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

2021 Latvia Baltic Way TST, P6
Tags: rectangle , rotation , domain , combinatorics
Let's call $1 \times 2$ rectangle, which can be a rotated, a domino. Prove that there exists polygon, who can be covered by dominoes in exactly $2021$ different ways.

2023 Sharygin Geometry Olympiad, 8.6
Tags: geometry
For which $n$ the plane may be paved by congruent figures bounded by $n$ arcs of circles?

2022 MIG, 1
Tags:
What is $4^0 - 3^1 - 2^2 - 1^3$? $\textbf{(A) }{-}8\qquad\textbf{(B) }{-}7\qquad\textbf{(C) }{-}5\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

2014 May Olympiad, 1
Tags: number theory , digit
A natural number $N$ is [i]good [/i] if its digits are $1, 2$, or $3$ and all $2$-digit numbers are made up of digits located in consecutive positions of $N$ are distinct numbers. Is there a good number of $10$ digits? Of $11$ digits?

2008 Danube Mathematical Competition, 2
Tags: midpoint , circles , chord , concurrency , geometry
In a triangle $ABC$ let $A_1$ be the midpoint of side $BC$. Draw circles with centers $A, A1$ and radii $AA_1, BC$ respectively and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$. Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.

1985 All Soviet Union Mathematical Olympiad, 398
Tags: coloring , polygon
You should paint all the sides and diagonals of the regular $n$-gon so, that every pair of segments, having the common point, would be painted with different colours. How many colours will you require?

2019 Belarusian National Olympiad, 10.2
Tags: circumcircle , geometry
A point $P$ is chosen in the interior of the side $BC$ of triangle $ABC$. The points $D$ and $C$ are symmetric to $P$ with respect to the vertices $B$ and $C$, respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point $C_1$ and the ray $AC$ intersects the segment $XE$ at the point $B_1$. Prove that the lines $BC$ and $B_1C_1$ are parallel. [i](A. Voidelevich)[/i]

1978 AMC 12/AHSME, 4
Tags:
If $a = 1,~ b = 10, ~c = 100$, and $d = 1000$, then \[(a+ b+ c-d) + (a + b- c+ d) +(a-b+ c+d)+ (-a+ b+c+d) \] is equal to $\textbf{(A) }1111\qquad\textbf{(B) }2222\qquad\textbf{(C) }3333\qquad\textbf{(D) }1212\qquad \textbf{(E) }4242$

1990 Tournament Of Towns, (270) 4
Tags: geometry , parallelogram
The sides $AB$, $BC$, $CD$ and $DA$ of the quadrilateral $ABCD$ are respectively equal to the sides $A'B'$, $B'C'$, $C'D' $ and $D'A'$ of the quadrilateral $A'B'CD$' and it is known that $AB \parallel CD$ and $B'C' \parallel D'A'$. Prove that both quadrilaterals are parallelograms. (V Proizvolov, Moscow)

2013 IMAC Arhimede, 4
Tags: number theory , greatest common divisor , floor function
Let $p,n$ be positive integers, such that $p$ is prime and $p <n$. If $p$ divides $n + 1$ and $ \left(\left[\frac{n}{p}\right], (p-1)!\right) = 1$, then prove that $p\cdot \left[\frac{n}{p}\right]^2$ divides ${n \choose p} -\left[\frac{n}{p}\right]$ . (Here $[x]$ represents the integer part of the real number $x$.)

