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

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.

2012 Grigore Moisil Intercounty, 2
Tags: real analysis
Let $ \left( x_n \right)_{n\ge 0} $ be a sequence of positive real numbers with $ x_0=1 $ and defined recursively: $$ x_{n+1}=x_n+\frac{x_0}{x_1+x_2+\cdots +x_n} $$ [b]a)[/b] Show that $ \lim_{n\to\infty } x_n=\infty . $ [b]b)[/b] Calculate $ \lim_{n\to\infty }\frac{x_n}{\sqrt{\ln n}} . $ [i]Ovidiu Furdui[/i]

1997 Croatia National Olympiad, Problem 4
Tags: triangle , geometry , parallelogram
On the sides of a triangle $ABC$ are constructed similar triangles $ABD,BCE,CAF$ with $k=AD/DB=BE/EC=CF/FA$ and $\alpha=\angle ADB=\angle BEC=\angle CFA$. Prove that the midpoints of the segments $AC,BC,CD$ and $EF$ form a parallelogram with an angle $\alpha$ and two sides whose ratio is $k$.

1985 Traian Lălescu, 1.2
Tags: geometry , perimeter , optimization , pure geometry
For the triangles of fixed perimeter, find the maximum value of the product of the radius of the incircle with the radius of the excircle.

1985 Dutch Mathematical Olympiad, 4
Tags: geometry
A convex hexagon $ ABCDEF$ is such that each of the diagonals $ AD,BE,CF$ divides the hexagon into two parts of equal area. Prove that these three diagonals are concurrent.

2016 Canadian Mathematical Olympiad Qualification, 6
Tags: number theory , greatest common divisor
Determine all ordered triples of positive integers $(x, y, z)$ such that $\gcd(x+y, y+z, z+x) > \gcd(x, y, z)$.