Unnamed: 0
int64 0
56.9k
| problem
stringlengths 16
7.44k
| ground_truth
stringlengths 1
942
| solved_percentage
float64 0
100
|
|---|---|---|---|
1,900 |
Find the largest real number $a$ such that \[\left\{ \begin{array}{l}
x - 4y = 1
ax + 3y = 1
\end{array} \right.
\] has an integer solution.
|
1
| 92.96875 |
1,901 |
Find the maximum number of different integers that can be selected from the set $ \{1,2,...,2013\}$ so that no two exist that their difference equals to $17$ .
|
1010
| 0 |
1,902 |
The radius $r$ of a circle with center at the origin is an odd integer.
There is a point ( $p^m, q^n$ ) on the circle, with $p,q$ prime numbers and $m,n$ positive integers.
Determine $r$ .
|
5
| 57.03125 |
1,903 |
Let $ABC$ be an isosceles right triangle with $\angle A=90^o$ . Point $D$ is the midpoint of the side $[AC]$ , and point $E \in [AC]$ is so that $EC = 2AE$ . Calculate $\angle AEB + \angle ADB$ .
|
135^\circ
| 61.71875 |
1,904 |
There is a $2n\times 2n$ rectangular grid and a chair in each cell of the grid. Now, there are $2n^2$ pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating at $(3,3)$ and $(2,5)$ respectively, then the distance between them is $|3-2|+|3-5|=3$ . Moreover, define the total distance to be the sum of the distance in each pair. Find the maximal total distance among all possibilities.
|
4n^3
| 3.90625 |
1,905 |
The area of the region in the $xy$ -plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$ , for some integer $k$ . Find $k$ .
*Proposed by Michael Tang*
|
210
| 37.5 |
1,906 |
Let $f(x) = x-\tfrac1{x}$ , and defi ne $f^1(x) = f(x)$ and $f^n(x) = f(f^{n-1}(x))$ for $n\ge2$ . For each $n$ , there is a minimal degree $d_n$ such that there exist polynomials $p$ and $q$ with $f^n(x) = \tfrac{p(x)}{q(x)}$ and the degree of $q$ is equal to $d_n$ . Find $d_n$ .
|
d_n = 2^n - 1
| 0 |
1,907 |
A store offers packages of $12$ pens for $\$ 10 $ and packages of $ 20 $ pens for $ \ $15$ . Using only these two types of packages of pens, find the greatest number of pens $\$ 173$ can buy at this store.
*Proposed by James Lin*
|
224
| 41.40625 |
1,908 |
Find all positive integers $n$ such that the inequality $$ \left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i $$ holds for any $n$ positive numbers $a_1, \dots, a_n$ .
|
n = 3
| 0 |
1,909 |
Let $b$ be a real number randomly sepected from the interval $[-17,17]$ . Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation \[x^4+25b^2=(4b^2-10b)x^2\] has $\textit{at least}$ two distinct real solutions. Find the value of $m+n$ .
|
63
| 96.875 |
1,910 |
Points $E$ and $F$ are chosen on sides $BC$ and $CD$ respectively of rhombus $ABCD$ such that $AB=AE=AF=EF$ , and $FC,DF,BE,EC>0$ . Compute the measure of $\angle ABC$ .
|
80^\circ
| 0 |
1,911 |
Let $\alpha\geq 1$ be a real number. Define the set $$ A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\} $$ Suppose that all the positive integers that **does not belong** to the $A(\alpha)$ are exactly the positive integers that have the same remainder $r$ in the division by $2021$ with $0\leq r<2021$ . Determine all the possible values of $\alpha$ .
|
\frac{2021}{2020}
| 11.71875 |
1,912 |
Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R} $$ For $f \in \mathcal{F},$ let $$ I(f)=\int_0^ef(x) dx $$ Determine $\min_{f \in \mathcal{F}}I(f).$ *Liviu Vlaicu*
|
\frac{3}{2}
| 0.78125 |
1,913 |
What is the smallest number that can be written as a sum of $2$ squares in $3$ ways?
|
325
| 89.0625 |
1,914 |
Let ABC be a right triangle with $\angle B = 90^{\circ}$ .Let E and F be respectively the midpoints of AB and AC.Suppose the incentre I of ABC lies on the circumcircle of triangle AEF,find the ratio BC/AB.
