idx uint32 0 19.1k | source_dataset large_stringclasses 1
value | source_idx large_stringlengths 36 36 | problem large_stringlengths 2 10.1k | answer large_stringlengths 1 7.66k ⌀ | category large_stringclasses 0
values | subcategory large_stringclasses 0
values | source large_stringclasses 24
values | is_aimo_format bool 1
class | reserve bool 1
class |
|---|---|---|---|---|---|---|---|---|---|
0 | math_hard | b360141a-c7fb-52a8-b9ce-a46404d1ece8 | "If a whole number $n$ is not prime, then the whole number $n-2$ is not prime." A value of $n$ which shows this statement to be false is
$\textbf{(A)}\ 9 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 23$ | A. 9.0 | null | null | di-zhang-fdu/AOPS | false | true |
1 | math_hard | 4c72d3f3-e08b-5ae6-8354-fa6affe00057 | $(1+11+21+31+41)+(9+19+29+39+49)=$
$\text{(A)}\ 150 \qquad \text{(B)}\ 199 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 249 \qquad \text{(E)}\ 250$ | E. 250.0 | null | null | di-zhang-fdu/AOPS | false | true |
2 | math_hard | aac23d98-b8f9-5756-9c7f-8474bd409548 | $(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193) =$
$\text{(A)}\ 167,400 \qquad \text{(B)}\ 172,050 \qquad \text{(C)}\ 181,071 \qquad \text{(D)}\ 199,300 \qquad \text{(E)}\ 362,142$ | A. 167400.0 | null | null | di-zhang-fdu/AOPS | false | true |
3 | math_hard | fde68796-6be6-5bb8-b3d2-5c27ce275159 | $(2\times 3\times 4)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right) =$
$\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 26$ | E. 26.0 | null | null | di-zhang-fdu/AOPS | false | true |
4 | math_hard | 44ac58cb-9e34-574d-a966-529ceb0d719c | $-15+9\times (6\div 3) =$
$\text{(A)}\ -48 \qquad \text{(B)}\ -12 \qquad \text{(C)}\ -3 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 12$ | D. 3.0 | null | null | di-zhang-fdu/AOPS | false | true |
5 | math_hard | 3aaf1e83-3107-5463-9ca7-a8619729111e | $.4+.02+.006=$
$\text{(A)}\ .012 \qquad \text{(B)}\ .066 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .24 \qquad \text{(E)} .426$ | E. 0.426 | null | null | di-zhang-fdu/AOPS | false | true |
6 | math_hard | b3c9f161-620b-54f1-914f-a2a1aecc3fe3 | $1,000,000,000,000-777,777,777,777=$
$\text{(A)}\ 222,222,222,222 \qquad \text{(B)}\ 222,222,222,223 \qquad \text{(C)}\ 233,333,333,333 \qquad \\ \text{(D)}\ 322,222,222,223 \qquad \text{(E)}\ 333,333,333,333$ | B. 222222222223.0 | null | null | di-zhang-fdu/AOPS | false | true |
7 | math_hard | ec70ad93-2f44-59a1-9d0b-8adb6f2f6858 | $1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=$
$\text{(A)}\ -998 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 998$ | C. 0.0 | null | null | di-zhang-fdu/AOPS | false | true |
8 | math_hard | 543ee2f1-8015-51fb-af4e-27d624c96fdc | $10\times10\times10$ grid of points consists of all points in space of the form $(i,j,k)$ , where $i$ $j$ , and $k$ are integers between $1$ and $10$ , inclusive. Find the number of different lines that contain exactly $8$ of these points. | 168.0 | null | null | di-zhang-fdu/AOPS | false | true |
9 | math_hard | a765c9a0-aada-5ff4-ab82-eb32c5195f38 | $150\times 324\times 375$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes | 768.0 | null | null | di-zhang-fdu/AOPS | false | true |
10 | math_hard | fec37a7c-9192-562f-b40e-bf56ea321f99 | $1990-1980+1970-1960+\cdots -20+10 =$
$\text{(A)}\ -990 \qquad \text{(B)}\ -10 \qquad \text{(C)}\ 990 \qquad \text{(D)}\ 1000 \qquad \text{(E)}\ 1990$ | D. 1000.0 | null | null | di-zhang-fdu/AOPS | false | true |
11 | math_hard | 761ee3ff-067d-5ce9-a9c0-a1ad34c7c522 | $2 \times 3$ rectangle and a $3 \times 4$ rectangle are contained within a square without overlapping at any point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
