with the following claim:
\begin{claim*}
If $a \mid b$ then $f(a) \mid f(b)$.
\end{claim*}
\begin{proof}
From applying the condition with $n=a$,
we find that the set $S_a = \{n \ge 2: a \mid f(n)\}$
is an arithmetic progression with common difference $f(a)$.
Similarly, the set $S_b = \{n \ge 2: b \m... |
sols-TSTST-2022_5 | Let $A_1$, \dots, $A_{2022}$ 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 th... |
sols-TSTST-2022_6 | Let $O$ and $H$ be the circumcenter and orthocenter,
respectively, of an acute scalene triangle $ABC$.
The perpendicular bisector of $\ol{AH}$ intersects
$\ol{AB}$ and $\ol{AC}$ at $X_A$ and $Y_A$ respectively.
Let $K_A$ denote the intersection of
the circumcircles of triangles $OX_AY_A$ and $BOC$ other than $O$.
Defi... | We present several approaches.
\paragraph{First solution, by author.}
Let $\odot OX_AY_A$ intersects $AB$, $AC$ again at $U$, $V$.
Then by Reim's theorem $UVCB$ are concyclic.
Hence the radical axis of $\odot OX_AY_A$, $\odot OBC$ and $\odot (UVCB)$
are concurrent, i.e.\ $OK_A$, $BC$, $UV$ are concurrent,
Denote the ... |
sols-TSTST-2022_7 | Let $ABCD$ be a parallelogram.
Point $E$ lies on segment $CD$ such that
\[ 2\angle AEB = \angle ADB+\angle ACB, \]
and point $F$ lies on segment $BC$ such that
\[ 2\angle DFA = \angle DCA + \angle DBA. \]
Let $K$ be the circumcenter of triangle $ABD$. Prove that $KE=KF$. | Let the circle through $A$, $B$, and $E$ intersect $CD$ again at $E'$,
and let the circle through $D$, $A$, and $F$ intersect $BC$ again at $F'$.
Now $ABEE'$ and $DAF'F$ are cyclic quadrilaterals with two parallel sides,
so they are isosceles trapezoids.
From $KA=KB$, it now follows that $KE=KE'$,
whereas from $KA=KD$ ... |
sols-TSTST-2022_8 | Find all functions $f \colon \NN \to \ZZ$ such that
\[ \left\lfloor \frac{f(mn)}{n} \right\rfloor = f(m) \]
for all positive integers $m$, $n$. | There are two families of functions that work:
for each $\alpha \in \RR$ the function $f(n) = \lfloor \alpha n \rfloor$,
and for each $\alpha \in \RR$ 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 work... |
sols-TSTST-2022_9 | 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:
\begin{itemize}
\item Each element of the sequence is between $1$ and $n$, inclusive.
\item For any two different contiguous subsequences of the seq... | 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... |
sols-TSTST-2023_1 | Let $ABC$ be a triangle with centroid $G$.
Points $R$ and $S$ are chosen on rays $GB$ and $GC$, respectively, such that
\[ \angle ABS = \angle ACR = 180^\circ - \angle BGC. \]
Prove that $\angle RAS + \angle BAC = \angle BGC$. | In all the following solutions,
let $M$ and $N$ denote the midpoints of $\ol{AC}$ and $\ol{AB}$, respectively.
\begin{center}
\begin{asy}
size(12cm);
pair A = dir(97);
pair B = dir(190);
pair C = dir(350);
pair M = midpoint(A--C);
pair N = midpoint(A--B);
pair G = extension(B, M, C, N);
draw(A--G, blue);
pair Y = A*d... |
sols-TSTST-2023_2 | Let $n \ge m \ge 1$ be integers.
Prove that
\[ \sum_{k=m}^n \left( \frac{1}{k^2} + \frac{1}{k^3} \right)
\ge m \cdot \left( \sum_{k=m}^n \frac{1}{k^2} \right)^2. \] | We show several approaches.
\paragraph{First solution (authors).}
By Cauchy-Schwarz, we have
\begin{align*}
\sum_{k=m}^n\frac{k+1}{k^3}
&= \sum_{k=m}^n\frac{\left(\frac1{k^2}\right)^2}{\frac1{k(k+1)}} \\
&\geq
\frac{
\left( \frac{1}{m^2} + \frac{1}{(m+1)^2} + \dots + \frac{1}{n^2} \right)^2
}
{
\frac... |
sols-TSTST-2023_3 | 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 \emph{good}. To show that all positive integers $n$ are good we prove the following:
\begin{itemize}
\item[(i)] If $n$ is good and $p$ is an odd prime, then $pn$ is good;
\item[(ii)] For every $k ... |
sols-TSTST-2023_4 | Let $n \ge 3$ be an integer and let $K_n$ be the complete graph on $n$ vertices.
Each edge of $K_n$ is colored either red, green, or blue.
