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

2008 Polish MO Finals, 5
Tags: geometry
Let $ R$ be a parallelopiped. Let us assume that areas of all intersections of $ R$ with planes containing centers of three edges of $ R$ pairwisely not parallel and having no common points, are equal. Show that $ R$ is a cuboid.

2020 IMO, 3
Tags: combinatorics , graph theory , petersen
There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: [list] [*]The total weights of both piles are the same. [*] Each pile contains two pebbles of each colour. [/list] [i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]

2024 Ukraine National Mathematical Olympiad, Problem 4
Tags: geometry , tangency
Points $E, F$ are selected on sides $AC, AB$ respectively of triangle $ABC$ with $AC=AB$ so that $AE = BF$. Point $D$ is chosen so that $D, A$ are in the same halfplane with respect to line $EF$, and $\triangle DFE \sim \triangle ABC$. Lines $EF, BC$ intersect at point $K$. Prove that the line $DK$ is tangent to the circumscribed circle of $\triangle ABC$. [i]Proposed by Fedir Yudin[/i]

2004 Harvard-MIT Mathematics Tournament, 9
Tags: number theory , greatest common divisor , modular arithmetic
A sequence of positive integers is defined by $a_0=1$ and $a_{n+1}=a_n^2+1$ for each $n\ge0$. Find $\text{gcd}(a_{999},a_{2004})$.

2022 IFYM, Sozopol, 1
Tags: number theory , divide
Are there natural numbers $n$ and $N$ such that $n > 10^{10}$, $$n^n < 2^{2^{\frac{8N}{\omega (N)}}}$$ and $n$ is divisible by $p^{2022(v_p(N)-1)}(p-1)$ for every prime divisor $p$ of $N$? (For a natural number $N$, we denote by $\omega (N)$ the number of its different prime divisors and with $v_p(N)$ the power of the prime number $p$ in its canonical representation.)

2023 LMT Fall, 22
Tags: geometry , 3d geometry , algebra
Consider all pairs of points $(a,b,c)$ and $(d,e, f )$ in the $3$-D coordinate system with $ad +be +c f = -2023$. What is the least positive integer that can be the distance between such a pair of points? [i]Proposed by William Hua[/i]

2012 IMO Shortlist, N5
Tags: algebra , number theory , prime number , polynomial
For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.

2016 AMC 10, 14
Tags:
How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008\cdot 2 + 0\cdot 3$ and $402\cdot 2 + 404\cdot 3$ are two such ways.) $\textbf{(A)}\ 236\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 337\qquad\textbf{(D)}\ 403\qquad\textbf{(E)}\ 672$

2004 China Team Selection Test, 2
Tags: perimeter , inequalities , trigonometry , trig identities , law of sines , geometry
Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.

2011 NZMOC Camp Selection Problems, 5
Tags: distance , square , geometry , collinear
Let a square $ABCD$ with sides of length $1$ be given. A point $X$ on $BC$ is at distance $d$ from $C$, and a point $Y$ on $CD$ is at distance $d$ from $C$. The extensions of: $AB$ and $DX$ meet at $P$, $AD$ and $BY$ meet at $Q, AX$ and $DC$ meet at $R$, and $AY$ and $BC$ meet at $S$. If points $P, Q, R$ and $S$ are collinear, determine $d$.

1936 Moscow Mathematical Olympiad, 023
Tags: rectangle , triangle construction , geometric construction , isosceles
All rectangles that can be inscribed in an isosceles triangle with two of their vertices on the triangle’s base have the same perimeter. Construct the triangle.

2008 China Team Selection Test, 3
Tags: probability , combinatorics
Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$

2021 BMT, 10
Tags: geometry
Triangle $\vartriangle ABC$ has side lengths $AB = AC = 27$ and $BC = 18$. Point $D$ is on $\overline{AB}$ and point $E$ is on $\overline{AC}$ such that $\angle BCD = \angle CBE = \angle BAC$. Compute $DE$.

2007 Czech and Slovak Olympiad III A, 5
Tags: geometry
In an acute-angled triangle $ABC$ ($AC\ne BC$), let $D$ and $E$ be points on $BC$ and $AC$, respectively, such that the points $A,B,D,E$ are concyclic and $AD$ intersects $BE$ at $P$. Knowing that $CP\bot AB$, prove that $P$ is the orthocenter of triangle $ABC$.

2011 Kyiv Mathematical Festival, 3
Tags: geometry , concurrency , square
$ABC$ is right triangle with right angle near vertex $B, M$ is the midpoint of $AC$. The square $BKLM$ is built on $BM$, such that segments $ML$ and $BC$ intersect. Segment $AL$ intersects $BC$ in point $E$. Prove that lines $AB,CL$ and$ KE$ intersect in one point.

2016 China Team Selection Test, 4
Tags: number theory , floor function
Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.

1991 Baltic Way, 3
Tags:
There are $20$ cats priced from $\$12$ to $\$15$ and $20$ sacks priced from $10$ cents to $\$1$ for sale, all of different prices. Prove that John and Peter can each buy a cat in a sack paying the same amount of money.

1965 All Russian Mathematical Olympiad, 070
Tags: polyhedron , geometry , 3d geometry
Prove that the sum of the lengths of the polyhedron edges exceeds its tripled diameter (distance between two farest vertices).

2024 HMNT, 3
Tags: team
Rectangle $R$ with area $20$ and diagonal of length $7$ is translated $2$ units in some direction to form a new rectangle $R'.$ The vertices of $R$ and $R'$ that are not contained in the other rectangle form a convex hexagon. Compute the maximum possible area of this hexagon.

2003 Silk Road, 2
Tags: geometry , perimeter , circumcircle , incenter , euler , parallelogram , trapezoid
Let $s=\frac{AB+BC+AC}{2}$ be half-perimeter of triangle $ABC$. Let $L$ and $N$be a point's on ray's $AB$ and $CB$, for which $AL=CN=s$. Let $K$ is point, symmetric of point $B$ by circumcenter of $ABC$. Prove, that perpendicular from $K$ to $NL$ passes through incenter of $ABC$. Solution for problem [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

2023 Sharygin Geometry Olympiad, 18
Tags: bicentral , arc midpoint , constructive geometry , geometry
Restore a bicentral quadrilateral $ABCD$ if the midpoints of the arcs $AB,BC,CD$ of its circumcircle are given.

2019 AMC 10, 18
Tags:
For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$? $\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17$

2017 Saudi Arabia BMO TST, 1
Tags: number theory , divide , divisible , factorial
Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.

2022 IMO Shortlist, A3
Tags: functional inequality
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying $$xf(y)+yf(x) \leq 2$$

2011 IMAC Arhimede, 5
Tags: number theory
Solve in set of integers the following equation $x^5+y^5+z^5+t^5=93$.