$\textbf{(A) } 16\qquad \textbf{(B) } 25\qquad \textbf{(C) } 36\qquad \... | B. 25.0 | null | null | di-zhang-fdu/AOPS | false | true |
12 | math_hard | 19bf7f44-e302-5644-98c5-7b0d0d3a7685 | $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green s... | B. 6.0 | null | null | di-zhang-fdu/AOPS | false | true |
13 | math_hard | 1912a4d4-c5bc-5b6a-b8c5-7db892189bd2 | $2(81+83+85+87+89+91+93+95+97+99)=$
$\text{(A)}\ 1600 \qquad \text{(B)}\ 1650 \qquad \text{(C)}\ 1700 \qquad \text{(D)}\ 1750 \qquad \text{(E)}\ 1800$ | E. 1800.0 | null | null | di-zhang-fdu/AOPS | false | true |
14 | math_hard | 7930878b-ac63-509f-a5ce-1a59664fc2cd | $25$ foot ladder is placed against a vertical wall of a building. The foot of the ladder is $7$ feet from the base of the building. If the top of the ladder slips $4$ feet, then the foot of the ladder will slide:
$\textbf{(A)}\ 9\text{ ft} \qquad \textbf{(B)}\ 15\text{ ft} \qquad \textbf{(C)}\ 5\text{ ft} \qquad \textb... | D. 8.0 | null | null | di-zhang-fdu/AOPS | false | true |
15 | math_hard | df281d8c-a6bb-51a2-8987-3d79b5d7a03e | $2\left(1-\dfrac{1}{2}\right) + 3\left(1-\dfrac{1}{3}\right) + 4\left(1-\dfrac{1}{4}\right) + \cdots + 10\left(1-\dfrac{1}{10}\right)=$
$\text{(A)}\ 45 \qquad \text{(B)}\ 49 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 54 \qquad \text{(E)}\ 55$ | A. 45.0 | null | null | di-zhang-fdu/AOPS | false | true |
16 | math_hard | d8c21f4f-1261-5328-9c80-fb38b0033d1c | $3 \times 7$ rectangle is covered without overlap by 3 shapes of tiles: $2 \times 2$ $1\times4$ , and $1\times1$ , shown below. What is the minimum possible number of $1\times1$ tiles used?
$\textbf{(A) } 1\qquad\textbf{(B)} 2\qquad\textbf{(C) } 3\qquad\textbf{(D) } 4\qquad\textbf{(E) } 5$ | E. 5.0 | null | null | di-zhang-fdu/AOPS | false | true |
17 | math_hard | d26732be-de99-5fef-8b97-f8e75dc16abd | $4(299)+3(299)+2(299)+298=$
$\text{(A)}\ 2889 \qquad \text{(B)}\ 2989 \qquad \text{(C)}\ 2991 \qquad \text{(D)}\ 2999 \qquad \text{(E)}\ 3009$ | B. 2989.0 | null | null | di-zhang-fdu/AOPS | false | true |
18 | math_hard | a9401ee2-e8d3-554c-832a-3efd306ae307 | $4\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?
$\begin{tabular}[t]{|c|c|c|c|} \multicolumn{4}{c}{}\\\hline 1&2&3&4\\\hline 8&9&... | B. 4.0 | null | null | di-zhang-fdu/AOPS | false | true |
19 | math_hard | 26dfcc62-8b04-5dbb-8a62-566d695a858b | $4\times 4\times 4$ cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch a side or the bottom of the box?
$\text{(A)}\ 48 \qquad \text{(B)}\ 52 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 80$ | B. 52.0 | null | null | di-zhang-fdu/AOPS | false | true |
20 | math_hard | 42cafa46-543b-5006-97f6-18f2e8561872 | $7\times 1$ board is completely covered by $m\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7\times 1$ board in which all three colors are used at least ... | 106.0 | null | null | di-zhang-fdu/AOPS | false | true |
21 | math_hard | 00536928-e12e-58eb-8a45-31bb27c5f7a6 | $90+91+92+93+94+95+96+97+98+99=$
$\text{(A)}\ 845 \qquad \text{(B)}\ 945 \qquad \text{(C)}\ 1005 \qquad \text{(D)}\ 1025 \qquad \text{(E)}\ 1045$ | B. 945.0 | null | null | di-zhang-fdu/AOPS | false | true |
22 | math_hard | e0bd114c-059c-53a3-8b21-6ced49528d16 | $A$ $B$ $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the grea... | E. 59.0 | null | null | di-zhang-fdu/AOPS | false | true |
23 | math_hard | 8fe98533-0893-5393-97ce-5d88414f1e2a | $A$ and $B$ together can do a job in $2$ days; $B$ and $C$ can do it in four days; and $A$ and $C$ in $2\frac{2}{5}$ days.