Let $A$ denote the number of triangles in $K_n$ with all edges of the same color, and
let $B$ denote the number of triangles in $K_n$ with all edges of different colors.
Prove that ... | Consider all unordered pairs of different edges which share exactly one vertex
(call these \emph{vees} for convenience).
Assign each vee a \emph{charge} of $+2$ if its edge colors are the same,
and a charge of $-1$ otherwise.
We compute the total charge in two ways.
\paragraph{Total charge by summing over triangles.}... |
sols-TSTST-2023_5 | 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
\[ p = (a+b+c) + \left( \frac1a+\frac1b+\frac1c \right)
\qquad\text{and}\qquad
q = \frac ab + \frac bc + \frac ca. \]
Given that both $p$ and $q$ are real numbers,
... | We show $(p,q) = (-3,3)$ is the only possible ordered pair.
\paragraph{First solution.}
\subparagraph{Setup for proof}
Let us denote $a = y/x$, $b = z/y$, $c = x/z$,
where $x$, $y$, $z$ are nonzero complex numbers.
Then
\begin{align*}
p + 3 &= 3 + \sum_{\text{cyc}} \left( \frac xy + \frac yx \right)
= 3 + \frac{x^... |
sols-TSTST-2023_6 | Let $ABC$ be a scalene triangle
and let $P$ and $Q$ be two distinct points in its interior.
Suppose that the angle bisectors of $\angle PAQ$, $\angle PBQ$,
and $\angle PCQ$ are the altitudes of triangle $ABC$.
Prove that the midpoint of $\ol{PQ}$ lies on the Euler line of $ABC$. | We present three approaches.
\paragraph{Solution 1 (Ankit Bisain).}
Let $H$ be the orthocenter of $ABC$, and construct $P'$ using the following claim.
\begin{claim*}
There is a point $P'$ for which \[\measuredangle APH + \measuredangle AP'H = \measuredangle BPH + \measuredangle BP'H = \measuredangle CPH + \measur... |
sols-TSTST-2023_7 | The Bank of Pittsburgh issues coins that have a heads side and a tails side.
Vera has a row of $2023$ such coins alternately tails-up and heads-up,
with the leftmost coin tails-up.
In a \emph{move}, Vera may flip over one of the coins in the row,
subject to the following rules:
\begin{itemize}
\item On the first mov... | The answer is $\boxed{4044}$.
In general, replacing $2023$ with $4n+3$, the answer is $8n+4$.
\paragraph{Bound.}
Observe that the first and last coins must be flipped,
and so every coin is flipped at least once.
Then, the $2n+1$ even-indexed coins must be flipped at least twice,
so they are flipped at least $4n+2$ tim... |
sols-TSTST-2023_8 | Let $ABC$ be an equilateral triangle with side length $1$.
Points $A_1$ and $A_2$ are chosen on side $BC$,
points $B_1$ and $B_2$ are chosen on side $CA$, and
points $C_1$ and $C_2$ are chosen on side $AB$
such that $BA_1 < BA_2$, $CB_1 < CB_2$, and $AC_1 < AC_2$.
Suppose that the three line segments $B_1C_2$, $C_1A_2... | The only possible value of the common perimeter, denoted $p$, is $1$.
\paragraph{Synthetic approach (from author).}
We prove the converse of the problem first:
\begin{claim*}
[$p=1$ implies concurrence]
Suppose the six points are chosen so that triangles
$AB_2C_1$, $BC_2A_1$, $CA_2B_1$ all have perimeter $1$.
... |
sols-TSTST-2023_9 | Let $p$ be a fixed prime and let $a \ge 2$ and $e \ge 1$ be fixed integers.
Given a function $f \colon \ZZ/a\ZZ \to \ZZ/p^e\ZZ$
and an integer $k \ge 0$, the \emph{$k$th finite difference},
denoted $\Delta^k f$, is the function from $\ZZ/a\ZZ$ to $\ZZ/p^e\ZZ$
defined recursively by
\begin{align*}
\Delta^0 f(n) &= f(n... | The answer is
\[ (p^e)^a \cdot p^{-ep^{\nu_p(a)}} = p^{e(a - p^{\nu_p(a)})}. \]
\paragraph{First solution by author.}
For convenience in what follows, set $d = \nu_p(a)$, let $a = p^d \cdot b$,
and let a function $f\colon \ZZ/a\ZZ \to \ZZ/p^e\ZZ$ be \emph{essential}
if it equals one of its iterated finite differences.... |
sols-TSTST-2024_1 | 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 for all i... | \paragraph{Answer.} $c\geq 4$.
\paragraph{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 l... |
sols-TSTST-2024_2 | 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
\[
1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 +
(p-1)x}}}=\frac{P(x)}{Q(x)}.
