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

PEN A Problems, 61
Tags: search , divisibility theory
For any positive integer $n>1$, let $p(n)$ be the greatest prime divisor of $n$. Prove that there are infinitely many positive integers $n$ with \[p(n)<p(n+1)<p(n+2).\]

2016 Hanoi Open Mathematics Competitions, 5
Tags: diophantine equation , diophantine , number theory
There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to (A): $2013$ (B): $2014$ (C): $2015$ (D): $2016$ (E): None of the above.

2003 AMC 10, 18
Tags: algebra , polynomial
What is the largest integer that is a divisor of \[ (n\plus{}1)(n\plus{}3)(n\plus{}5)(n\plus{}7)(n\plus{}9) \]for all positive even integers $ n$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 165$

2009 Harvard-MIT Mathematics Tournament, 3
Tags: trigonometry
If $\tan x + \tan y = 4$ and $\cot x + \cot y = 5$, compute $\tan(x + y)$.

Russian TST 2021, P1
Tags: combinatorics , recursion
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

2021 239 Open Mathematical Olympiad, 5
Tags: complex number , inequalities
Let $a,b,c$ be some complex numbers. Prove that $$|\dfrac{a^2}{ab+ac-bc}| + |\dfrac{b^2}{ba+bc-ac}| + |\dfrac{c^2}{ca+cb-ab}| \ge \dfrac{3}{2}$$ if the denominators are not 0

2016 CMIMC, 7
Tags: number theory
Determine the smallest positive prime $p$ which satisfies the congruence \[p+p^{-1}\equiv 25\pmod{143}.\] Here, $p^{-1}$ as usual denotes multiplicative inverse.

2010 Saint Petersburg Mathematical Olympiad, 3
Tags: geometry
$M,N$ are midpoints of $AB$ and $CD$ for convex quadrilateral $ABCD$. Points $X$ and $Y$ are on $ AD$ and $BC$ and $XD=3AX,YC=3BY$. $\angle MXA=\angle MYB = 90$. Prove that $\angle XMN=\angle ABC$

2022 Sharygin Geometry Olympiad, 8.3
Tags: geometry
A circle $\omega$ and a point $P$ not lying on it are given. Let $ABC$ be an arbitrary equilateral triangle inscribed into $\omega$ and $A', B', C'$ be the projections of $P$ to $BC, CA, AB$. Find the locus of centroids of triangles $A' B'C'$.

2005 QEDMO 1st, 1 (Z4)
Tags: geometry , 3d geometry , induction , modular arithmetic , number theory
Prove that every integer can be written as sum of $5$ third powers of integers.

2014 District Olympiad, 3
Tags: trigonometry , incenter , angle bisector , similar triangles , geometry
Let $ABC$ be a triangle in which $\measuredangle{A}=135^{\circ}$. The perpendicular to the line $AB$ erected at $A$ intersects the side $BC$ at $D$, and the angle bisector of $\angle B$ intersects the side $AC$ at $E$. Find the measure of $\measuredangle{BED}$.

2008 Indonesia MO, 4
Tags: combinatorics
Let $ A \equal{} \{1,2,\ldots,2008\}$ a) Find the number of subset of $ A$ which satisfy : the product of its elements is divisible by 7 b) Let $ N(i)$ denotes the number of subset of $ A$ which sum of its elements remains $ i$ when divided by 7. Prove that $ N(0) \minus{} N(1) \plus{} N(2) \minus{} N(3) \plus{} N(4) \minus{} N(5) \plus{} N(6)\minus{}N(7) \equal{} 0$ EDITED : thx for cosinator.. BTW, your statement and my correction give 80% hint of the solution :D

2023 Ukraine National Mathematical Olympiad, 9.7
Tags: number theory
You are given $n \ge 2$ distinct positive integers. Let's call a pair of these integers [i]elegant[/i] if their sum is an integer power of $2$. For every $n$ find the largest possible number of elegant pairs. [i]Proposed by Oleksiy Masalitin[/i]

2025 6th Memorial "Aleksandar Blazhevski-Cane", P3
Tags: concave , sequence , algebra
A sequence of real numbers $(a_k)_{k \ge 0}$ is called [i]log-concave[/i] if for every $k \ge 1$, the inequality $a_{k - 1}a_{k + 1} \le a_k^2$ holds. Let $n, l \in \mathbb{N}$. Prove that the sequence $(a_k)_{k \ge 0}$ with general term \[a_k = \sum_{i = k}^{k + l} {n \choose i}\] is log-concave. Proposed by [i]Svetlana Poznanovikj[/i]