The number of days required for A to do the job alone is:
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 2.8$ | B. 3.0 | null | null | di-zhang-fdu/AOPS | false | true |
24 | math_hard | d42e5527-7ccd-5fad-8f8b-2cef1fc445ca | $ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n,$ where $m$ and $n$ are relatively prime posi... | 293.0 | null | null | di-zhang-fdu/AOPS | false | true |
25 | math_hard | 493d82ac-972f-50a8-b9a4-c47c2d9facdd | $ABCD$ is a square of side length $\sqrt{3} + 1$ . Point $P$ is on $\overline{AC}$ such that $AP = \sqrt{2}$ . The square region bounded by $ABCD$ is rotated $90^{\circ}$ counterclockwise with center $P$ , sweeping out a region whose area is $\frac{1}{c} (a \pi + b)$ , where $a$ $b$ , and $c$ are positive integers and ... | C. 19.0 | null | null | di-zhang-fdu/AOPS | false | true |
26 | math_hard | 4a1d7713-20cf-567e-ae14-c91105024f9c | $ABCDE$ is a regular pentagon. $AP, AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD, CB$ extended and $DE$ extended,
respectively. Let $O$ be the center of the pentagon. If $OP = 1$ , then $AO + AQ + AR$ equals
[asy] defaultpen(fontsize(10pt)+linewidth(.8pt)); pair O=origin, A=2*dir(90), B=2*dir(18), C=2... | C. 4.0 | null | null | di-zhang-fdu/AOPS | false | true |
27 | math_hard | 8034824e-de13-518e-b7a8-cdecd800656b | $R$ varies directly as $S$ and inversely as $T$ . When $R = \frac{4}{3}$ and $T = \frac {9}{14}$ $S = \frac37$ . Find $S$ when $R = \sqrt {48}$ and $T = \sqrt {75}$
$\textbf{(A)}\ 28\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 60$ | B. 30.0 | null | null | di-zhang-fdu/AOPS | false | true |
28 | math_hard | 7b468ae5-d0d5-55b5-bce2-16901da98ef7 | $[x-(y-z)] - [(x-y) - z] =$
$\textbf{(A)}\ 2y \qquad \textbf{(B)}\ 2z \qquad \textbf{(C)}\ -2y \qquad \textbf{(D)}\ -2z \qquad \textbf{(E)}\ 0$ | B. 2.0 | null | null | di-zhang-fdu/AOPS | false | true |
29 | math_hard | e4d40c5b-afdb-5e80-a01b-a58cb4724451 | $\angle 1 + \angle 2 = 180^\circ$
$\angle 3 = \angle 4$
Find $\angle 4.$
[asy] pair H,I,J,K,L; H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0); draw(H--I--J--cycle); draw(K--L--J); draw(arc((0,0),dir(70),(1,0),CW)); label("$70^\circ$",dir(35),NE); draw(arc(I,I+dir(250),I+dir(290),CC... | D. 35.0 | null | null | di-zhang-fdu/AOPS | false | true |
30 | math_hard | 29d7f9a7-0b6b-5e86-a5e7-f0cf013a17ae | $\dfrac{10-9+8-7+6-5+4-3+2-1}{1-2+3-4+5-6+7-8+9}=$
$\text{(A)}\ -1 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 10$ | B. 1.0 | null | null | di-zhang-fdu/AOPS | false | true |
31 | math_hard | 6a8c96a1-9374-56f0-9106-a45eb85e1690 | $\dfrac{1}{10} + \dfrac{9}{100} + \dfrac{9}{1000} + \dfrac{7}{10000} =$
$\text{(A)}\ 0.0026 \qquad \text{(B)}\ 0.0197 \qquad \text{(C)}\ 0.1997 \qquad \text{(D)}\ 0.26 \qquad \text{(E)}\ 1.997$ | C. 0.1997 | null | null | di-zhang-fdu/AOPS | false | true |
32 | math_hard | 64be6f0c-e010-57f4-a174-b7448e7b9fe9 | $\dfrac{1}{10}+\dfrac{2}{10}+\dfrac{3}{10}+\dfrac{4}{10}+\dfrac{5}{10}+\dfrac{6}{10}+\dfrac{7}{10}+\dfrac{8}{10}+\dfrac{9}{10}+\dfrac{55}{10}=$
$\text{(A)}\ 4\dfrac{1}{2} \qquad \text{(B)}\ 6.4 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11$ | D. 10.0 | null | null | di-zhang-fdu/AOPS | false | true |
33 | math_hard | 8007ca84-3302-5d9e-98ad-a994205734ea | $\diamondsuit$ and $\Delta$ are whole numbers and $\diamondsuit \times \Delta =36$ . The largest possible value of $\diamondsuit + \Delta$ is
$\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 20\ \qquad \text{(E)}\ 37$ | E. 37.0 | null | null | di-zhang-fdu/AOPS | false | true |
34 | math_hard | 6c118559-3587-5453-9254-044f5767a665 | $\emph{triangular number}$ is a positive integer that can be expressed in the form $t_n = 1+2+3+\cdots+n$ , for some positive integer $n$ . The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$ $t_8 = 36 = 6^2$ , and $t_{49} = 1225 = 35^2$ . What is the sum of the digits of the fourth ... | D. 18.0 | null | null | di-zhang-fdu/AOPS | false | true |
35 | math_hard | a44affe1-93a4-5b52-9636-3806bcee8544 | $\frac{(3!)!}{3!}=$
