id
int64 0
10k
| old_math_orz_id
int64 1
56.9k
| input
stringlengths 20
3.72k
| target
stringlengths 1
18
| qwen2.5-3b-instruct-pass-at-10
int64 0
10
|
|---|---|---|---|---|
0
| 47,080
|
Find all positive integers \( k \) such that the polynomial \( x^{2k+1} + x + 1 \) is divisible by \( x^k + x + 1 \). For each \( k \) that satisfies this condition, find the positive integers \( n \) such that \( x^n + x + 1 \) is divisible by \( x^k + x + 1 \).
(British Mathematical Olympiad, 1991)
|
2
| 0
|
1
| 52,076
|
All natural numbers whose digit sum is equal to 5 are arranged in ascending order. Which number is in the 125th position?
|
41000
| 0
|
2
| 25,966
|
In a rectangle $ABCD,$ where $AB=15$ and diagonal $AC=17,$ find the area and the perimeter of rectangle $ABCD.$
[asy]
draw((0,0)--(15,0)--(15,8)--(0,8)--cycle,black+linewidth(1));
draw((0,8)--(15,0),black+linewidth(1));
label("$A$",(0,8),NW);
label("$B$",(15,8),NE);
label("$C$",(15,0),SE);
label("$D$",(0,0),SW);
label("15",(0,8)--(15,8),N);
label("17",(0,8)--(15,0),SW);
[/asy]
|
46
| 0
|
3
| 49,284
|
An ant crawls along the edges of a cube with side length 1 unit. Starting from one of the vertices, in each minute the ant travels from one vertex to an adjacent vertex. After crawling for 7 minutes, the ant is at a distance of \(\sqrt{3}\) units from the starting point. Find the number of possible routes the ant has taken.
|
546
| 0
|
4
| 11,822
|
What is the perimeter of pentagon $ABCDE$ in this diagram? [asy]
pair cis(real r,real t) { return (r*cos(t),r*sin(t)); }
pair a=(0,0);
pair b=cis(1,-pi/2);
pair c=cis(sqrt(2),-pi/4);
pair d=cis(sqrt(3),-pi/4+atan(1/sqrt(2)));
pair e=cis(2,-pi/4+atan(1/sqrt(2))+atan(1/sqrt(3)));
dot(a); dot(b); dot(c); dot(d); dot(e);
draw(a--b--c--d--e--a);
draw(a--c); draw(a--d);
draw(0.86*b--0.86*b+0.14*(c-b)--b+0.14*(c-b));
draw(0.9*c--0.9*c+0.14*(d-c)--c+0.14*(d-c));
draw(0.92*d--0.92*d+0.14*(e-d)--d+0.14*(e-d));
label("$A$",a,NW);
label("$B$",b,SW);
label("$C$",c,SSE);
label("$D$",d,ESE);
label("$E$",e,NE);
label("1",(a+b)/2,W);
label("1",(b+c)/2,S);
label("1",(c+d)/2,SE);
label("1",(d+e)/2,E);
[/asy]
|
6
| 0
|
5
| 80
|
For any positive integer $a, \sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$.
|
125
| 0
|
6
| 1,662
|
Let $M$ be a set of six distinct positive integers whose sum is $60$ . These numbers are written on the faces of a cube, one number to each face. A *move* consists of choosing three faces of the cube that share a common vertex and adding $1$ to the numbers on those faces. Determine the number of sets $M$ for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
|
84
| 0
|
7
| 48,681
|
For positive integer \( n \), let \( f(n) \) denote the unit digit of \( 1+2+3+\cdots+n \). Find the value of \( f(1)+f(2)+\cdots+f(2011) \).
|
7046
| 0
|
8
| 20,923
|
Suppose we have 12 dogs and we want to divide them into three groups, one with 4 dogs, one with 5 dogs, and one with 3 dogs. How many ways can we form the groups such that Fluffy is in the 4-dog group and Nipper is in the 5-dog group?
|
4200
| 0
|
9
| 25,636
|
Find the number of integers $n$ that satisfy
\[30 < n^2 < 200.\]
|
18
| 0
|
10
| 53,702
|
Given that among any 3 out of $n$ people, at least 2 know each other, if there are always 4 people who all know each other, find the minimum value of $n$.
|
9
| 0
|
11
| 1,383
|
At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61{,}629{,}875}$ and $n = \sqrt[7]{170{,}859{,}375}$ . Given that $m$ and $n$ are both integers, compute $100m+n$ .