|
\frac{4}{3}
| 3.125 |
1,915 |
$56$ lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly $594$ points, what is the maximum number of them that could have the same slope?
|
44
| 23.4375 |
1,916 |
A polyhedron has faces that all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces?
|
4:3
| 0 |
1,917 |
Let $S$ be the set of all integers $k$ , $1\leq k\leq n$ , such that $\gcd(k,n)=1$ . What is the arithmetic mean of the integers in $S$ ?
|
\frac{n}{2}
| 96.09375 |
1,918 |
Let $H$ be a rectangle with angle between two diagonal $\leq 45^{0}$ . Rotation $H$ around the its center with angle $0^{0}\leq x\leq 360^{0}$ we have rectangle $H_{x}$ . Find $x$ such that $[H\cap H_{x}]$ minimum, where $[S]$ is area of $S$ .
|
x = 45^\circ
| 0 |
1,919 |
Let $a$ , $b$ , $c$ be positive integers such that $abc + bc + c = 2014$ . Find the minimum possible value of $a + b + c$ .
|
40
| 46.09375 |
1,920 |
There are $n$ rooms in a sauna, each has unlimited capacity. No room may be attended by a female and a male simultaneously. Moreover, males want to share a room only with males that they don't know and females want to share a room only with females that they know. Find the biggest number $k$ such that any $k$ couples can visit the sauna at the same time, given that two males know each other if and only if their wives know each other.
|
n-1
| 0 |
1,921 |
Find the integer $n$ such that
\[n + \left\lfloor\sqrt{n}\right\rfloor + \left\lfloor\sqrt{\sqrt{n}}\right\rfloor = 2017.\] Here, as usual, $\lfloor\cdot\rfloor$ denotes the floor function.
|
1967
| 92.1875 |
1,922 |
In a trapezoid $ABCD$ , the internal bisector of angle $A$ intersects the base $BC$ (or its extension) at the point $E$ . Inscribed in the triangle $ABE$ is a circle touching the side $AB$ at $M$ and side $BE$ at the point $P$ . Find the angle $DAE$ in degrees, if $AB:MP=2$ .
|
60^\circ
| 12.5 |
1,923 |
For all pairs $(m, n)$ of positive integers that have the same number $k$ of divisors we define the operation $\circ$ . Write all their divisors in an ascending order: $1=m_1<\ldots<m_k=m$ , $1=n_1<\ldots<n_k=n$ and set $$ m\circ n= m_1\cdot n_1+\ldots+m_k\cdot n_k. $$ Find all pairs of numbers $(m, n)$ , $m\geqslant n$ , such that $m\circ n=497$ .
|
(18, 20)
| 0 |
1,924 |
Juca has decided to call all positive integers with 8 digits as $sextalternados$ if it is a multiple of 30 and its consecutive digits have different parity. At the same time, Carlos decided to classify all $sextalternados$ that are multiples of 12 as $super sextalternados$ .
a) Show that $super sextalternados$ numbers don't exist.
b) Find the smallest $sextalternado$ number.
|
10101030
| 3.90625 |
1,925 |
Find the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$ .
|
420
| 94.53125 |
1,926 |
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$ . Determine $f (2014)$ . $N_0=\{0,1,2,...\}$
|
671
| 69.53125 |
1,927 |
Find all $3$ -digit numbers $\overline{abc}$ ( $a,b \ne 0$ ) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$
|
627
| 96.09375 |
1,928 |
A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$ . What is the least number of dominoes that are entirely inside some square $2 \times 2$ ?
|
100
| 0 |
1,929 |
The side lengths $a,b,c$ of a triangle $ABC$ are positive integers. Let:
\[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\]
for any positive integer $n$ . If $\frac{T_{2}}{2T_{1}}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $ABC$ .
|
49
| 0 |
1,930 |
Find the integer $n \ge 48$ for which the number of trailing zeros in the decimal representation of $n!$ is exactly $n-48$ .
*Proposed by Kevin Sun*
|
62
| 100 |
1,931 |
The product of $10$ integers is $1024$ . What is the greatest possible sum of these $10$ integers?
|
1033
| 1.5625 |
1,932 |
Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$ ?