$\text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 6\qquad\text{(D)}\ 40\qquad\text{(E)}\ 120$ | E. 120.0 | null | null | di-zhang-fdu/AOPS | false | true |
36 | math_hard | 6a58197c-6268-5baa-9ccd-91fb168f555c | $\frac{10^7}{5\times 10^4}=$
$\text{(A)}\ .002 \qquad \text{(B)}\ .2 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 200 \qquad \text{(E)}\ 2000$ | D. 200.0 | null | null | di-zhang-fdu/AOPS | false | true |
37 | math_hard | 14a3a332-7cb4-5432-a538-96c747fcb1a6 | $\frac{16+8}{4-2}=$
$\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20$ | C. 12.0 | null | null | di-zhang-fdu/AOPS | false | true |
38 | math_hard | 2b4d8b39-4f13-5131-8051-324656b82cee | $\frac{1}{10}+\frac{2}{20}+\frac{3}{30} =$
$\text{(A)}\ .1 \qquad \text{(B)}\ .123 \qquad \text{(C)}\ .2 \qquad \text{(D)}\ .3 \qquad \text{(E)}\ .6$ | D. 0.3 | null | null | di-zhang-fdu/AOPS | false | true |
39 | math_hard | 3a3737a8-8840-5667-bc10-9b97749b94bc | $\frac{2}{10}+\frac{4}{100}+\frac{6}{1000}=$
$\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$ | D. 0.246 | null | null | di-zhang-fdu/AOPS | false | true |
40 | math_hard | e3b32e6a-d1a0-55df-a391-520e31330eb1 | $\frac{2}{25}=$
$\text{(A)}\ .008 \qquad \text{(B)}\ .08 \qquad \text{(C)}\ .8 \qquad \text{(D)}\ 1.25 \qquad \text{(E)}\ 12.5$ | B. 0.08 | null | null | di-zhang-fdu/AOPS | false | true |
41 | math_hard | c64fcf56-01df-5b74-aa87-270814721e28 | $\left(\frac{(x+1)^{2}(x^{2}-x+1)^{2}}{(x^{3}+1)^{2}}\right)^{2}\cdot\left(\frac{(x-1)^{2}(x^{2}+x+1)^{2}}{(x^{3}-1)^{2}}\right)^{2}$ equals:
$\textbf{(A)}\ (x+1)^{4}\qquad\textbf{(B)}\ (x^{3}+1)^{4}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ [(x^{3}+1)(x^{3}-1)]^{2}$ $\textbf{(E)}\ [(x^{3}-1)^{2}]^{2}$ | C. 1.0 | null | null | di-zhang-fdu/AOPS | false | true |
42 | math_hard | a0c63e8d-899e-5844-993d-e423d0bf5208 | $\log p+\log q=\log(p+q)$ only if:
$\textbf{(A) \ }p=q=\text{zero} \qquad \textbf{(B) \ }p=\frac{q^2}{1-q} \qquad \textbf{(C) \ }p=q=1 \qquad$
$\textbf{(D) \ }p=\frac{q}{q-1} \qquad \textbf{(E) \ }p=\frac{q}{q+1}$ | D. 1.0 | null | null | di-zhang-fdu/AOPS | false | true |
43 | math_hard | c1e8442f-6bae-51cc-a371-1c102951c651 | $\text{palindrome}$ , such as $83438$ , is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$
$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 22 \qquad \textbf{(D)}\ 23 \qqu... | E. 24.0 | null | null | di-zhang-fdu/AOPS | false | true |
44 | math_hard | 8af45d76-cf54-5dea-8d50-d01ebfd974dd | $\triangle ABC$ has a right angle at $C$ and $\angle A = 20^\circ$ . If $BD$ $D$ in $\overline{AC}$ ) is the bisector of $\angle ABC$ , then $\angle BDC =$
$\textbf{(A)}\ 40^\circ \qquad \textbf{(B)}\ 45^\circ \qquad \textbf{(C)}\ 50^\circ \qquad \textbf{(D)}\ 55^\circ\qquad \textbf{(E)}\ 60^\circ$ | D. 55.0 | null | null | di-zhang-fdu/AOPS | false | true |
45 | math_hard | 08ecaa27-c6ce-50a7-8bf7-7b09b4e8b238 | $\triangle$ or $\bigcirc$ is placed in each of the nine squares in a $3$ -by- $3$ grid. Shown below is a sample configuration with three $\triangle$ s in a line. [asy] //diagram size(5cm); defaultpen(linewidth(1.5)); real r = 0.37; path equi = r * dir(-30) -- (r+0.03) * dir(90) -- r * dir(210) -- cycle; draw((0,0)--(0,... | D. 84.0 | null | null | di-zhang-fdu/AOPS | false | true |
46 | math_hard | fd63b51d-6040-596d-9c42-d328d6c6c0c4 | A "stair-step" figure is made of alternating black and white squares in each row. Rows $1$ through $4$ are shown. All rows begin and end with a white square. The number of black squares in the $37\text{th}$ row is
[asy] draw((0,0)--(7,0)--(7,1)--(0,1)--cycle); draw((1,0)--(6,0)--(6,2)--(1,2)--cycle); draw((2,0)--(5,0)-... | C. 36.0 | null | null | di-zhang-fdu/AOPS | false | true |
47 | math_hard | 08c17b0c-0394-5c94-a293-18d5b279e429 | A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the... | 52.0 | null | null | di-zhang-fdu/AOPS | false | true |
48 | math_hard | 84a67173-e317-51f9-893d-3e498a7025d4 | A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$ -coordinate or the $y$ -coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 \le x \le 2$ $-2 \le y \le 2$ at each step?