*Proposed by Evan Chen*
|
39515
| 0
|
12
| 10,939
|
An $8$-cm-by-$8$-cm square is partitioned as shown. Points $A$ and $B$ are the midpoints of two opposite sides of the square. What is the area of the shaded region?
[asy]
draw((0,0)--(10,0));
draw((10,0)--(10,10));
draw((10,10)--(0,10));
draw((0,0)--(0,10));
draw((0,0)--(5,10));
draw((5,10)--(10,0));
draw((0,10)--(5,0));
draw((5,0)--(10,10));
fill((5,0)--(7.5,5)--(5,10)--(2.5,5)--cycle,gray);
label("A",(5,10),N);
label("B",(5,0),S);
[/asy]
|
16
| 0
|
13
| 45,481
|
A state issues car license plates consisting of 6 digits (each digit ranging from $0$ to $9$), with the condition that any two license plate numbers must differ in at least two places. (For example, license numbers 027592 and 020592 cannot both be used). Determine the maximum number of license plate numbers possible under this condition. Provide a proof.
|
100000
| 0
|
14
| 2,071
|
Let $\mathcal{P}$ be the set of all polynomials $p(x)=x^4+2x^2+mx+n$ , where $m$ and $n$ range over the positive reals. There exists a unique $p(x) \in \mathcal{P}$ such that $p(x)$ has a real root, $m$ is minimized, and $p(1)=99$ . Find $n$ .
*Proposed by **AOPS12142015***
|
56
| 0
|
15
| 40,441
|
Find the largest constant $n$, such that for any positive real numbers $a,$ $b,$ $c,$ $d,$ and $e,$
\[\sqrt{\frac{a}{b + c + d + e}} + \sqrt{\frac{b}{a + c + d + e}} + \sqrt{\frac{c}{a + b + d + e}} + \sqrt{\frac{d}{a + b + c + e}} + \sqrt{\frac{e}{a + b + c + d}} > n.\]
|
2
| 0
|
16
| 34,851
|
Let $a_1$, $a_2$, $a_3$, $d_1$, $d_2$, and $d_3$ be real numbers such that for every real number $x$, the equation
\[
x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 = (x^2 + a_1 x + d_1)(x^2 + a_2 x + d_2)(x^2 + a_3 x + d_3)
\]
holds. Compute $a_1 d_1 + a_2 d_2 + a_3 d_3$.
|
1
| 0
|
17
| 28,780
|
Suppose we want to divide 12 dogs into three groups, one with 4 dogs, one with 5 dogs, and one with 3 dogs. How many ways can we form the groups such that Fluffy is in the 4-dog group and Nipper is in the 5-dog group?
|
4200
| 0
|
18
| 42,777
|
Let $b_n$ be the integer obtained by writing all the integers from 1 to $n$ from left to right. For example, $b_4 = 1234$ and $b_{13} = 12345678910111213$. Compute the remainder when $b_{21}$ is divided by $12$.
|
9
| 0
|
19
| 4,489
|
Find the smallest integer $n$ such that each subset of $\{1,2,\ldots, 2004\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$ .
|
337
| 0
|
20
| 48,537
|
Specify the smallest number that ends in 37, has a digit sum of 37, and is divisible by 37.
|
99937
| 0
|
21
| 24,184
|
Let $g$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[3g(a^2 + b^2) = [g(a)]^2 + 2[g(b)]^2\] for all nonnegative integers $a$ and $b.$
Let $n$ be the number of possible values of $g(25),$ and let $s$ be the sum of the possible values of $g(25).$ Find $n \times s.$
|
2
| 0
|
22
| 675
|
Find the number of pairs $(a,b)$ of natural nunbers such that $b$ is a 3-digit number, $a+1$ divides $b-1$ and $b$ divides $a^{2} + a + 2$ .
