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 Iran MO (3rd Round), 3
Tags: algebra , functional equation
Find all functions $f : \mathbb R \to \mathbb R$ such that for all $x, y,$ \[f(f(x) + y) = f(x^2 - y) + 4f(x)y.\]

2003 Iran MO (3rd Round), 19
Tags: number theory
An integer $ n$ is called a good number if and only if $ |n|$ is not square of another intger. Find all integers $ m$ such that they can be written in infinitely many ways as sum of three different good numbers and product of these three numbers is square of an odd number.

2023 ELMO Shortlist, C5
Tags: combinatorics
Define the [i]mexth[/i] of \(k\) sets as the \(k\)th smallest positive integer that none of them contain, if it exists. Does there exist a family \(\mathcal F\) of sets of positive integers such that [list] [*]for any nonempty finite subset \(\mathcal G\) of \(\mathcal F\), the mexth of \(\mathcal G\) exists, and [*]for any positive integer \(n\), there is exactly one nonempty finite subset \(\mathcal G\) of \(\mathcal F\) such that \(n\) is the mexth of \(\mathcal G\). [/list] [i]Proposed by Espen Slettnes[/i]

1976 Euclid, 2
Tags: geometric sequence , arithmetic sequence
Source: 1976 Euclid Part B Problem 2 ----- Given that $x$, $y$, and $2$ are in geometric progression, and that $x^{-1}$, $y^{-1}$, and $9x^{-2}$ are in are in arithmetic progression, then find the numerical value of $xy$.

1994 Putnam, 5
Tags: limit
Let $(r_n)_{n\ge 0}$ be a sequence of positive real numbers such that $\lim_{n\to \infty} r_n = 0$. Let $S$ be the set of numbers representable as a sum \[ r_{i_1} + r_{i_2} +\cdots + r_{i_{1994}} ,\] with $i_1 < i_2 < \cdots< i_{1994}.$ Show that every nonempty interval $(a, b)$ contains a nonempty subinterval $(c, d)$ that does not intersect $S$.

1974 IMO Longlists, 25
Tags: trigonometry , limit , algebra
Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$ \[x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).\] Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$

2007 Princeton University Math Competition, 7
Tags: geometry
$A, B, C$, and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, what is the area of $ABCD$?

2024 Israel TST, P3
Tags: combinatorics
Let $0<c<1$ and $n$ a positive integer. Alice and Bob are playing a game. Bob writes $n$ integers on the board, not all equal. On a player's turn, they erase two numbers from the board and write their arithmetic mean instead. Alice starts and performs at most $cn$ moves. After her, Bob makes moves until there are only two numbers left on the board. Alice wins if these two numbers are different, and otherwise, Bob wins. For which values of $c$ does Alice win for all large enough $n$?

2022 Israel TST, 2
Tags: function , algebra , polynomial
Let $f: \mathbb{Z}^2\to \mathbb{R}$ be a function. It is known that for any integer $C$ the four functions of $x$ \[f(x,C), f(C,x), f(x,x+C), f(x, C-x)\] are polynomials of degree at most $100$. Prove that $f$ is equal to a polynomial in two variables and find its maximal possible degree. [i]Remark: The degree of a bivariate polynomial $P(x,y)$ is defined as the maximal value of $i+j$ over all monomials $x^iy^j$ appearing in $P$ with a non-zero coefficient.[/i]

2016 AMC 8, 21
Tags: probability
A box contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn? $\textbf{(A) }\frac{3}{10}\qquad\textbf{(B) }\frac{2}{5}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{3}{5}\qquad \textbf{(E) }\frac{7}{10}$

2006 Moldova Team Selection Test, 2
Tags: analytic geometry , trigonometry , geometry
Consider a right-angled triangle $ABC$ with the hypothenuse $AB=1$. The bisector of $\angle{ACB}$ cuts the medians $BE$ and $AF$ at $P$ and $M$, respectively. If ${AF}\cap{BE}=\{P\}$, determine the maximum value of the area of $\triangle{MNP}$.

1983 IMO Longlists, 2
Tags: combinatorics
Seventeen cities are served by four airlines. It is noted that there is direct service (without stops) between any two cities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.