*Proposed by Giacomo Rizzo*
|
444
| 52.34375 |
1,933 |
Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$ . The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$ . Find the least number of tiles that Shenelle can have.
|
94
| 78.90625 |
1,934 |
Find the mathematical expectation of the area of the projection of a cube with edge of length $1$ onto a plane with an isotropically distributed random direction of projection.
|
\frac{3}{2}
| 17.1875 |
1,935 |
Find $k$ where $2^k$ is the largest power of $2$ that divides the product \[2008\cdot 2009\cdot 2010\cdots 4014.\]
|
2007
| 99.21875 |
1,936 |
For a positive integer $n$ , let $\omega(n)$ denote the number of positive prime divisors of $n$ . Find the smallest positive tinteger $k$ such that $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$ .
|
5
| 32.03125 |
1,937 |
Each side of a triangle is extended in the same clockwise direction by the length of the given side as shown in the figure. How many times the area of the triangle, obtained by connecting the endpoints, is the area of the original triangle?
|
7
| 35.15625 |
1,938 |
Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$ .
|
509
| 87.5 |
1,939 |
Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer.
*Note: A monic polynomial has a leading coefficient equal to 1.*
*(Greece - Panagiotis Lolas and Silouanos Brazitikos)*
|
f(x) = x - 3
| 2.34375 |
1,940 |
Let $ABC$ be a triangle with sides $51, 52, 53$ . Let $\Omega$ denote the incircle of $\bigtriangleup ABC$ . Draw tangents to $\Omega$ which are parallel to the sides of $ABC$ . Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed $r_1 + r_2 + r_3$ .
|
15
| 28.90625 |
1,941 |
Find the derivative of the solution of the equation $\ddot{x} = \dot{x}^2 + x^3$ with initial condition $x(0) = 0$ , $\dot{x}(0) = A$ with respect to $A$ for $A = 0$ .
|
t
| 47.65625 |
1,942 |
In a certain tournament bracket, a player must be defeated three times to be eliminated. If 512 contestants enter the tournament, what is the greatest number of games that could be played?
|
1535
| 5.46875 |
1,943 |
On $x-y$ plane, let $C: y=2006x^{3}-12070102x^{2}+\cdots.$
Find the area of the region surrounded by the tangent line of $C$ at $x=2006$ and the curve $C.$
|
\frac{1003}{6}
| 0.78125 |
1,944 |
Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$ , respectively, and are externally tangent at point $A$ . Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$ . Points $B$ and $C$ lie on the same side of $\ell$ , and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .
[asy]
import cse5;
pathpen=black; pointpen=black;
size(6cm);
pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689);
filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7));
filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7));
D(CR((0,1),1)); D(CR((4,4),4,150,390));
D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5));
D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0));
D(MP("E",E,N));
[/asy]
|
129
| 0 |
1,945 |
Let $n$ be a positive integer, and let $S_n = \{1, 2, \ldots, n\}$ . For a permutation $\sigma$ of $S_n$ and an integer $a \in S_n$ , let $d(a)$ be the least positive integer $d$ for which \[\underbrace{\sigma(\sigma(\ldots \sigma(a) \ldots))}_{d \text{ applications of } \sigma} = a\](or $-1$ if no such integer exists). Compute the value of $n$ for which there exists a permutation $\sigma$ of $S_n$ satisfying the equations \[\begin{aligned} d(1) + d(2) + \ldots + d(n) &= 2017, \frac{1}{d(1)} + \frac{1}{d(2)} + \ldots + \frac{1}{d(n)} &= 2. \end{aligned}\]
*Proposed by Michael Tang*
|
53
| 0 |
1,946 |
Let $f : [0, 1] \to \mathbb R$ be continuous and satisfy:
\[ \begin{cases}bf(2x) = f(x), &\mbox{ if } 0 \leq x \leq 1/2, f(x) = b + (1 - b)f(2x - 1), &\mbox{ if } 1/2 \leq x \leq 1,\end{cases}\]
where $b = \frac{1+c}{2+c}$ , $c > 0$ . Show that $0 < f(x)-x < c$ for every $x, 0 < x < 1.$
|
0 < f(x) - x < c
| 82.8125 |
1,947 |
Bianca has a rectangle whose length and width are distinct primes less than 100. Let $P$ be the perimeter of her rectangle, and let $A$ be the area of her rectangle. What is the least possible value of $\frac{P^2}{A}$ ?
|
\frac{82944}{5183}
| 4.6875 |
1,948 |
For positive integers $n,$ let $s(n)$ be the sum of the digits of $n.$ Over all four-digit positive integers $n,$ which value of $n$ maximizes the ratio $\frac{s(n)}{n}$ ?