$\textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qqua... | D. 1698.0 | null | null | di-zhang-fdu/AOPS | false | true |
49 | math_hard | b2b3cfd9-e97a-5061-937e-30a6059d3697 | A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$ . What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?
$\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)} 96$ | D. 90.0 | null | null | di-zhang-fdu/AOPS | false | true |
50 | math_hard | 012b1bd6-1c10-5b82-93a5-9da5f4befcbc | A bag contains only blue balls and green balls. There are $6$ blue balls. If the probability of drawing a blue ball at random from this bag is $\frac{1}{4}$ , then the number of green balls in the bag is
$\text{(A)}\ 12 \qquad \text{(B)}\ 18 \qquad \text{(C)}\ 24 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 36$ | B. 18.0 | null | null | di-zhang-fdu/AOPS | false | true |
51 | math_hard | 4a5450d1-e4b8-5b4e-a28f-5a304f7e3b0d | A ball is dropped from a height of $3$ meters. On its first bounce it rises to a height of $2$ meters. It keeps falling and bouncing to $\frac{2}{3}$ of the height it reached in the previous bounce. On which bounce will it not rise to a height of $0.5$ meters?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C... | C. 5.0 | null | null | di-zhang-fdu/AOPS | false | true |
52 | math_hard | 89f98f65-8450-5a0e-a0aa-7adad53bb93c | A bank charges $\textdollar{6}$ for a loan of $\textdollar{120}$ . The borrower receives $\textdollar{114}$ and repays the loan in $12$ easy installments of $\textdollar{10}$ a month. The interest rate is approximately:
$\textbf{(A)}\ 5 \% \qquad \textbf{(B)}\ 6 \% \qquad \textbf{(C)}\ 7 \% \qquad \textbf{(D)}\ 9\% \qq... | A. 5.0 | null | null | di-zhang-fdu/AOPS | false | true |
53 | math_hard | f692649d-0d4c-5b2f-9e36-fd43b7ae7421 | A bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly $25\%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter?
[asy] size(250); void bargraph(... | A. 2.5 | null | null | di-zhang-fdu/AOPS | false | true |
54 | math_hard | 71ec20e9-0da1-52e3-9f44-69f53422dadf | A barn with a roof is rectangular in shape, $10$ yd. wide, $13$ yd. long and $5$ yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is:
$\mathrm{(A) \ } 360 \qquad \mathrm{(B) \ } 460 \qquad \mathrm{(C) \ } 490 \qquad \mathrm{(D... | D. 590.0 | null | null | di-zhang-fdu/AOPS | false | true |
55 | math_hard | b21eb127-cd58-53cb-b0df-3a8385d14829 | A base-10 three digit number $n$ is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of $n$ are both three-digit numerals?
$\mathrm{(A) \ } 0.3\qquad \mathrm{(B) \ } 0.4\qquad \mathrm{(C) \ } 0.5\qquad \mathrm{(D) \ } 0.6\qquad \mathr... | E. 0.7 | null | null | di-zhang-fdu/AOPS | false | true |
56 | math_hard | 1caef5a0-9966-558b-b21d-33d1d0f7df36 | A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$ . Each team plays a $76$ game schedule. How many games does a team play within its own division?
$\textbf{(A) } 36 \qquad \t... | B. 48.0 | null | null | di-zhang-fdu/AOPS | false | true |
57 | math_hard | e25571e2-98b2-56e1-938d-bd69d08365de | A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_n$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10} = .4$ and $a_n\le.4$ for all $n$ such that $1\le n\le9$ is given to be $p^aq^br/\left(s^c\right)$ where $p... | 660.0 | null | null | di-zhang-fdu/AOPS | false | true |
58 | math_hard | 87b8fc5c-0d86-5382-b6aa-51d7451181b2 | A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player?
$\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 3\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 5\qquad\mathrm{(E)}\ 6$ | E. 6.0 | null | null | di-zhang-fdu/AOPS | false | true |
59 | math_hard | 52bf4399-fe93-5b0b-8768-b95ab8a00015 | A basketball team's players were successful on 50% of their two-point shots and 40% of their three-point shots, which resulted in 54 points. They attempted 50% more two-point shots than three-point shots. How many three-point shots did they attempt?