|
16
| 0
|
23
| 1,139
|
The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$ . The students are initially divided up into $20$ groups of $101$ , with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as $\frac{a}{b}$ , where $a, b$ are relatively prime integers, determine $a + b$ .
|
2123
| 0
|
24
| 572
|
A positive integer $n$ is called $\textit{good}$ if $2 \mid \tau(n)$ and if its divisors are $$ 1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n, $$ then $d_{k+1}-d_k=2$ and $d_{k+2}-d_{k-1}=65$ . Find the smallest $\textit{good}$ number.
|
2024
| 0
|
25
| 4,390
|
A semicircle is joined to the side of a triangle, with the common edge removed. Sixteen points are arranged on the figure, as shown below. How many non-degenerate triangles can be drawn from the given points?
[asy]
draw((0,-2)--arc((0,0),1,0,180)--cycle);
dot((-0.8775,-0.245));
dot((-0.735,-0.53));
dot((-0.5305,-0.939));
dot((-0.3875,-1.225));
dot((-0.2365,-1.527));
dot((0.155,-1.69));
dot((0.306,-1.388));
dot((0.4,-1.2));
dot((0.551,-0.898));
dot((0.837,-0.326));
dot(dir(25));
dot(dir(50));
dot(dir(65));
dot(dir(100));
dot(dir(115));
dot(dir(140));
[/asy]
|
540
| 0
|
26
| 36,404
|
The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \frac {b_n + 4030} {1 + b_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $b_1 + b_2$.
|
127
| 0
|
27
| 2,963
|
Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $ n < 60$ (both Kelly and Jason know that $ n < 60$ , but that they don't know what the value of $ n$ is). The computer tells Kelly the unit digit of $ n$ , and it tells Jason the number of divisors of $ n$ . Then, Kelly and Jason have the following dialogue:
Kelly: I don't know what $ n$ is, and I'm sure that you don't know either. However, I know that $ n$ is divisible by at least two different primes.
Jason: Oh, then I know what the value of $ n$ is.
Kelly: Now I also know what $ n$ is.
Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of $ n$ ?
|
10
| 0
|
28
| 43,572
|
A solid rectangular block is created by gluing together \(N\) 1-cm cube units. When this block is situated such that three faces are visible, \(462\) of the 1-cm cubes cannot be seen. Determine the smallest possible value of \(N\).
|
672
| 0
|
29
| 44,001
|
In an exam with 3 questions, four friends checked their answers after the test and found that they got 3, 2, 1, and 0 questions right, respectively. When the teacher asked how they performed, each of them made 3 statements as follows:
Friend A: I got two questions correct, and I did better than B, C scored less than D.
Friend B: I got all questions right, C got them all wrong, and A did worse than D.
Friend C: I got one question correct, D got two questions right, B did worse than A.
Friend D: I got all questions right, C did worse than me, A did worse than B.
If each person tells as many true statements as the number of questions they got right, let \(A, B, C, D\) represent the number of questions each of A, B, C, and D got right, respectively. Find the four-digit number \(\overline{\mathrm{ABCD}}\).
|
1203
| 0
|
30
| 16,486
|
The student locker numbers at Liberty High are numbered consecutively starting from locker number $1$. Each digit costs three cents. If it costs $206.91 to label all the lockers, how many lockers are there?
|
2001
| 0
|
31
| 53,394
|
In a competition consisting of $n$ true/false questions, 8 participants are involved. It is known that for any ordered pair of true/false questions $(A, B)$, there are exactly two participants whose answers are (true, true); exactly two participants whose answers are (true, false); exactly two participants whose answers are (false, true); and exactly two participants whose answers are (false, false). Find the maximum value of $n$ and explain the reasoning.
|
7
| 0
|
32
| 13,775
|
Given the weights of boxes in pounds with three different combinations, $135$, $139$, $142$, and $145$, determine the combined weight of the four boxes.
|
187
| 0
|
33
| 5,990
|
A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the substitutes. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let $n$ be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when $n$ is divided by $1000$.