*Proposed by Michael Tang*
|
1099
| 83.59375 |
1,949 |
The positive numbers $a, b, c,d,e$ are such that the following identity hold for all real number $x$ : $(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$ . Find the smallest value of $d$ .
|
1
| 99.21875 |
1,950 |
Let $n$ be a positive integer. A sequence of $n$ positive integers (not necessarily distinct) is called **full** if it satisfies the following condition: for each positive integer $k\geq2$ , if the number $k$ appears in the sequence then so does the number $k-1$ , and moreover the first occurrence of $k-1$ comes before the last occurrence of $k$ . For each $n$ , how many full sequences are there ?
|
n!
| 26.5625 |
1,951 |
For any real number $t$ , let $\lfloor t \rfloor$ denote the largest integer $\le t$ . Suppose that $N$ is the greatest integer such that $$ \left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4 $$ Find the sum of digits of $N$ .
|
24
| 53.90625 |
1,952 |
At a meeting of $ 12k$ people, each person exchanges greetings with exactly $ 3k\plus{}6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?
|
36
| 14.0625 |
1,953 |
Find the number of trailing zeros at the end of the base- $10$ representation of the integer $525^{25^2}
\cdot 252^{52^5}$ .
|
1250
| 99.21875 |
1,954 |
In $ISI$ club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does $ISI$ club have????
|
10
| 75.78125 |
1,955 |
Integers $1, 2, 3, ... ,n$ , where $n > 2$ , are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be $17$ . What is the maximum sum of the two removed numbers?
|
51
| 71.875 |
1,956 |
Find the least possible cardinality of a set $A$ of natural numbers, the smallest and greatest of which are $1$ and $100$ , and having the property that every element of $A$ except for $1$ equals the sum of two elements of $A$ .
|
9
| 9.375 |
1,957 |
Let $a$ be a complex number, and set $\alpha$ , $\beta$ , and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$ . Suppose \[(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.\] Compute the product of all possible values of $a$ .
|
2009
| 2.34375 |
1,958 |
Express the number 1988 as the sum of some positive integers in such a way that the product of these positive integers is maximal.
|
2 \cdot 3^{662}
| 0 |
1,959 |
Find largest possible constant $M$ such that, for any sequence $a_n$ , $n=0,1,2,...$ of real numbers, that satisfies the conditions :
i) $a_0=1$ , $a_1=3$ ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$ to be true that $$ \frac{a_{n+1}}{a_n} >M $$ for any integer $n\ge 0$ .
|
M = 2
| 0 |
1,960 |
Let $P(x) = x^2 + ax + b$ be a quadratic polynomial. For how many pairs $(a, b)$ of positive integers where $a, b < 1000$ do the quadratics $P(x+1)$ and $P(x) + 1$ have at least one root in common?
|
30
| 10.15625 |
1,961 |
$n \ge 4$ real numbers are arranged in a circle. It turned out that for any four consecutive numbers $a, b, c, d$ , that lie on the circle in this order, holds $a+d = b+c$ . For which $n$ does it follow that all numbers on the circle are equal?
*Proposed by Oleksiy Masalitin*
|
n
| 25 |
1,962 |
Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$ . Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$ . What is the minimum possible value of $A$ ?
*Proposed by Ray Li*
|
28
| 9.375 |
1,963 |
I have $8$ unit cubes of different colors, which I want to glue together into a $2\times 2\times 2$ cube. How many distinct $2\times 2\times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.
|
1680
| 72.65625 |
1,964 |
Determine the maximal value of $k$ such that the inequality $$ \left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right)
\le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left( \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right) $$ holds for all positive reals $a, b, c$ .
|
\sqrt[3]{9} - 1
| 92.1875 |
1,965 |
Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \dfrac{193}{137}.\] If $a+b=m/n$ , where $m$ and $n$ are relatively prime positive integers, compute $m+n$ .