$\textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textb... | C. 20.0 | null | null | di-zhang-fdu/AOPS | false | true |
60 | math_hard | 05210e1a-d9d2-5eb7-91f5-8d9eb9f68da5 | A beam of light strikes $\overline{BC}\,$ at point $C\,$ with angle of incidence $\alpha=19.94^\circ\,$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\overline{AB}\,$ and $\overline{BC}\,$ according to the rule: angle of incidence equals angle... | 71.0 | null | null | di-zhang-fdu/AOPS | false | true |
61 | math_hard | 6bf7cbdd-f0ea-59e9-ac45-abc09468c375 | A bee starts flying from point $P_0$ . She flies $1$ inch due east to point $P_1$ . For $j \ge 1$ , once the bee reaches point $P_j$ , she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$ . When the bee reaches $P_{2015}$ she is exactly $a \sqrt{b} + c \sqrt{d}$ inches away fr... | B. 2024.0 | null | null | di-zhang-fdu/AOPS | false | true |
62 | math_hard | aea48f8f-2007-50b5-bfcf-c98980d20a37 | A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$ . (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsui... | A. 109.0 | null | null | di-zhang-fdu/AOPS | false | true |
63 | math_hard | 95fba5b3-c49d-520c-ba2c-6684c9bbbb11 | A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 25% of these fish ... | 840.0 | null | null | di-zhang-fdu/AOPS | false | true |
64 | math_hard | d3e6bdd8-e8ac-53d2-bc4e-77899de6f896 | A block of cheese in the shape of a rectangular solid measures $10$ cm by $13$ cm by $14$ cm. Ten slices are cut from the cheese. Each slice has a width of $1$ cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic... | 729.0 | null | null | di-zhang-fdu/AOPS | false | true |
65 | math_hard | 96811a09-912b-5ad4-904f-bebeb6fe83e9 | A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$ , and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\circ$ . The block is then sliced in half a... | 53.0 | null | null | di-zhang-fdu/AOPS | false | true |
66 | math_hard | 58a600ca-0d10-5f18-aee4-a5757e4f0011 | A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wa... | D. 353.0 | null | null | di-zhang-fdu/AOPS | false | true |
67 | math_hard | a135973b-7bfb-5646-8f44-eafa2fd357d8 | A board game spinner is divided into three regions labeled $A$ $B$ and $C$ . The probability of the arrow stopping on region $A$ is $\frac{1}{3}$ and on region $B$ is $\frac{1}{2}$ . The probability of the arrow stopping on region $C$ is:
$\text{(A)}\ \frac{1}{12}\qquad\text{(B)}\ \frac{1}{6}\qquad\text{(C)}\ \frac{1}{... | B. 16.0 | null | null | di-zhang-fdu/AOPS | false | true |
68 | math_hard | 9d77444c-1d6e-5140-9f6e-cc7ca66505f9 | A book that is to be recorded onto compact discs takes $412$ minutes to read aloud. Each disc can hold up to $56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?
$\mathrm{(A)}\ 50.2 ... | B. 51.5 | null | null | di-zhang-fdu/AOPS | false | true |
69 | math_hard | 627260f4-7af0-524e-9ad2-be689fd0314c | A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$ . He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he en... | 342.0 | null | null | di-zhang-fdu/AOPS | false | true |
70 | math_hard | 8ad4ef9c-1cf7-5cd3-90b0-80a264443669 | A bowl is formed by attaching four regular hexagons of side $1$ to a square of side $1$ . The edges of the adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl? [asy] import three; size(225); ... | B. 7.0 | null | null | di-zhang-fdu/AOPS | false | true |
71 | math_hard | 9c7ec036-cbeb-561b-a2fb-3e37b28d0b71 | A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$
$\textbf{(A)}\ 120\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 200\qquad\textbf{... | D. 240.0 | null | null | di-zhang-fdu/AOPS | false | true |
72 | math_hard | 71696a7e-9222-548c-9a55-76d9aa25d5bd | A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn?
$\textbf{(A) } 75 \qquad\textbf{(B) }... | B. 76.0 | null | null | di-zhang-fdu/AOPS | false | true |
73 | math_hard | e31da722-688a-5429-a8d4-14b238adc5cd | A box contains a collection of triangular and square tiles. There are $25$ tiles in the box, containing $84$ edges total. How many square tiles are there in the box?