|
122
| 0
|
34
| 44,901
|
8. Andrey likes all numbers that are not divisible by 3, and Tanya likes all numbers that do not contain digits that are divisible by 3.
a) How many four-digit numbers are liked by both Andrey and Tanya?
b) Find the total sum of the digits of all such four-digit numbers.
|
14580
| 0
|
35
| 19,183
|
Let $\mathcal{S}$ be the set $\lbrace1,2,3,\ldots,12\rbrace$. Let $n$ be the number of ordered pairs of two non-empty disjoint subsets of $\mathcal{S}$. Find the remainder obtained when $n$ is divided by $1000$.
|
250
| 0
|
36
| 2,304
|
Points $C\neq D$ lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$ , $BC=AD=10$ , and $CA=DB=17$ . The intersection of these two triangular regions has area $\tfrac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .
|
59
| 0
|
37
| 2,898
|
Let $f(n)$ denote the largest odd factor of $n$ , including possibly $n$ . Determine the value of
\[\frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048},\]
rounded to the nearest integer.
|
1365
| 0
|
38
| 48,963
|
All digits in the 6-digit natural numbers $a$ and $b$ are even, and any number between them contains at least one odd digit. Find the largest possible value of the difference $b-a$.
|
111112
| 0
|
39
| 50,691
|
Let \( x \) be a non-zero real number such that
\[ \sqrt[5]{x^{3}+20 x}=\sqrt[3]{x^{5}-20 x} \].
Find the product of all possible values of \( x \).
|
-5
| 0
|
40
| 15,669
|
Given that vertices P, Q, R, and S of a quadrilateral have coordinates (a, a), (a, -a), (-a, -a), and (-a, a), and the area of the quadrilateral PQRS is 36, calculate the value of a + b.
|
6
| 0
|
41
| 5,972
|
Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
|
200
| 0
|
42
| 4,138
|
Consider digits $\underline{A}, \underline{B}, \underline{C}, \underline{D}$ , with $\underline{A} \neq 0,$ such that $\underline{A} \underline{B} \underline{C} \underline{D} = (\underline{C} \underline{D} ) ^2 - (\underline{A} \underline{B})^2.$ Compute the sum of all distinct possible values of $\underline{A} + \underline{B} + \underline{C} + \underline{D}$ .
*Proposed by Kyle Lee*
|
21
| 0
|
43
| 22,623
|
Find the number of triples $(x,y,z)$ of real numbers that satisfy
\begin{align*}
x &= 2023 - 2024 \operatorname{sign}(y + z), \\
y &= 2023 - 2024 \operatorname{sign}(x + z), \\
z &= 2023 - 2024 \operatorname{sign}(x + y).
\end{align*}
|
3
| 0
|
44
| 2,499
|
Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob?
*Author: Ray Li*
|
2
| 0
|
45
| 44,859
|
Given \( n = p \cdot q \cdot r \cdot s \), where \( p, q, r, s \) are distinct primes such that:
1. \( s = p + r \)
2. \( p(p + q + r + s) = r(s - q) \)
3. \( qs = 1 + qr + s \)
Find \( n \).
|
2002
| 0
|
46
| 52,822
|
Let \( a < b < c < d < e \) be real numbers. Among the 10 sums of the pairs of these numbers, the least three are 32, 36, and 37, while the largest two are 48 and 51. Find all possible values of \( e \).
|
27.5
| 0
|
47
| 49,297
|
The product of the digits of any multi-digit number is always less than this number. If we calculate the product of the digits of a given multi-digit number, then the product of the digits of this product, and so on, we will necessarily reach a single-digit number after some number of steps. This number of steps is called the persistence of the number. For example, the number 723 has a persistence of 2 because $7 \cdot 2 \cdot 3 = 42$ (1st step) and $4 \cdot 2 = 8$ (2nd step).
