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

2019 LIMIT Category B, Problem 5
Tags: geometry
A polygon has twice as many diagonals as it has sides. How many sides does it have?

2014 HMNT, 3
Tags: hmmt , inequalities , triangle inequality
The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of $42,$ and another is a multiple of $72$. What is the minimum possible length of the third side?

2024-IMOC, G2
Tags: geometry
Triangle $ABC$ has circumcenter $O$. $D$ is an arbitrary point on $BC$, and $AD$ intersects $\odot(ABC)$ at $E$. $S$ is a point on $\odot(ABC)$ such that $D, O, E, S$ are colinear. $AS$ intersects $BC$ at $P$. $Q$ is a point on $BC$ such that $D, O, A, Q$ are concylic. Prove that $\odot(ABC)$ is tangent to $\odot (APQ)$. [i]Proposed by chengbilly[/i]

2016 PAMO, 4
Tags: inequality , algebra
Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that $\frac{1}{(x+1)^2+y^2+1}$ $+$ $\frac{1}{(y+1)^2+z^2+1}$ $+$ $\frac{1}{(z+1)^2+x^2+1}$ $\leq$ ${\frac{1}{2}}$.

1990 AMC 12/AHSME, 13
Tags: arithmetic series
If the following instructions are carried out by a computer, which of $X$ will be printed because of instruction $5$? $1.$ Start $X$ at $3$ and $S$ at $0$ $2.$ Increase the value of $X$ by $2$. $3.$ Increase the value of $S$ by the value of $X$. $4.$ If $S$ is at least $10000$, then go to instsruction $5$; otherwise, go to instruction $2$ and proceed from there. $5.$ Print the value of $X$. $6.$ Stop. $\text{(A)} \ 19 \qquad \text{(B)} \ 21 \qquad \text{(C)} \ 23 \qquad \text{(D)} \ 199 \qquad \text{(E)} \ 201$

2019 Turkey Junior National Olympiad, 1
Tags: gcd or factors , number theory , power of prime
Solve $2a^2+3a-44=3p^n$ in positive integers where $p$ is a prime.

2009 India IMO Training Camp, 6
Tags: function , combinatorics
Prove The Following identity: $ \sum_{j \equal{} 0}^n \left ({3n \plus{} 2 \minus{} j \choose j}2^j \minus{} {3n \plus{} 1 \minus{} j \choose j \minus{} 1}2^{j \minus{} 1}\right ) \equal{} 2^{3n}$. The Second term on left hand side is to be regarded zero for j=0.

2013 Purple Comet Problems, 10
Tags:
Find the least positive integer $k$ so that the mean of the numbers $k,k + 1,k + 2,k + 3,\ldots,2k$ exceeds $200$.

2015 Indonesia MO Shortlist, G3
Tags: tangent , geometry , circumcircle , incircle
Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.

2019 Belarusian National Olympiad, 9.7
Tags: algebra , polynomial
Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients such that $P(Q(x)^2)=P(x)\cdot Q(x)^2$. [i](I. Voronovich)[/i]

Champions Tournament Seniors - geometry, 2001.4
Tags: pentagon , equal angles , equal segments , geometry
Given a convex pentagon $ABCDE$ in which $\angle ABC = \angle AED = 90^o$, $\angle BAC= \angle DAE$. Let $K$ be the midpoint of the side $CD$, and $P$ the intersection point of lines $AD$ and $BK$, $Q$ be the intersection point of lines $AC$ and $EK$. Prove that $BQ = PE$.

2002 Bundeswettbewerb Mathematik, 1
Tags:
Planet Ypsilon has a calendar similar to ours: A year consists of $365$ days, and every month has $28$, $30$ or $31$ days. Prove that on Planet Ypsilon, a year must have $12$ months.

2022 Saudi Arabia BMO + EGMO TST, 2.1
Tags: geometry , circumcenter
Let $ABC$ be an acute-angled triangle. Point $P$ is such that $AP = AB$ and $PB \parallel AC$. Point $Q$ is such that $AQ = AC$ and $CQ \parallel AB$. Segments $CP$ and $BQ$ meet at point $X$. Prove that the circumcenter of triangle $ABC$ lies on the circumcircle of triangle $PXQ$.

