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

1998 Baltic Way, 6
Tags: polynomial , algebra
Let $P$ be a polynomial of degree $6$ and let $a,b$ be real numbers such that $0<a<b$. Suppose that $P(a)=P(-a),P(b)=P(-b),P'(0)=0$. Prove that $P(x)=P(-x)$ for all real $x$.

1999 India Regional Mathematical Olympiad, 7
Tags: quadratic , algebra , polynomial
Find the number of quadratic polynomials $ax^2 + bx +c$ which satisfy the following: (a) $a,b,c$ are distinct; (b) $a,b,c \in \{ 1,2,3,\cdots 1999 \}$; (c) $x+1$ divides $ax^2 + bx+c$.

2013-2014 SDML (High School), 6
Tags:
The total number of edges in two regular polygons is $2014$, and the total number of diagonals is $1,014,053$. How many edges does the polygon with the smaller number [of] edges have?

2016 Macedonia JBMO TST, 4
Tags: algebra , inequality
Let $x$, $y$, and $z$ be positive real numbers. Prove that $\sqrt {\frac {xy}{x^2 + y^2 + 2z^2}} + \sqrt {\frac {yz}{y^2 + z^2 + 2x^2}}+\sqrt {\frac {zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}$. When does equality hold?

1961 AMC 12/AHSME, 21
Tags: geometry
Medians $AD$ and and $CE$ of triangle $ABC$ intersect in $M$. The midpoint of $AE$ is $N$. Let the area of triangle $MNE$ be $k$ times the area of triangle $ABC$. Then $k$ equals: ${{ \textbf{(A)}\ \frac{1}{6} \qquad\textbf{(B)}\ \frac{1}{8} \qquad\textbf{(C)}\ \frac{1}{9} \qquad\textbf{(D)}\ \frac{1}{12} }\qquad\textbf{(E)}\ \frac{1}{16} } $

2012 Princeton University Math Competition, A6
Tags: combinatorics
Two white pentagonal pyramids, with side lengths all the same, are glued to each other at their regular pentagon bases. Some of the resulting $10$ faces are colored black. How many rotationally distinguishable colorings may result?

2022-2023 OMMC, 22
Tags: abstract algebra
Find the number of ordered pairs of integers $(x, y)$ with $0 \le x, y \le 40$ where $$\frac{x^2-xy^2+1}{41}$$ is an integer.

2013 Dutch Mathematical Olympiad, 2
Tags: algebra , system of equations
Find all triples $(x, y, z)$ of real numbers satisfying: $x + y - z = -1$ , $x^2 - y^2 + z^2 = 1$ and $- x^3 + y^3 + z^3 = -1$

2013 Philippine MO, 1
Tags:
1. Determine, with proof, the least positive integer $n$ for which there exist $n$ distinct positive integers, $\left(1-\frac{1}{x_1}\right)\left(1-\frac{1}{x_2}\right)......\left(1-\frac{1}{x_n}\right)=\frac{15}{2013}$

2023 China National Olympiad, 2
Tags: geometry
Let $\triangle ABC$ be an equilateral triangle of side length 1. Let $D,E,F$ be points on $BC,AC,AB$ respectively, such that $\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}$. Let $X,Y,Z$ be on lines $BC,CA,AB$ respectively, such that $XY\perp DE, YZ\perp EF, ZX\perp FD$. Find all possible values of $\frac{1}{[DEF]} + \frac{1}{[XYZ]}$.

2002 Canada National Olympiad, 5
Tags: function , modular arithmetic , functional equation , algebra
Let $\mathbb N = \{0,1,2,\ldots\}$. Determine all functions $f: \mathbb N \to \mathbb N$ such that \[ xf(y) + yf(x) = (x+y) f(x^2+y^2) \] for all $x$ and $y$ in $\mathbb N$.

2013 Purple Comet Problems, 11
Tags: percent
After Jennifer walked $r$ percent of the way from her home to the store, she turned around and walked home, got on her bicycle, and bicycled to the store and back home. Jennifer bicycles two and a half times faster than she walks. Find the largest value of $r$ so that returning home for her bicycle was not slower than her walking all the way to and from the store without her bicycle.

