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

1997 Kurschak Competition, 1
Tags: geometry , linear algebra , matrix , number theory
Let $p>2$ be a prime number and let $L=\{0,1,\dots,p-1\}^2$. Prove that we can find $p$ points in $L$ with no three of them collinear.

2010 JBMO Shortlist, 2
Tags: algebra , number theory , equation
[b]Determine all four digit numbers [/b]$\bar{a}\bar{b}\bar{c}\bar{d}$[b] such that[/b] $$a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\bar{a}\bar{b}\bar{c}\bar{d}$$

2022 Turkey MO (2nd round), 2
Tags: number theory , function
For positive integers $k$ and $n$, we know $k \geq n!$. Prove that $ \phi (k) \geq (n-1)!$

2007 Federal Competition For Advanced Students, Part 1, 3
Tags: induction , combinatorics
Let $ M(n )\equal{}\{\minus{}1,\minus{}2,\ldots,\minus{}n\}$. For every non-empty subset of $ M(n )$ we consider the product of its elements. How big is the sum over all these products?

2011 JBMO Shortlist, 4
Tags: combinatorics , construct
In a group of $n$ people, each one had a different ball. They performed a sequence of swaps, in each swap, two people swapped the ball they had at that moment. Each pair of people performed at least one swap. In the end each person had the ball he/she had at the start. Find the least possible number of swaps, if: a) $n = 5$, b) $n = 6$.

2009 Iran Team Selection Test, 11
Tags: modular arithmetic , number theory
Let $n$ be a positive integer. Prove that \[ 3^{\dfrac{5^{2^n}-1}{2^{n+2}}} \equiv (-5)^{\dfrac{3^{2^n}-1}{2^{n+2}}} \pmod{2^{n+4}}. \]

2014 India National Olympiad, 6
Tags: induction , combinatorics
Let $n>1$ be a natural number. Let $U=\{1,2,...,n\}$, and define $A\Delta B$ to be the set of all those elements of $U$ which belong to exactly one of $A$ and $B$. Show that $|\mathcal{F}|\le 2^{n-1}$, where $\mathcal{F}$ is a collection of subsets of $U$ such that for any two distinct elements of $A,B$ of $\mathcal{F}$ we have $|A\Delta B|\ge 2$. Also find all such collections $\mathcal{F}$ for which the maximum is attained.

2015 Turkmenistan National Math Olympiad, 2
Tags: algebra
Find $ \lim_{n\to\infty}(\sum_{i=0}^{n}\frac{1}{n+i})$

1971 IMO Longlists, 46
Tags: combinatorics , modular arithmetic
Natural numbers from $1$ to $99$ (not necessarily distinct) are written on $99$ cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by $100$. Show that all the cards contain the same number.

2003 AMC 12-AHSME, 11
Tags: perimeter , area of a triangle , geometry
A square and an equilateral triangle have the same perimeter. Let $ A$ be the area of the circle circumscribed about the square and $ B$ be the area of the circle circumscribed about the triangle. Find $ A/B$. $ \textbf{(A)}\ \frac{9}{16} \qquad \textbf{(B)}\ \frac{3}{4} \qquad \textbf{(C)}\ \frac{27}{32} \qquad \textbf{(D)}\ \frac{3\sqrt{6}}{8} \qquad \textbf{(E)}\ 1$

2018 Math Prize for Girls Problems, 13
Tags:
A circle overlaps an equilateral triangle of side length $100\sqrt{3}$. The three chords in the circle formed by the three sides of the triangle have lengths 6, 36, and 60, respectively. What is the area of the circle?

2009 AMC 10, 19
Tags: search
Circle $ A$ has radius $ 100$. Circle $ B$ has an integer radius $ r<100$ and remains internally tangent to circle $ A$ as it rolls once around the circumference of circle $ A$. The two circles have the same points of tangency at the beginning and end of circle $ B$'s trip. How many possible values can $ r$ have? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 90$

2011 Switzerland - Final Round, 3
Tags: number theory
For positive integers $m$ and $n$, find the smalles possible value of $|2011^m-45^n|$. [i](Swiss Mathematical Olympiad, Final round, problem 3)[/i]

2007 Greece Junior Math Olympiad, 1
Tags: incenter , circumcircle , angle bisector , geometry
In a triangle $ABC$ with the incentre $I,$ the angle bisector $AD$ meets the circumcircle of triangle $BIC$ at point $N\neq I$. a) Express the angles of $\triangle BCN$ in terms of the angles of triangle $ABC$. b) Show that the circumcentre of triangle $BIC$ is at the intersection of $AI$ and the circumcentre of $ABC$.

2014 Moldova Team Selection Test, 2
Tags: function , algebra
Find all functions $f:R \rightarrow R$, which satisfy the equality for any $x,y \in R$: $f(xf(y)+y)+f(xy+x)=f(x+y)+2xy$,

2001 India IMO Training Camp, 2
Tags: function , polynomial , algebra
Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.

2023 CCA Math Bonanza, L2.2
Tags:
For a positive integer $n$ let $f(n)$ denote the number of ways to put $n$ objects into pairs if the only thing that matters is which object each object gets paired with. Find the sum of all $f(f(2k))$, where $k$ ranges from 1 to 2023. [i]Lightning 2.2[/i]

2007 China Team Selection Test, 1
Tags: polynomial , geometry , algebra
When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.

1993 APMO, 4
Tags: number theory
Determine all positive integers $n$ for which the equation \[ x^n + (2+x)^n + (2-x)^n = 0 \] has an integer as a solution.

1990 IMO, 1
Tags: function , number theory , algebra , functional equation
Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

2009 QEDMO 6th, 4
Tags: combination , sum , algebra
Let $a$ and $b$ be two real numbers and let $n$ be a nonnegative integer. Then prove that $$\sum_{k=0}^{n} {n \choose k} (a + k)^k (b - k)^{n-k} = \sum_{k=0}^{n} \frac{n!}{t!} (a + b)^t $$

2007 National Olympiad First Round, 22
Tags: modular arithmetic
Let $n$ and $m$ be integers such that $n\leq 2007 \leq m$ and $n^n \equiv -1 \equiv m^m \pmod 5$. What is the least possible value of $m-n$? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 8 $

2014 CentroAmerican, 3
Tags: modular arithmetic , number theory
A positive integer $n$ is [i]funny[/i] if for all positive divisors $d$ of $n$, $d+2$ is a prime number. Find all funny numbers with the largest possible number of divisors.

2012 India Regional Mathematical Olympiad, 1
Tags: ratio , midpoint , geometry
Let $ABC$ be a triangle and $D$ be a point on the segment $BC$ such that $DC = 2BD$. Let $E$ be the mid-point of $AC$. Let $AD$ and $BE$ intersect in $P$. Determine the ratios $BP:PE$ and $AP:PD$.

2013 Canadian Mathematical Olympiad Qualification Repechage, 5
Tags: number theory
For each positive integer $k$, let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$. Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$. Determine the number of digits in $n$.