1997 AMC 12/AHSME, 1
Tags:
If $a$ and $b$ are digits for which \begin{tabular}{ccc} & 2 & a\\ $\times$ & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{tabular} Then $a+b =$ A. 3 B. 4 C. 7 D. 9 E. 12

2014 Harvard-MIT Mathematics Tournament, 4
Tags: geometry , geometric transformation , reflection , trigonometry , trig identities , law of sines , congruent triangles
In quadrilateral $ABCD$, $\angle DAC = 98^{\circ}$, $\angle DBC = 82^\circ$, $\angle BCD = 70^\circ$, and $BC = AD$. Find $\angle ACD.$

2013 ELMO Shortlist, 2
Tags: algebra , polynomial , geometry , 3d geometry , modular arithmetic , number theory
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

2020/2021 Tournament of Towns, P5
Tags: combinatorics
There are 101 coins in a circle, each weights 10g or 11g. Prove that there exists a coin such that the total weight of the $k{}$ coins to its left is equal to the total weight of the $k{}$ coins to its right where a) $k = 50$ and b) $k = 49$. [i]Alexandr Gribalko[/i]

2010 Romanian Masters In Mathematics, 1
Tags: induction , modular arithmetic , combinatorics
For a finite non empty set of primes $P$, let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$. (i) Show that $|P|\le m(P)$, with equality if and only if $\min(P)>|P|$. (ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$. (The number $|P|$ is the size of set $P$) [i]Dan Schwarz, Romania[/i]

2019 BAMO, 4
Tags: function , functional equation , algebra
Let $S$ be a finite set of nonzero real numbers, and let $f : S\to S$ be a function with the following property: for each $x \in S$, either $f ( f (x)) = x+ f (x)$ or $f ( f (x)) = \frac{x+ f (x)}{2}$. Prove that $f (x) = x$ for all $x \in S$.

2014 Contests, 3
Tags: geometry
From the point $P$ outside a circle $\omega$ with center $O$ draw the tangents $PA$ and $PB$ where $A$ and $B$ belong to $\omega$.In a random point $M$ in the chord $AB$ we draw the perpendicular to $OM$, which intersects $PA$ and $PB$ in $C$ and $D$. Prove that $M$ is the midpoint $CD$.

2025 Macedonian Balkan MO TST, 4
Tags: number theory , sum of squares , prime
Let $n$ be a positive integer. Prove that for every odd prime $p$ dividing $n^2 + n + 2$, there exist integers $a, b$ such that $p = a^2 + 7b^2$.

2017 CCA Math Bonanza, TB1
Tags:
Compute \[12^3+4\times56+7\times8+9.\] [i]2017 CCA Math Bonanza Tiebreaker Round #1[/i]

2010 Miklós Schweitzer, 8
Tags: topology , function , harmonic functions
Let $ D \subset \mathbb {R} ^ {2} $ be a finite Lebesgue measure of a connected open set and $ u: D \rightarrow \mathbb {R} $ a harmonic function. Show that it is either a constant $ u $ or for almost every $ p \in D $ $$ f ^ {\prime} (t) = (\operatorname {grad} u) (f (t)), \quad f (0) = p $$has no initial value problem(differentiable everywhere) solution to $ f:[0,\infty) \rightarrow D $.

2010 Contests, 4
Tags: function , inequalities
Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]

2013 Purple Comet Problems, 20
Tags: ratio , geometric series , algebra , complex number
Let $z$ be a complex number satisfying $(z+\tfrac{1}{z})(z+\tfrac{1}{z}+1)=1$. Evaluate $(3z^{100}+\tfrac{2}{z^{100}}+1)(z^{100}+\tfrac{2}{z^{100}}+3)$.

2009 Tournament Of Towns, 3
Tags: symmetry , number theory
Find all positive integers $a$ and $b$ such that $(a + b^2)(b + a^2) = 2^m$ for some integer $m.$ [i](6 points)[/i]

2010 Kazakhstan National Olympiad, 6
Tags: trigonometry , geometry
Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$ Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$

2024 Sharygin Geometry Olympiad, 9.7
Tags: geometry , geo
Let $P$ and $Q$ be arbitrary points on the side $BC$ of triangle ABC such that $BP = CQ$. The common points of segments $AP$ and $AQ$ with the incircle form a quadrilateral $XYZT$. Find the locus of common points of diagonals of such quadrilaterals.