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 11$ | D. 9.0 | null | null | di-zhang-fdu/AOPS | false | true |
74 | math_hard | 9463074b-a029-54ef-86dc-f393716184d2 | A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips, and at most one third the number of red chips. The number which are white or blue is at least $55$ . The minimum number of red chips is
$\textbf{(A) }24\qquad \textbf{(B) }33\qquad \textbf{(C... | 57.0 | null | null | di-zhang-fdu/AOPS | false | true |
75 | math_hard | 8bcfa93d-d731-5b9b-8800-c98c75d73e9a | A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven p... | A. 0.0 | null | null | di-zhang-fdu/AOPS | false | true |
76 | math_hard | 5099507d-eec1-5c05-8d68-86010fa50af7 | A boy buys oranges at $3$ for $10$ cents. He will sell them at $5$ for $20$ cents. In order to make a profit of $$1.00$ , he must sell:
$\textbf{(A)}\ 67 \text{ oranges} \qquad \textbf{(B)}\ 150 \text{ oranges} \qquad \textbf{(C)}\ 200\text{ oranges}\\ \textbf{(D)}\ \text{an infinite number of oranges}\qquad \textbf{(... | B. 150.0 | null | null | di-zhang-fdu/AOPS | false | true |
77 | math_hard | 60b1f578-dcd6-5ee8-9956-517f622be40c | A bug crawls along a number line, starting at $-2$ . It crawls to $-6$ , then turns around and crawls to $5$ . How many units does the bug crawl altogether?
$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15$ | E. 15.0 | null | null | di-zhang-fdu/AOPS | false | true |
78 | math_hard | f90617c2-cebc-5d10-93bb-46f11fd229bd | A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are... | 683.0 | null | null | di-zhang-fdu/AOPS | false | true |
79 | math_hard | 2ad9157e-8644-581f-8cd4-e7bda0eae543 | A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?
[asy] size(10cm); draw((0.0,0.0)--(1.0,1.732050807568... | E. 2400.0 | null | null | di-zhang-fdu/AOPS | false | true |
80 | math_hard | 12371aa3-007d-55e4-828c-46e06db82a36 | A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$ -axis or $y$ -axis. Let $A = (-3, 2)$ and $B = (3, -2)$ . Consider all possible paths of the bug from $A$ to $B$ of length at most $20$ . How many points with integer coordinates lie on at least one of these paths?
$\textbf{... | C. 195.0 | null | null | di-zhang-fdu/AOPS | false | true |
81 | math_hard | 4f1689cd-4e93-5142-9a07-0bdac36960e4 | A bug walks all day and sleeps all night. On the first day, it starts at point $O$ , faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the po... | 103.0 | null | null | di-zhang-fdu/AOPS | false | true |
82 | math_hard | 7f9b52a5-4490-591b-b494-459ef978fe41 | A burger at Ricky C's weighs $120$ grams, of which $30$ grams are filler.
What percent of the burger is not filler?
$\mathrm{(A)}\ 60\% \qquad\mathrm{(B)}\ 65\% \qquad\mathrm{(C)}\ 70\% \qquad\mathrm{(D)}\ 75\% \qquad\mathrm{(E)}\ 90\%$ | D. 75.0 | null | null | di-zhang-fdu/AOPS | false | true |
83 | math_hard | 395b5749-314e-5422-a70e-224e80d5c84b | A bus takes $2$ minutes to drive from one stop to the next, and waits $1$ minute at each stop to let passengers board. Zia takes $5$ minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwi... | A. 17.0 | null | null | di-zhang-fdu/AOPS | false | true |
84 | math_hard | 39208e5e-3340-5169-a475-9b7304e8bade | A can of soup can feed $3$ adults or $5$ children. If there are $5$ cans of soup and $15$ children are fed, then how many adults would the remaining soup feed?
$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$ | B. 6.0 | null | null | di-zhang-fdu/AOPS | false | true |
85 | math_hard | 741bbfd7-cce3-5b34-99f3-9a2a42a85379 | A car travels $120$ miles from $A$ to $B$ at $30$ miles per hour but returns the same distance at $40$ miles per hour. The average speed for the round trip is closest to:
$\textbf{(A)}\ 33\text{ mph}\qquad\textbf{(B)}\ 34\text{ mph}\qquad\textbf{(C)}\ 35\text{ mph}\qquad\textbf{(D)}\ 36\text{ mph}\qquad\textbf{(E)}\ 37... | B. 34.0 | null | null | di-zhang-fdu/AOPS | false | true |
86 | math_hard | 2d0f07ac-9773-5ebd-9ed7-62beee9b0157 | A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0$ , the center of the storm is $110$ miles due north of the car. At time $t=t_1$ minutes, the car enters the s... | 198.0 | null | null | di-zhang-fdu/AOPS | false | true |
87 | math_hard | 77e713d9-5265-583f-b86a-0b6e1eb96d79 | A cart rolls down a hill, travelling $5$ inches the first second and accelerating so that during each successive $1$ -second time interval, it travels $7$ inches more than during the previous $1$ -second interval. The cart takes $30$ seconds to reach the bottom of the hill. How far, in inches, does it travel?
$\textbf{... | D. 3195.0 | null | null | di-zhang-fdu/AOPS | false | true |
88 | math_hard | 792e5632-f0fe-5a62-b144-1663d9a4aeae | A cell phone plan costs $20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay?