1985 IMO Longlists, 31
Tags: ellipse , concurrency , conic , geometry
Let $E_1, E_2$, and $E_3$ be three mutually intersecting ellipses, all in the same plane. Their foci are respectively $F_2, F_3; F_3, F_1$; and $F_1, F_2$. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.

2007 Stanford Mathematics Tournament, 7
Tags:
Find the minimum value of $xy+x+y+\frac{1}{xy}+\frac{1}{x}+\frac{1}{y}$ for $x, y>0$ real.

1992 Tournament Of Towns, (349) 1
Tags: geometry , 3d geometry , combinatorics , combinatorial geometry
We are given a cube with edges of length $n$ cm. At our disposal is a long piece of insulating tape of width $1$ cm. It is required to stick this tape to the cube. The tape may freely cross an edge of the cube on to a different face but it must always be parallel to an edge of the cube. It may not overhang the edge of a face or cross over a vertex. How many pieces of the tape are necessary in order to completely cover the cube? (You may assume that $n$ is an integer.) (A Spivak)

2004 Baltic Way, 17
Tags: rectangle , geometry , inequalities
Consider a rectangle with sidelengths 3 and 4, pick an arbitrary inner point on each side of this rectangle. Let $x, y, z$ and $u$ denote the side lengths of the quadrilateral spanned by these four points. Prove that $25 \leq x^2+y^2+z^2+u^2 \leq 50$.

1998 Bulgaria National Olympiad, 3
Tags: trigonometry , complex number , geometry
On the sides of a non-obtuse triangle $ABC$ a square, a regular $n$-gon and a regular $m$-gon ($m$,$n > 5$) are constructed externally, so that their centers are vertices of a regular triangle. Prove that $m = n = 6$ and find the angles of $\triangle ABC$.

2018 Polish MO Finals, 5
Tags: radical axis , projective geometry , geometry
An acute triangle $ABC$ in which $AB<AC$ is given. Points $E$ and $F$ are feet of its heights from $B$ and $C$, respectively. The line tangent in point $A$ to the circle escribed on $ABC$ crosses $BC$ at $P$. The line parallel to $BC$ that goes through point $A$ crosses $EF$ at $Q$. Prove $PQ$ is perpendicular to the median from $A$ of triangle $ABC$.

2007 AMC 12/AHSME, 25
Tags:
Call a set of integers [i]spacy[/i] if it contains no more than one out of any three consecutive integers. How many subsets of $ \{1,2,3,\ldots,12\},$ including the empty set, are spacy? $ \textbf{(A)}\ 121 \qquad \textbf{(B)}\ 123 \qquad \textbf{(C)}\ 125 \qquad \textbf{(D)}\ 127 \qquad \textbf{(E)}\ 129$

2014 Contests, 2
Tags:
Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$?

2009 Costa Rica - Final Round, 2
Tags: induction , number theory
Prove that for that for every positive integer $ n$, the smallest integer that is greater than $ (\sqrt {3} \plus{} 1)^{2n}$ is divisible by $ 2^{n \plus{} 1}$.

2013 Argentina National Olympiad, 6
Tags: number theory
A positive integer $n$ is called [i]pretty[/i] if there exists two divisors $d_1,d_2$ of $n$ $(1\leq d_1,d_2\leq n)$ such that $d_2-d_1=d$ for each divisor $d$ of $n$ (where $1<d<n$). Find the smallest pretty number larger than $401$ that is a multiple of $401$.

2009 Romania National Olympiad, 3
Tags: function , find all functions , algebra
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation $$ f\left( x^3+y^3 \right) =xf\left( y^2 \right) + yf\left( x^2 \right) , $$ for all real numbers $ x,y. $

1966 IMO Longlists, 24
Tags: graph theory , vertex degree , combinatorics
There are $n\geq 2$ people at a meeting. Show that there exist two people at the meeting who have the same number of friends among the persons at the meeting. (It is assumed that if $A$ is a friend of $B,$ then $B$ is a friend of $A;$ moreover, nobody is his own friend.)

2004 Greece Junior Math Olympiad, 1
Tags: number theory
The numbers $203$ and $298$ divided with the positive integer $x$ give both remainder $13$. Which are the possible values of $x$ ?