1. Find the largest odd number with distinct digits that has a persistence of 1.
2. Find the largest even number with distinct nonzero digits that has a persistence of 1.
3. Find the smallest natural number that has a persistence of 3.
|
39
| 0
|
48
| 41,747
|
Determine the value of $m$ modulo 9, where $0 \leq m < 9$, for the sum $$2+33+444+5555+66666+777777+8888888+99999999.$$
|
6
| 0
|
49
| 18,829
|
Find the sum of all integral values of \( c \) with \( c \leq 30 \) for which the equation \( y = x^2 - 9x - c \) has two rational roots.
|
-28
| 0
|
50
| 23,681
|
How many of the divisors of \(10!\) are larger than \(9!\)?
|
9
| 0
|
51
| 38,745
|
Cindy now wishes to arrange her coins into $X$ piles, each containing the same number of coins $Y$. Just as before, each pile will have more than one coin and no pile will have all the coins. If there are 19 possible values for $Y$ given all of the restrictions, what is the smallest number of coins Cindy could have now?
|
576
| 0
|
52
| 32,710
|
Let $x$ and $y$ be integers such that $xy = 144.$ Find the minimum value of $x + y.$
|
-145
| 0
|
53
| 4,548
|
The number $734{,}851{,}474{,}594{,}578{,}436{,}096$ is equal to $n^6$ for some positive integer $n$ . What is the value of $n$ ?
|
3004
| 0
|
54
| 50,340
|
$[a]$ denotes the greatest integer less than or equal to $a$. Given that $\left(\left[\frac{1}{7}\right]+1\right) \times\left(\left[\frac{2}{7}\right]+1\right) \times\left(\left[\frac{3}{7}\right]+1\right) \times \cdots \times$ $\left(\left[\frac{\mathrm{k}}{7}\right]+1\right)$ leaves a remainder of 7 when divided by 13, find the largest positive integer $k$ not exceeding 48.
|
45
| 0
|
55
| 51,368
|
A function \( f \) is defined on the positive integers by: \( f(1) = 1 \); \( f(3) = 3 \); \( f(2n) = f(n) \), \( f(4n + 1) = 2f(2n + 1) - f(n) \), and \( f(4n + 3) = 3f(2n + 1) - 2f(n) \) for all positive integers \( n \). Determine the number of positive integers \( n \) less than or equal to 1988 for which \( f(n) = n \).
|
92
| 0
|
56
| 26,258
|
How many numbers in the sequence $\{7, 17, 27, 37, \ldots\}$ up to 100 can be written as the difference of two prime numbers?
|
5
| 0
|
57
| 50,924
|
Given the sequence $\left\{a_{n}\right\}$ that satisfies $a_{1}=p, a_{2}=p+1, a_{n+2}-2 a_{n+1}+a_{n}=n-20$, where $p$ is a given real number and $n$ is a positive integer, find the value of $n$ that makes $a_{n}$ minimal.
|
40
| 0
|
58
| 44,982
|
Let the set \( M = \{1, 2, \cdots, 12\} \). A three-element subset \( A = \{a, b, c\} \) satisfies \( A \subset M \) and \( a + b + c \) is a perfect square. Determine the number of such sets \( A \).
|
26
| 0
|
59
| 14,989
|
The real number \(x\) satisfies the equation \(x + \frac{1}{x} = \sqrt{3}\). Evaluate the expression \(x^{7} - 5x^{5} + x^{2}\).
|
-1
| 0
|
60
| 2,261
|
Define the *hotel elevator cubic*as the unique cubic polynomial $P$ for which $P(11) = 11$ , $P(12) = 12$ , $P(13) = 14$ , $P(14) = 15$ . What is $P(15)$ ?
*Proposed by Evan Chen*
|
13
| 0
|
61
| 53,989
|
The sequence \( a_n \) is defined as follows:
\( a_1 = 2, a_{n+1} = a_n + \frac{2a_n}{n} \) for \( n \geq 1 \). Find \( a_{200} \).
|
40200
| 0
|
62
| 26,367
|
A math conference is hosting a series of lectures by seven distinct lecturers. Dr. Smith's lecture depends on Dr. Jones’s lecture, and additionally, Dr. Brown's lecture depends on Dr. Green’s lecture. How many valid orders can these seven lecturers be scheduled, given these dependencies?
|
1260
| 0
|
63
| 4,517
|
Let $p,q,r$ be distinct prime numbers and let
\[A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} \]
Find the least $n\in\mathbb{N}$ such that for any $B\subset A$ where $|B|=n$ , has elements $x$ and $y$ such that $x$ divides $y$ .