1904 Eotvos Mathematical Competition, 2
Tags: number theory , diophantine
If a is a natural number, show that the number of positive integral solutions of the indeterminate equation $$x_1 + 2x_2 + 3x_3 + ... + nx_n = a \ \ (1) $$ is equal to the number of non-negative integral solutions of $$y_1 + 2y_2 + 3y_3 + ... + ny_n = a - \frac{n(n + 1)}{2} \ \ (2)$$ [By a solution of equation (1), we mean a set of numbers $\{x_1, x_2,..., x_n\}$ which satisfies equation (1)].

2017 BMT Spring, 10
Tags: geometry
Colorado and Wyoming are both defined to be $4$ degrees tall in latitude and $7$ degree wide in longitude. In particular, Colorado is defined to be at $37^o N$ to $41^o N$, and $102^o03' W$ to $109^o03' W$, whereas Wyoming is defined to be $41^o N$ to $45^o N$, and $104^o 03' W$ to $111^o 03' W$. Assuming Earth is a perfect sphere with radius $R$, what is the ratio of the areas of Wyoming to Colorado, in terms of $R$?

2022 Cyprus JBMO TST, 2
Tags: geometry , square , area , parallelogram
Let $ABCD$ be a square. Let $E, Z$ be points on the sides $AB, CD$ of the square respectively, such that $DE\parallel BZ$. Assume that the triangles $\triangle EAD, \triangle ZCB$ and the parallelogram $BEDZ$ have the same area. If the distance between the parallel lines $DE$ and $BZ$ is equal to $1$, determine the area of the square.

2000 Harvard-MIT Mathematics Tournament, 7
Tags: geometry , 3d geometry
A number $n$ is called multiplicatively perfect if the product of all the positive divisors of $n$ is $n^2$. Determine the number of positive multiplicatively perfect numbers less than $100$.

1972 Putnam, A3
Tags: limit , sequence
A sequence $(x_{i})$ is said to have a [i]Cesaro limit[/i] exactly if $\lim_{n\to\infty} \frac{x_{1}+\ldots+x_{n}}{n}$ exists. Find all real-valued functions $f$ on the closed interval $[0, 1]$ such that $(f(x_i))$ has a Cesaro limit if and only if $(x_i)$ has a Cesaro limit.

2019 Latvia Baltic Way TST, 12
Tags: perimeter , geometric transformation , reflection , circumcircle , geometry
Let $AX$, $AY$ be tangents to circle $\omega$ from point $A$. Le $B$, $C$ be points inside $AX$ and $AY$ respectively, such that perimeter of $\triangle ABC$ is equal to length of $AX$. $D$ is reflection of $A$ over $BC$. Prove that circumcircle $\triangle BDC$ and $\omega$ are tangent to each other.

2023 AMC 12/AHSME, 23
Tags: equation
How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation \[(1+2a)(2+2b)(2a+b) = 32ab?\] $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{an infinite number}$

2008 Moldova MO 11-12, 5
Tags: polynomial , algebra
Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$.

2022 Canadian Mathematical Olympiad Qualification, 8
Tags: inequalities
Let $\{m, n, k\}$ be positive integers. $\{k\}$ coins are placed in the squares of an $m \times n$ grid. A square may contain any number of coins, including zero. Label the $\{k\}$ coins $C_1, C_2, · · · C_k$. Let $r_i$ be the number of coins in the same row as $C_i$, including $C_i$ itself. Let $s_i$ be the number of coins in the same column as $C_i$, including $C_i$ itself. Prove that \[\sum_{i=1}^k \frac{1}{r_i+s_i} \leq \frac{m+n}{4}\]

1996 Brazil National Olympiad, 4
Tags: geometry , circumcircle , vector , ratio , trigonometry , perpendicular bisector , geometric transformation
$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$, $O_2$ is the circumcenter of $ACD$, and $O$ is the circumcenter of $AO_1O_2$. Find the locus of $O$.

2024 MMATHS, 4
Tags:
Consider a pattern of squares and triangles. The first move of the pattern is to place an isosceles right triangle with side lengths $1, 1, \sqrt{2}.$ For each subsequent move, you need to attach a square to every non-hypotenuse side of a triangle and attach the same isosceles right triangle to every side of a square. After $2024$ moves, what is smallest possible area of the resulting shape?