|
28009
| 68.75 |
1,966 |
Given $a,x\in\mathbb{R}$ and $x\geq 0$ , $a\geq 0$ . Also $\sin(\sqrt{x+a})=\sin(\sqrt{x})$ . What can you say about $a$ ??? Justify your answer.
|
a = 0
| 0 |
1,967 |
Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$ . Of all such sequences, find the one with the smallest possible common difference.
|
32
| 59.375 |
1,968 |
Points $E$ and $F$ lie inside rectangle $ABCD$ with $AE=DE=BF=CF=EF$ . If $AB=11$ and $BC=8$ , find the area of the quadrilateral $AEFB$ .
|
32
| 2.34375 |
1,969 |
In how many ways can six marbles be placed in the squares of a $6$ -by- $6$ grid such that no two marbles lie in the same row or column?
|
720
| 100 |
1,970 |
Let $n$ be an odd integer greater than $11$ ; $k\in \mathbb{N}$ , $k \geq 6$ , $n=2k-1$ .
We define \[d(x,y) = \left | \{ i\in \{1,2,\dots, n \} \bigm | x_i \neq y_i \} \right |\] for $T=\{ (x_1, x_2, \dots, x_n) \bigm | x_i \in \{0,1\}, i=1,2,\dots, n \}$ and $x=(x_1,x_2,\dots, x_n), y=(y_1, y_2, \dots, y_n) \in T$ .
Show that $n=23$ if $T$ has a subset $S$ satisfying
[list=i]
[*] $|S|=2^k$ [*]For each $x \in T$ , there exists exacly one $y\in S$ such that $d(x,y)\leq 3$ [/list]
|
n = 23
| 0 |
1,971 |
Determine the least integer $n$ such that for any set of $n$ lines in the 2D plane, there exists either a subset of $1001$ lines that are all parallel, or a subset of $1001$ lines that are pairwise nonparallel.
*Proposed by Samuel Wang*
<details><summary>Solution</summary>*Solution.* $\boxed{1000001}$ Since being parallel is a transitive property, we note that in order for this to not exist, there must exist at most $1001$ groups of lines, all pairwise intersecting, with each group containing at most $1001$ lines. Thus, $n = 1000^2 + 1 =
\boxed{1000001}$ .</details>
|
1000001
| 92.96875 |
1,972 |
An equilateral triangle of side $n$ is divided into equilateral triangles of side $1$ . Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle.
|
n(n+1)
| 12.5 |
1,973 |
The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$ , erases them and she writes instead the number $x + y + xy$ . She continues to do this until only one number is left on the board. What are the possible values of the final number?
|
2012
| 96.875 |
1,974 |
$p$ is a prime number that is greater than $2$ . Let $\{ a_{n}\}$ be a sequence such that $ na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}$ .
Show that if $a_{1}=5$ , the $16 \mid a_{81}$ .
|
16 \mid a_{81}
| 42.96875 |
1,975 |
A sequence $a_1, a_2, a_3, \ldots$ of positive integers satisfies $a_1 > 5$ and $a_{n+1} = 5 + 6 + \cdots + a_n$ for all positive integers $n$ . Determine all prime numbers $p$ such that, regardless of the value of $a_1$ , this sequence must contain a multiple of $p$ .
|
p = 2
| 0 |
1,976 |
The student population at one high school consists of freshmen, sophomores, juniors, and seniors. There are 25 percent more freshmen than juniors, 10 percent fewer sophomores than freshmen, and 20 percent of the students are seniors. If there are 144 sophomores, how many students attend the school?
|
540
| 53.125 |
1,977 |
Let $BE$ and $CF$ be altitudes in triangle $ABC$ . If $AE = 24$ , $EC = 60$ , and $BF = 31$ , determine $AF$ .
|
32
| 1.5625 |
1,978 |
Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$ . What is the product of its inradius and circumradius?
|
10
| 47.65625 |
1,979 |
Let $S$ be the locus of all points $(x,y)$ in the first quadrant such that $\dfrac{x}{t}+\dfrac{y}{1-t}=1$ for some $t$ with $0<t<1$ . Find the area of $S$ .
|
\frac{1}{6}
| 0 |
1,980 |
$22$ football players took part in the football training. They were divided into teams of equal size for each game ( $11:11$ ). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training.
|
5
| 3.125 |
1,981 |
Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...,$ find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$ .