*Ioan Tomescu*
|
28
| 0
|
64
| 14,742
|
Two pictures, each 2 feet across, are hung in the center of a wall that is 25 feet wide with 1 foot of space between them. Calculate the distance from the end of the wall to the nearest edge of the first picture.
|
10
| 0
|
65
| 38,869
|
Find the phase shift, amplitude, and vertical shift of the graph of \( y = 2\sin(4x - \frac{\pi}{2}) + 1 \).
|
1
| 0
|
66
| 25,691
|
How many non-empty subsets $T$ of $\{1,2,3,\ldots,17\}$ have the following two properties?
$(1)$ No two consecutive integers belong to $T$.
$(2)$ If $T$ contains $k$ elements, then $T$ contains no number less than $k+1$.
|
594
| 0
|
67
| 6,088
|
Let $N$ denote the number of permutations of the $15$-character string $AAAABBBBBCCCCCC$ such that
None of the first four letters is an $A$.
None of the next five letters is a $B$.
None of the last six letters is a $C$.
Find the remainder when $N$ is divided by $1000$.
|
320
| 0
|
68
| 54,030
|
Swap the digit in the hundreds place with the digit in the units place of a three-digit number while keeping the digit in the tens place unchanged. The new number obtained is equal to the original number. How many such numbers are there? How many of these numbers are divisible by 4?
|
20
| 0
|
69
| 14,357
|
The number of eighth graders and fifth graders who bought a pencil can be represented as variables x and y, respectively. Given that some eighth graders each bought a pencil and paid a total of $2.34 dollars, and some of the $50$ fifth graders each bought a pencil and paid a total of $3.25 dollars, determine the difference in the number of fifth graders and eighth graders who bought a pencil.
|
7
| 0
|
70
| 18,177
|
Given that the larger rectangular fort is $15$ feet long, $12$ feet wide, and $6$ feet high, with the floor, ceillings, and all four walls made with blocks that are one foot thick, determine the total number of blocks used to build the entire fort.
|
560
| 0
|
71
| 35,960
|
A deck of fifty-two cards consists of four $1$'s, four $2$'s,..., up to four $13$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards from the remaining deck also form a pair, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
|
1298
| 0
|
72
| 4,607
|
The convex quadrilateral $ABCD$ has area $1$ , and $AB$ is produced to $E$ , $BC$ to $F$ , $CD$ to $G$ and $DA$ to $H$ , such that $AB=BE$ , $BC=CF$ , $CD=DG$ and $DA=AH$ . Find the area of the quadrilateral $EFGH$ .
|
5
| 0
|
73
| 56,130
|
A visual artist is creating a painting that incorporates elements of light and shadow. The artist uses 12 different shades of gray to depict the shadows in the painting and 8 different shades of yellow to depict the light. If the artist decides to use 3 shades of gray for each shadowed area and 2 shades of yellow for each lit area, how many total areas of light and shadow can the artist create in the painting?
|
248
| 0
|
74
| 3,158
|
Let $ABC$ be a triangle with $\angle BAC = 90^\circ$ . Construct the square $BDEC$ such as $A$ and the square are at opposite sides of $BC$ . Let the angle bisector of $\angle BAC$ cut the sides $[BC]$ and $[DE]$ at $F$ and $G$ , respectively. If $|AB|=24$ and $|AC|=10$ , calculate the area of quadrilateral $BDGF$ .
|
338
| 0
|
75
| 50,698
|
Each cell in a \(5 \times 5\) grid contains a natural number written in invisible ink. It is known that the sum of all the numbers is 200, and the sum of any three numbers inside any \(1 \times 3\) rectangle is 23. What is the value of the central number in the grid?
|
16
| 0
|
76
| 50,738
|
Consider an equilateral triangle \(ABC\), where \(AB = BC = CA = 2011\). Let \(P\) be a point inside \(\triangle ABC\). Draw line segments passing through \(P\) such that \(DE \parallel BC\), \(FG \parallel CA\), and \(HI \parallel AB\). Suppose \(DE : FG : HI = 8 : 7 : 10\). Find \(DE + FG + HI\).
|
4022
| 0
|
77
| 45,582
|
How many numbers between 100 and 999 (inclusive) have digits that form an arithmetic progression when read from left to right?