2019 Stanford Mathematics Tournament, 7
Tags: geometry
Let $G$ be the centroid of triangle $ABC$ with $AB = 9$, $BC = 10,$ and $AC = 17$. Denote $D$ as the midpoint of $BC$. A line through $G$ parallel to $BC$ intersects $AB$ at $M$ and $AC$ at $N$. If $BG$ intersects $CM$ at $E$ and $CG$ intersects $BN$ at $F$, compute the area of triangle $DEF$.

Novosibirsk Oral Geo Oly VIII, 2020.2
Tags: geometry , perimeter , chessboard , combinatorics
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?

2003 Romania National Olympiad, 2
Tags: function , real analysis , integration , definite integral , titu andreescu
Let be an odd natural number $ n\ge 3. $ Find all continuous functions $ f:[0,1]\longrightarrow\mathbb{R} $ that satisfy the following equalities. $$ \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n,\quad\forall k\in\{ 1,2,\ldots ,n-1\} $$ [i]Titu Andreescu[/i]

2005 Tournament of Towns, 5
Tags: geometry , 3d geometry
Prove that if a regular icosahedron and a regular dodecahedron have a common circumsphere, then they have a common insphere. [i](7 points)[/i]

2002 Tournament Of Towns, 3
Tags: modular arithmetic , algebra , polynomial , quadratic , number theory
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.

2012 France Team Selection Test, 1
Tags: function , algebra
Let $k>1$ be an integer. A function $f:\mathbb{N^*}\to\mathbb{N^*}$ is called $k$-[i]tastrophic[/i] when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$: \[f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)\] For which $k$ does there exist a $k$-tastrophic function?

Russian TST 2016, P2
Tags: algebra , inequalities
Prove that \[1+\frac{2^1}{1-2^1}+\frac{2^2}{(1-2^1)(1-2^2)}+\cdots+\frac{2^{2016}}{(1-2^1)\cdots(1-2^{2016})}>0.\]

Kyiv City MO 1984-93 - geometry, 1993.8.3
Tags: geometry , angle bisector , equal segments
In the triangle $ABC$, $\angle .ACB = 60^o$, and the bisectors $AA_1$ and $BB_1$ intersect at the point $M$. Prove that $MB_1 = MA_1$.

2021 Science ON grade VII, 1
Tags: number theory , set
Supoose $A$ is a set of integers which contains all integers that can be written as $2^a-2^b$, $a,b\in \mathbb{Z}_{\ge 1}$ and also has the property that $a+b\in A$ whenever $a,b\in A$. Prove that if $A$ contains at least an odd number, then $A=\mathbb{Z}$. [i] (Andrei Bâra)[/i]

2014 Online Math Open Problems, 6
Tags: modular arithmetic , least common multiple
Let $L_n$ be the least common multiple of the integers $1,2,\dots,n$. For example, $L_{10} = 2{,}520$ and $L_{30} = 2{,}329{,}089{,}562{,}800$. Find the remainder when $L_{31}$ is divided by $100{,}000$. [i]Proposed by Evan Chen[/i]

2019 IOM, 3
Tags: circumcircle , incenter , geometry
In a non-equilateral triangle $ABC$ point $I$ is the incenter and point $O$ is the circumcenter. A line $s$ through $I$ is perpendicular to $IO$. Line $\ell$ symmetric to like $BC$ with respect to $s$ meets the segments $AB$ and $AC$ at points $K$ and $L$, respectively ($K$ and $L$ are different from $A$). Prove that the circumcenter of triangle $AKL$ lies on the line $IO$. [i]Dušan Djukić[/i]

2018 PUMaC Geometry B, 8
Tags: geometry , parallelogram , angle bisector
Let $ABCD$ be a parallelogram such that $AB = 35$ and $BC = 28$. Suppose that $BD \perp BC$. Let $\ell_1$ be the reflection of $AC$ across the angle bisector of $\angle BAD$, and let $\ell_2$ be the line through $B$ perpendicular to $CD$. $\ell_1$ and $\ell_2$ intersect at a point $P$. If $PD$ can be expressed in simplest form as $\frac{m}{n}$, find $m + n$.

1999 Harvard-MIT Mathematics Tournament, 1
Tags: calculus , function , integration
Find all twice differentiable functions $f(x)$ such that $f^{\prime \prime}(x)=0$, $f(0)=19$, and $f(1)=99$.