|
1026
| 32.03125 |
1,982 |
Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$ , respectively, and meet perpendicularly at $T$ . $Q$ is on $AT$ , $S$ is on $BT$ , and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$ . Determine the radius of the circle.
|
29
| 2.34375 |
1,983 |
Let $n$ be a positive integer and let $d_{1},d_{2},,\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\ldots <d_{k}=n$ . Find all values of $n$ for which $k\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$ .
|
130
| 100 |
1,984 |
A triangle $\triangle ABC$ satisfies $AB = 13$ , $BC = 14$ , and $AC = 15$ . Inside $\triangle ABC$ are three points $X$ , $Y$ , and $Z$ such that:
- $Y$ is the centroid of $\triangle ABX$
- $Z$ is the centroid of $\triangle BCY$
- $X$ is the centroid of $\triangle CAZ$
What is the area of $\triangle XYZ$ ?
*Proposed by Adam Bertelli*
|
\frac{84}{13}
| 1.5625 |
1,985 |
Consider an $8\times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner?
|
70
| 0.78125 |
1,986 |
The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
|
481^2
| 0 |
1,987 |
Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$ . Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$ |f(A)\cap f(B)|=|A\cap B| $$ whenever $A$ and $B$ are two distinct subsets of $X$ .
*(Sergiu Novac)*
|
n!
| 56.25 |
1,988 |
A positive integer is *bold* iff it has $8$ positive divisors that sum up to $3240$ . For example, $2006$ is bold because its $8$ positive divisors, $1$ , $2$ , $17$ , $34$ , $59$ , $118$ , $1003$ and $2006$ , sum up to $3240$ . Find the smallest positive bold number.
|
1614
| 15.625 |
1,989 |
There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$ . What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?
|
8
| 70.3125 |
1,990 |
Let the complex number $z = \cos\tfrac{1}{1000} + i \sin\tfrac{1}{1000}.$ Find the smallest positive integer $n$ so that $z^n$ has an imaginary part which exceeds $\tfrac{1}{2}.$
|
524
| 100 |
1,991 |
Determine all integers $a$ for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions $x, \;y$ .
|
a = 2
| 0 |
1,992 |
If $x > 10$ , what is the greatest possible value of the expression
\[
{( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ?
\]
All the logarithms are base 10.
|
0
| 39.84375 |
1,993 |
Define the determinant $D_1$ = $|1|$ , the determinant $D_2$ = $|1 1|$ $|1 3|$
, and the determinant $D_3=$
|1 1 1|
|1 3 3|
|1 3 5|
.
In general, for positive integer n, let the determinant $D_n$ have 1s in every position of its first row and first
column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that $D_n$ $\geq$ 2015.
|
12
| 57.8125 |
1,994 |
Let $C_1$ be a circle with centre $O$ , and let $AB$ be a chord of the circle that is not a diameter. $M$ is the midpoint of $AB$ . Consider a point $T$ on the circle $C_2$ with diameter $OM$ . The tangent to $C_2$ at the point $T$ intersects $C_1$ at two points. Let $P$ be one of these points. Show that $PA^2+PB^2=4PT^2$ .
|
PA^2 + PB^2 = 4PT^2
| 76.5625 |
1,995 |
We define the polynomial $$ P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x. $$ Find the largest prime divisor of $P (2)$ .
|
61
| 2.34375 |
1,996 |
Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$ . Determine max $\frac{AP}{PE}$ , over all such configurations.
|
5
| 0 |
1,997 |
For positive integers $n$ , let $S_n$ be the set of integers $x$ such that $n$ distinct lines, no three concurrent, can divide a plane into $x$ regions (for example, $S_2=\{3,4\}$ , because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum $i$ such that $S_i$ contains at least 4 elements?
|
4
| 64.0625 |
1,998 |
Consider the sum $$ S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|. $$ The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$ , satisfying $2c < d$ . Find the value of $c + d$ .
|
3031
| 36.71875 |
1,999 |
Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$ . For a given positive integer $n$ , $n\geq 2$ consider the system of equations: \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$ , $x_1=\lambda y_1$ , $\ldots$ , $x_n= \lambda y_n$ . How many nonequivalent solutions does the system have?
*Mircea Becheanu*
|
1
| 19.53125 |
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