A sequence of three numbers \( a, b, c \) is said to form an arithmetic progression if \( a + c = 2b \).
A correct numerical answer without justification will earn 4 points. For full points, a detailed reasoning is expected.
|
45
| 0
|
78
| 930
|
A four-digit positive integer is called *virtual* if it has the form $\overline{abab}$ , where $a$ and $b$ are digits and $a \neq 0$ . For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$ , for some positive integer $n$ .
|
8282
| 0
|
79
| 2,460
|
Find the sum of all positive integers whose largest proper divisor is $55$ . (A proper divisor of $n$ is a divisor that is strictly less than $n$ .)
|
550
| 0
|
80
| 55,145
|
Let \( ABCD - A_1B_1C_1D_1 \) be a cube with edge length 1. The endpoint \( M \) of the line segment \( MN \) is on the ray \( AA_1 \), and the point \( N \) is on the ray \( BC \). Furthermore, \( MN \) intersects the edge \( C_1D_1 \) at point \( L \). Determine the minimum length of \( MN \).
|
3
| 0
|
81
| 43,832
|
Rózsa, Ibolya, and Viola have decided to solve all the problems in their problem collection. Rózsa solves \(a\), Ibolya \(b\), and Viola \(c\) problems daily. (Only one of them works on a problem.) If Rózsa solved 11 times, Ibolya 7 times, and Viola 9 times more problems daily, they would finish in 5 days; whereas if Rózsa solved 4 times, Ibolya 2 times, and Viola 3 times more problems daily, they would finish in 16 days. How many days will it take for them to complete the solutions?
|
40
| 0
|
82
| 10,118
|
Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$ where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$.
|
108
| 0
|
83
| 48,876
|
Through the origin, lines (including the coordinate axes) are drawn that divide the coordinate plane into angles of $1^{\circ}$. Find the sum of the x-coordinates of the points of intersection of these lines with the line $y = 100 - x$.
|
8950
| 0
|
84
| 8,785
|
Let $x = \cos \frac{2 \pi}{7} + i \sin \frac{2 \pi}{7}.$ Compute the value of
\[(2x + x^2)(2x^2 + x^4)(2x^3 + x^6)(2x^4 + x^8)(2x^5 + x^{10})(2x^6 + x^{12}).\]
|
43
| 0
|
85
| 16,113
|
The sum of the first 1234 terms of the sequence $1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 3,\ldots$ can be found.
|
3604
| 0
|
86
| 18,979
|
Define an ordered triple $(D, E, F)$ of sets to be minimally intersecting if $|D \cap E| = |E \cap F| = |F \cap D| = 1$ and $D \cap E \cap F = \emptyset$. Let $M$ be the number of such ordered triples where each set is a subset of $\{1,2,3,4,5,6,7,8\}$. Find $M$ modulo $1000$.
|
064
| 0
|
87
| 26,883
|
Twelve 6-sided dice are rolled. What is the probability that exactly four of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.
|
0.089
| 0
|
88
| 30,683
|
Find the largest possible value of $k$ for which $3^{12}$ is expressible as the sum of $k$ consecutive positive integers.
|
729
| 0
|
89
| 3,098
|
Call a positive integer $n\geq 2$ *junk* if there exist two distinct $n$ digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that
- $a_1+a_2=b_1+b_2,$
- $a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}$ for all $2\leq i\leq n-1,$ and
- $a_{n-1}+a_n=b_{n-1}+b_n$ .
Find the number of junk positive integers less than or equal to $2016$ .