\]
Show that all coefficients of $P$ except for... | \paragraph{Solution 1.}
We first make some general observations about rational functions represented
through continued fractions.
\begin{claim*}
Let $a_1$, $a_2$, \dots, be a sequence of nonzero integers. Define the
sequence of polynomials $P_1(x) = 1$, $P_2(x) = 1 + a_1x$, and
\[P_{k+1}(x) = P_k(x) + a_kxP_{k-1... |
sols-TSTST-2024_3 | Let $A = \{a_1, \dots, a_{2024}\}$ be a set of $2024$ pairwise distinct real numbers.
Assume that there exist positive integers $b_1, b_2,\dotsc,b_{2024}$
such that \[ a_1b_1 + a_2b_2 + \dots + a_{2024}b_{2024} = 0. \]
Prove that one can choose $a_{2025}, a_{2026}, a_{2027}, \dots$ such that
$a_k \in A$ for all $k \ge ... | It will be convenient to use $0$-based indexing here,
i.e.\ $A = \{a_0, \dotsc, a_{2023}\}$ 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 $a_0, \dotsc, a_{m-1}$ such that $a_0+\dotsb+a_{m-1} = 0$.
\paragraph{Solution by direct construction.... |
sols-TSTST-2024_4 | 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 $C$ meet at $P$. The
tangent to $\omega_1$... | \paragraph{Solution 1.}
Let $R = \ol{AD} \cap \ol{BC}$ (possibly at infinity,
but we'll see it's an Euclidean point later).
\begin{center}
\begin{asy}
size(12cm);
pair A = dir(190), B = dir(350), C = dir(80), D = dir(120), O = (0, 0);
pair E = extension(A, C, B, D), R = extension(A, D, B, C);
pair P =... |
sols-TSTST-2024_5 | For a positive integer $k$,
let $s(k)$ denote the number of $1$s in the binary representation of $k$.
Prove that for any positive integer $n$,
\[ \sum_{i=1}^{n}(-1)^{s(3i)} > 0. \] | \paragraph{Solution 1.}
Given a set of positive integers $S$, define
\begin{align*}
f(S) = \sum_{k\in S}(-1)^{s(k)}.
\end{align*}
We also define
\begin{align*}
S_{\mathrm{even}} &= \{k\in S \mid k\text{ is even}\} \\
S_{\mathrm{odd}} &= \{k\in S \mid k\text{ is odd}\}
\end{align*}
and apply functions on sets poin... |
sols-TSTST-2024_6 | Determine whether there exists a function $f \colon \ZZ_{> 0} \to \ZZ_{>0}$
such that for all positive integers $m$ and $n$,
\[ f(m+nf(m))=f(n)^m+2024! \cdot m. \] | The answer is no. Let $P(m,n)$ denote the given FE.
\paragraph{Solution 1 (Gopal Goel).}
Suppose there was a function $f$, and let $r=f(1)$. Note that $P(1,n)$ gives
\[ f(1+rn) = f(n)+2024!. \]
Iterating this result gives
\[ f(1+r+\dotsb+r^k) = r+k\cdot 2024! \]
for all $k \in \ZZ_{\ge 0}$.
If $r=1$, this implies $f(k... |
sols-TSTST-2024_7 | An infinite sequence $a_1$, $a_2$, $a_3$, \dots\ of real numbers satisfies
\[ a_{2n-1} + a_{2n} > a_{2n+1} + a_{2n+2} \qquad
\mbox{and} \qquad a_{2n} + a_{2n+1} < a_{2n+2} + a_{2n+3} \]
for every positive integer $n$.
Prove that there exists a real number $C$ such
that $a_{n} a_{n+1} < C$ for every positive integer $... | 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 $d_1,d_2,\dotsc$ is strictly increasing.
We consider the following cases.
\begin{itemize}
\ii Suppose that $d_k > 0$ for some $k$. Then,
\[ a_{2n+1} = a_1 + (d_1+d_3+\dotsb+d_{2n-1}) \]
... |
sols-TSTST-2024_8 | 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 $S$ and lines $BY$ and $CX$ meet at $... | \paragraph{Solution by characterizing $\boldsymbol X$ and $\boldsymbol Y$.}
We first state an important property of $X$ and $Y$.
\begin{claim*}
Points $X$ and $Y$ are isogonal conjugates with respect to $\triangle ABC$.
\end{claim*}
\begin{center}
\begin{asy}
unitsize(120);
pair A = dir(125);
pair C = ... |
sols-TSTST-2024_9 | Let $n \ge 2$ 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 vertices... | The largest possible value of $N$ is $\tfrac1{12}n^2(n^2-1)$.
Call these rectangles \emph{wobbly}.
We defer the construction until the proof is complete,
since the proof suggests the construction.
\paragraph{Proof of bound.}
Call a triple of integers $(a,b,c)$ an \emph{elbow}
if $a$ and $b$ are in the same row, $b$ an... |
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