$\textbf{(A)}\ 24.00 \qquad \textbf{(B)}\ 24.50 \qquad \textbf{(C)}\ 25.50 ... | D. 28.0 | null | null | di-zhang-fdu/AOPS | false | true |
89 | math_hard | c1f4f07a-f460-5f19-8534-5157958c5098 | A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," w... | B. 9.0 | null | null | di-zhang-fdu/AOPS | false | true |
90 | math_hard | 9e0009ea-2a6b-58cd-bed6-e1585dc710c0 | A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$ , and that $f(x) = 1-|x-2|$ for $1\le x \le 3$ . Find the smallest $x$ for which $f(x) = f(2001)$ | 429.0 | null | null | di-zhang-fdu/AOPS | false | true |
91 | math_hard | 95459808-4973-5788-80a0-6778b418bca4 | A checkerboard consists of one-inch squares. A square card, $1.5$ inches on a side, is placed on the board so that it covers part or all of the area of each of $n$ squares. The maximum possible value of $n$ is
$\text{(A)}\ 4\text{ or }5 \qquad \text{(B)}\ 6\text{ or }7\qquad \text{(C)}\ 8\text{ or }9 \qquad \text{(D)... | E. 12.0 | null | null | di-zhang-fdu/AOPS | false | true |
92 | math_hard | 0dc1fdf5-3b6e-57d9-b893-5c2a66fe5d0b | A checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $1,2,\ldots,17$ , the second row $18,19,\ldots,34$ , and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1,2,\ldots,13,$ , ... | D. 555.0 | null | null | di-zhang-fdu/AOPS | false | true |
93 | math_hard | 5bea085f-604c-5443-95b8-4a6022573278 | A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a $3$ $3$ grid attacks all $8$ other squares, as shown below. Suppose a white king and a black king are placed on different squares of a $3$ $3$ grid so that they d... | E. 32.0 | null | null | di-zhang-fdu/AOPS | false | true |
94 | math_hard | 3645f04b-6786-58d6-bd0e-7391cb1e9f29 | A child builds towers using identically shaped cubes of different colors. How many different towers with a height $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)
$\textbf{(A) } 24 \qquad\textbf{(B) } 288 \qquad\textbf{(C) } 312 \qquad\textbf{(D) } 1,26... | D. 1260.0 | null | null | di-zhang-fdu/AOPS | false | true |
95 | math_hard | 19ff9fba-e335-5ed1-957e-5ef7f365817e | A child has a set of $96$ distinct blocks. Each block is one of $2$ materials (plastic, wood), $3$ sizes (small, medium, large), $4$ colors (blue, green, red, yellow), and $4$ shapes (circle, hexagon, square, triangle). How many blocks in the set differ from the 'plastic medium red circle' in exactly $2$ ways? (The 'wo... | 29.0 | null | null | di-zhang-fdu/AOPS | false | true |
96 | math_hard | 713bf123-3385-5da5-95bd-80135642d2e2 | A child's wading pool contains 200 gallons of water. If water evaporates at the rate of 0.5 gallons per day and no other water is added or removed, how many gallons of water will be in the pool after 30 days?
$\text{(A)}\ 140 \qquad \text{(B)}\ 170 \qquad \text{(C)}\ 185 \qquad \text{(D)}\ 198.5 \qquad \text{(E)}\ 199... | C. 185.0 | null | null | di-zhang-fdu/AOPS | false | true |
97 | math_hard | 263da4d2-fbd0-5bc8-93ae-fbda37a1b772 | A circle centered at $A$ with a radius of 1 and a circle centered at $B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. What is the radius of the third circle?
[asy] unitsize(0.75cm); pair A=(0,1), B=(4,4); dot(A); dot(B); dra... | 49.0 | null | null | di-zhang-fdu/AOPS | false | true |
98 | math_hard | 2acf0ab3-cb42-5950-9172-3fcb150252fd | A circle has a chord of length $10$ , and the distance from the center of the circle to the chord is $5$ . What is the area of the circle?
$\textbf{(A) }25\pi \qquad \textbf{(B) }50\pi \qquad \textbf{(C) }75\pi \qquad \textbf{(D) }100\pi \qquad \textbf{(E) }125\pi \qquad$ | B. 50.0 | null | null | di-zhang-fdu/AOPS | false | true |
99 | math_hard | dbded014-32ae-596d-af86-7b56776e802e | A circle has center $(-10, -4)$ and has radius $13$ . Another circle has center $(3, 9)$ and radius $\sqrt{65}$ . The line passing through the two points of intersection of the two circles has equation $x+y=c$ . What is $c$
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 3\sqrt{3}\qquad\textbf{(C)}\ 4\sqrt{2}\qquad\textbf{(D)}\ 6\... | A. 3.0 | null | null | di-zhang-fdu/AOPS | false | true |
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