[i]Proposed by Nathan Ramesh
|
672
| 0
|
90
| 56,520
|
Emily is a statistician who loves using numbers to help her community. She is working on a project to measure the impact of a new community center in her town. At the start of the year, the town had 240 children attending various after-school programs. After the community center opened, 5 new programs were introduced, each attracting 18 children. Additionally, 12 children from the original programs switched to the new ones. How many children are now attending after-school programs in the town?
|
330
| 0
|
91
| 43,882
|
Let a three-digit number \( n = \overline{abc} \), where \( a \), \( b \), and \( c \) can form an isosceles (including equilateral) triangle as the lengths of its sides. How many such three-digit numbers \( n \) are there?
|
165
| 0
|
92
| 52,453
|
In 2014, during the Beijing APEC meeting, measures such as traffic restrictions and suspension of polluting industries were implemented in Beijing-Tianjin-Hebei to ensure good air quality. After experiencing three bouts of smog in a month, Beijing's air quality reached level one (excellent) on November 3rd, a phenomenon dubbed the "APEC Blue". In 2013, the number of days with good air quality in Beijing accounted for only 47.9\%. In the first half of 2014, with a 30\% reduction in emissions, the number of days with good air quality increased by 20 days compared to the same period in 2013. To achieve the target of increasing the number of days with good air quality by 20\% for the entire year, how many more days with good air quality are needed in the second half of the year compared to the same period in 2013?
|
15
| 0
|
93
| 46,175
|
In a sports complex, there is a rectangular area \(ABCD\) with the longer side \(AB\). The diagonals \(AC\) and \(BD\) intersect at an angle of \(60^\circ\). Runners train on the large circuit \(ACBDA\) or on the small track \(AD\). Mojmír ran ten times around the large circuit, and Vojta ran fifteen times around the small track, in one direction and then fifteen times in the opposite direction. Together they ran \(4.5 \text{ km}\). How long is the diagonal \(AC\)?
|
100
| 0
|
94
| 34,405
|
The function $g$ defined by $g(x)= \frac{px+q}{rx+s}$, where $p$, $q$, $r$, and $s$ are nonzero real numbers, has the properties $g(13)=13$, $g(31)=31$ and $g(g(x))=x$ for all values except $\frac{-s}{r}$. Find the unique number that is not in the range of $g$.
|
22
| 0
|
95
| 2,471
|
Let $ABC$ be a triangle such that $AB = 7$ , $BC = 8$ , and $CA = 9$ . There exists a unique point $X$ such that $XB = XC$ and $XA$ is tangent to the circumcircle of $ABC$ . If $XA = \tfrac ab$ , where $a$ and $b$ are coprime positive integers, find $a + b$ .
*Proposed by Alexander Wang*
|
61
| 0
|
96
| 50,178
|
A \(15 \times 15\) square is divided into \(1 \times 1\) small squares. From these small squares, several were chosen, and in each chosen square, one or two diagonals were drawn. It turned out that no two drawn diagonals have a common endpoint. What is the maximum number of diagonals that can be drawn? (In the solution, provide the answer, the method of drawing the diagonals, and proof that this number of diagonals is indeed the maximum possible.)
|
128
| 0
|
97
| 38,286
|
A notebook and its cover together cost $\$3.30$. The notebook costs \$2 more than its cover. If the displayed price includes a 10% sales tax, what is the pre-tax cost of the notebook?
|
2.5
| 0
|
98
| 54,043
|
A monkey becomes happy when it eats three different fruits. What is the maximum number of monkeys that can be made happy if there are 20 pears, 30 bananas, 40 peaches, and 50 mandarins? Justify your answer.
|
45
| 0
|
99
| 55,930
|
Mrs. Green is a librarian who loves to recommend the best book-to-film adaptations. This week, she decided to organize a special event at the library showcasing these adaptations. She has a collection of 25 different book-to-film adaptations. She plans to display 5 adaptations each day over the course of several days.
However, she realizes that some adaptations are more popular than others, so she decides to display 3 of the most popular adaptations every day, and rotate the remaining adaptations throughout the event. If she wants to ensure that each adaptation is displayed at least once, how many days will the event need to last?
|
11
| 0
|
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