2002 Italy TST, 3
Tags: polynomial , vieta , relatively prime , number theory , algebra
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

1987 IMO Longlists, 18
Tags: geometric inequality , 3d geometry , polyhedron , geometry
Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality \[AF + AH + AC \leq AB + AD + AE + AG.\] In what cases does equality hold? [i]Proposed by France.[/i]

2018 Online Math Open Problems, 1
Tags:
Leonhard has five cards. Each card has a nonnegative integer written on it, and any two cards show relatively prime numbers. Compute the smallest possible value of the sum of the numbers on Leonhard's cards. Note: Two integers are relatively prime if no positive integer other than $1$ divides both numbers. [i]Proposed by ABCDE and Tristan Shin

1954 Moscow Mathematical Olympiad, 286
Tags: number theory , combinatorics , sum , 10-digit , subset
Consider the set of all $10$-digit numbers expressible with the help of figures $1$ and $2$ only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two $3$’s.

1961 Putnam, B5
Tags: factorial , inequalities
Let $k$ be a positive integer, and $n$ a positive integer greater than $2$. Define $$f_{1}(n)=n,\;\; f_{2}(n)=n^{f_{1}(n)},\;\ldots\;, f_{j+1}(n)=n^{f_{j}(n)}.$$ Prove either part of the inequality $$f_{k}(n) < n!! \cdots ! < f_{k+1}(n),$$ where the middle term has $k$ factorial symbols.

2023 BMT, Tie 1
Tags: combinatorics
Mataio has a weighted die numbered $1$ to $6$, where the probability of rolling a side $n$ for $1 \le n \le 6$ is inversely proportional to the value of $n$. If Mataio rolls the die twice, what is the probability that the sum of the two rolls is $7$?

2008 All-Russian Olympiad, 3
Tags: euler , circumcircle , vector , geometry
In a scalene triangle $ ABC, H$ and $ M$ are the orthocenter an centroid respectively. Consider the triangle formed by the lines through $ A,B$ and $ C$ perpendicular to $ AM,BM$ and $ CM$ respectively. Prove that the centroid of this triangle lies on the line $ MH$.

2024 Bosnia and Herzegovina Junior BMO TST, 1.
Tags: algebra
Let $a$,$b$,$c$ be real numbers different from 0 for which $ab$ + $bc$+ $ca$ = 0 holds a) Prove that ($a$+$b$)($b$+$c$)($c$+$a$)≠ 0 b) Let $X$ = $a$ + $b$ + $c$ and $Y$ = $\frac{1}{a+b}$ + $\frac{1}{b+c}$ + $\frac{1}{c+a}$. Prove that numbers $X$ and $Y$ are both positive or both negative.

2002 China Team Selection Test, 3
Tags: number theory
For positive integers $a,b,c$ let $ \alpha, \beta, \gamma$ be pairwise distinct positive integers such that \[ \begin{cases}{c} \displaystyle a &= \alpha + \beta + \gamma, \\ b &= \alpha \cdot \beta + \beta \cdot \gamma + \gamma \cdot \alpha, \\ c^2 &= \alpha\beta\gamma. \end{cases} \] Also, let $ \lambda$ be a real number that satisfies the condition \[\lambda^4 -2a\lambda^2 + 8c\lambda + a^2 - 4b = 0.\] Prove that $\lambda$ is an integer if and only if $\alpha, \beta, \gamma$ are all perfect squares.

2015 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: number theory , prime
Find all triplets $(p,a,b)$ of positive integers such that $$p=b\sqrt{\frac{a-8b}{a+8b}}$$ is prime

2000 All-Russian Olympiad Regional Round, 8.3
Tags: combinatorics , combinatorial geometry , geometry
What is the smallest number of sides that an polygon can have (not necessarily convex), which can be cut into parallelograms?

2014 China Team Selection Test, 6
Tags: combinatorics
Let $n\ge 2$ be a positive integer. Fill up a $n\times n$ table with the numbers $1,2,...,n^2$ exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most $n$. Show that there exist a $2\times 2$ square of adjacent cells such that the diagonally opposite pairs sum to the same number.