This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

1995 Abels Math Contest (Norwegian MO), 4
Tags: inequalities , algebra
Let $x_i,y_i$ be positive real numbers, $i = 1,2,...,n$. Prove that $$\left( \sum_{i=1}^n (x_i +y_i)^2\right)\left( \sum_{i=1}^n\frac{1}{x_iy_i}\right)\ge 4n^2$$

2018 IMO Shortlist, N1
Tags: number theory , arithmetic function , number of divisors , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

1998 USAMTS Problems, 2
Tags: greatest common divisor
Prove that there are infinitely many ordered triples of positive integers $(a,b,c)$ such that the greatest common divisor of $a,b,$ and $c$ is $1$, and the sum $a^2b^2+b^2c^2+c^2a^2$ is the square of an integer.

2017 ASDAN Math Tournament, 18
Tags:
Find the sum of all integers $0\le a \le124$ so that $a^3-2$ is a multiple of $125$.

2012 HMNT, 9
Tags: geometry
Triangle $ABC$ satisfies $\angle B > \angle C$. Let $M$ be the midpoint of $BC$, and let the perpendicular bisector of $BC$ meet the circumcircle of $\vartriangle ABC$ at a point $D$ such that points $A$, $D$, $C$, and $B$ appear on the circle in that order. Given that $\angle ADM = 68^o$ and $\angle DAC = 64^o$ , find $\angle B$.

1990 Swedish Mathematical Competition, 2
Tags: sum , geometric inequalities , geometry , collinear
The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.

1951 AMC 12/AHSME, 15
Tags: calculus , integration
The largest number by which the expression $ n^3 \minus{} n$ is divisible for all possible integral values of $ n$, is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$

2025 India STEMS Category C, 4
Tags: function , calculus
Does there exist a function $f:[0,1]\rightarrow (0,\infty)$ such that [list] [*]$f$ is differentiable on $[0,1]$ [*] It's derivative $f'$ is continuous on $[0,1]$. [*] $(f'(x))^3-x^{\frac{1}{3}}>6(1-f(x)^{\frac{1}{5}})$ for all $x\in [0,1]$. [*] $f(1)=1$ [/list] [i]Proposed by Medhansh Tripathi[/i]

2006 Estonia National Olympiad, 4
Tags: trigonometry , geometry
Let O be the circumcentre of an acute triangle ABC and let A′, B′ and C′ be the circumcentres of triangles BCO, CAO and ABO, respectively. Prove that the area of triangle ABC does not exceed the area of triangle A′B′C′.

2023 Germany Team Selection Test, 1
Tags: algebra
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

2003 AMC 10, 14
Tags:
Given that $ 3^8\cdot5^2 \equal{} a^b$, where both $ a$ and $ b$ are positive integers, find the smallest possible value for $ a \plus{} b$. $ \textbf{(A)}\ 25 \qquad \textbf{(B)}\ 34 \qquad \textbf{(C)}\ 351 \qquad \textbf{(D)}\ 407 \qquad \textbf{(E)}\ 900$

2016 Ecuador Juniors, 3
Tags: geometry
Let $P_1P_2 . . . P_{2016 }$ be a cyclic polygon of $2016$ sides. Let $K$ be a point inside the polygon and let $M$ be the midpoint of the segment $P_{1000}P_{2000}$. Knowing that $KP_1 = KP_{2011} = 2016$ and $KM$ is perpendicular to $P_{1000}P_{2000}$, find the length of segment $KP_{2016}$.

2010 Silk Road, 3
Tags: inequalities
For positive real numbers $a, b, c, d,$ satisfying the following conditions: $a(c^2 - 1)=b(b^2+c^2)$ and $d \leq 1$, prove that : $d(a \sqrt{1-d^2} + b^2 \sqrt{1+d^2}) \leq \frac{(a+b)c}{2}$

1984 Austrian-Polish Competition, 7
Tags: linear algebra , matrix
A $m\times n$ matrix $(a_{ij})$ of real numbers satisfies $|a_{ij}| <1$ and $\sum_{i=1}^m a_{ij}= 0$ for all$ j$. Show that one can permute the entries in each column in such a way that the obtained matrix $(b_{ij})$ satisfies $\sum_{j=1}^n b_{ij} < 2$ for all $i$.

1986 IMO Longlists, 43
Tags: probability , combinatorics , game , game strategy , counting
Three persons $A,B,C$, are playing the following game: A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$. For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)

2016 ASDAN Math Tournament, 5
Tags:
Find $$\lim_{x\rightarrow0}\frac{\sin(x)-x}{x\cos(x)-x}.$$

2011 Croatia Team Selection Test, 2
Tags: analytic geometry , combinatorics
There are lamps in every field of $n\times n$ table. At start all the lamps are off. A move consists of chosing $m$ consecutive fields in a row or a column and changing the status of that $m$ lamps. Prove that you can reach a state in which all the lamps are on only if $m$ divides $n.$

2018 India PRMO, 17
Tags: geometry , equal angles
Triangles $ABC$ and $DEF$ are such that $\angle A = \angle D, AB = DE = 17, BC = EF = 10$ and $AC - DF = 12$. What is $AC + DF$?

2024 ELMO Shortlist, A2
Tags: combinatorics , floor function
Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$ [i]Andrew Carratu[/i]

2010 Junior Balkan Team Selection Tests - Romania, 4
Tags: lattice , combinatorics , combinatorial geometry
The plan considers $51$ points of integer coordinates, so that the distances between any two points are natural numbers. Show that at least $49\%$ of the distances are even.

2021 239 Open Mathematical Olympiad, 1
Tags: integer , polynomial , integer polynomial , algebra , prime
You are given $n$ different primes $p_1, p_2,..., p_n$. Consider the polynomial $$x^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n$$, where $a_i$ is the product of the first $i$ given prime numbers. For what $n$ can it have an integer root?

2020 AMC 10, 25
Tags: probability
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice? $\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}$

1992 Mexico National Olympiad, 2
Tags: number theory , prime
Given a prime number $p$, how many $4$-tuples $(a, b, c, d)$ of positive integers with $0 \le a, b, c, d \le p-1$ satisfy $ad = bc$ mod $p$?

2022 Sharygin Geometry Olympiad, 13
Tags: geometry
Eight points in a general position are given in the plane. The areas of all $56$ triangles with vertices at these points are written in a row. Prove that it is possible to insert the symbols "$+$" and "$-$" between them in such a way that the obtained sum is equal to zero.

2003 Romania Team Selection Test, 5
Tags: algebra , polynomial , calculus , integration , function , complex number , absolute value
Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial over the ring of integer polynomials, such that $|f(0)|$ is not a perfect square. Prove that if the leading coefficient of $f$ is 1 (the coefficient of the term having the highest degree in $f$) then $f(X^2)$ is also irreducible in the ring of integer polynomials. [i]Mihai Piticari[/i]