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

2015 Balkan MO Shortlist, N4

Find all pairs of positive integers $(x,y)$ with the following property: If $a,b$ are relative prime and positive divisors of $ x^3 + y^3$, then $a+b - 1$ is divisor of $x^3+y^3$. (Cyprus)

2013 India IMO Training Camp, 3

We define an operation $\oplus$ on the set $\{0, 1\}$ by \[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\] For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$. For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.

1987 Bundeswettbewerb Mathematik, 3

Prove that for every convex polygon, we can choose three of its consecutive vertices, such that the circle, defined by them, covers the the entire polygon. (proposed by J. Tabov)

1994 All-Russian Olympiad Regional Round, 9.7

Find all prime numbers $p,q,r,s$ such that their sum is a prime number and $p^2+qs$ and $p^2 +qr$ are squares of integers.

2024 MMATHS, 10

Tags:
Find the sum of all prime numbers $p$ such that $\binom{20242024p}{p}\equiv 2024\pmod{p}.$

1989 IMO Longlists, 60

A family of sets $ A_1, A_2, \ldots ,A_n$ has the following properties: [b](i)[/b] Each $ A_i$ contains 30 elements. [b](ii)[/b] $ A_i \cap A_j$ contains exactly one element for all $ i, j, 1 \leq i < j \leq n.$ Determine the largest possible $ n$ if the intersection of all these sets is empty.

1969 AMC 12/AHSME, 21

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If the graph of $x^2+y^2=m$ is tangent to that of $x+y=\sqrt{2m}$, then: $\textbf{(A) }m\text{ must equal }\tfrac12\qquad \textbf{(B) }m\text{ must equal }\tfrac1{\sqrt2}\qquad$ $\textbf{(C) }m\text{ must equal }\sqrt2\qquad \textbf{(D) }m\text{ must equal }2\qquad$ $\textbf{(E) }m\text{ may be any nonnegative real number}$

JBMO Geometry Collection, 2007

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.

2006 District Olympiad, 1

Let $x>0$ be a real number and $A$ a square $2\times 2$ matrix with real entries such that $\det {(A^2+xI_2 )} = 0$. Prove that $\det{ (A^2+A+xI_2) } = x$.

2003 Rioplatense Mathematical Olympiad, Level 3, 3

An $8\times 8$ chessboard is to be tiled (i.e., completely covered without overlapping) with pieces of the following shapes: [asy] unitsize(.6cm); draw(unitsquare,linewidth(1)); draw(shift(1,0)*unitsquare,linewidth(1)); draw(shift(2,0)*unitsquare,linewidth(1)); label("\footnotesize $1\times 3$ rectangle",(1.5,0),S); draw(shift(8,1)*unitsquare,linewidth(1)); draw(shift(9,1)*unitsquare,linewidth(1)); draw(shift(10,1)*unitsquare,linewidth(1)); draw(shift(9,0)*unitsquare,linewidth(1)); label("\footnotesize T-shaped tetromino",(9.5,0),S); [/asy] The $1\times 3$ rectangle covers exactly three squares of the chessboard, and the T-shaped tetromino covers exactly four squares of the chessboard. [list](a) What is the maximum number of pieces that can be used? (b) How many ways are there to tile the chessboard using this maximum number of pieces?[/list]

2003 Gheorghe Vranceanu, 1

Find all nonnegative numbers $ n $ which have the property that $ a_{2}\neq 9, $ where $ \sum_{i=1}^{\infty } a_i10^{-i} $ is the decimal representation of the fractional part of $ \sqrt{n(n+1)} . $

2003 AIME Problems, 6

In triangle $ABC,$ $AB=13,$ $BC=14,$ $AC=15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area if the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$

2024 Kyiv City MO Round 2, Problem 2

Mykhailo wants to arrange all positive integers from $1$ to $2024$ in a circle so that each number is used exactly once and for any three consecutive numbers $a, b, c$ the number $a + c$ is divisible by $b + 1$. Can he do it? [i]Proposed by Fedir Yudin[/i]

2015 India National Olympiad, 3

Find all real functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2+yf(x))=xf(x+y)$.

LMT Team Rounds 2010-20, 2020.S20

Tags:
Let $c_1<c_2<c_3$ be the three smallest positive integer values of $c$ such that the distance between the parabola $y=x^2+2020$ and the line $y=cx$ is a rational multiple of $\sqrt{2}$. Compute $c_1+c_2+c_3$.

2011 Belarus Team Selection Test, 2

Two different points $X,Y$ are marked on the side $AB$ of a triangle $ABC$ so that $\frac{AX \cdot BX}{CX^2}=\frac{AY \cdot BY}{CY^2}$ . Prove that $\angle ACX=\angle BCY$. I.Zhuk

2007 ISI B.Math Entrance Exam, 7

Let $ 0\leq \theta\leq \frac{\pi}{2}$ . Prove that $\sin \theta \geq \frac{2\theta}{\pi}$.

2011 Romania Team Selection Test, 3

Given a positive integer number $n$, determine the maximum number of edges a simple graph on $n$ vertices may have such that it contain no cycles of even length.

2021 Alibaba Global Math Competition, 8

Let $f(z)$ be a holomorphic function in $\{\vert z\vert \le R\}$ ($0<R<\infty$). Define \[M(r,f)=\max_{\vert z\vert=r} \vert f(z)\vert, \quad A(r,f)=\max_{\vert z\vert=r} \text{Re}\{f(z)\}.\] Show that \[M(r,f) \le \frac{2r}{R-r}A(R,f)+\frac{R+r}{R-r} \vert f(0)\vert, \quad \forall 0 \le r<R.\]

2014 Mexico National Olympiad, 4

Problem 4 Let $ABCD$ be a rectangle with diagonals $AC$ and $BD$. Let $E$ be the intersection of the bisector of $\angle CAD$ with segment $CD$, $F$ on $CD$ such that $E$ is midpoint of $DF$, and $G$ on $BC$ such that $BG = AC$ (with $C$ between $B$ and $G$). Prove that the circumference through $D$, $F$ and $G$ is tangent to $BG$.

2002 Moldova National Olympiad, 1

Consider the real numbers $ a\ne 0,b,c$ such that the function $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ satisfies $ |f(x)|\le 1$ for all $ x\in [0,1]$. Find the greatest possible value of $ |a| \plus{} |b| \plus{} |c|$.

2013 National Olympiad First Round, 22

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For how many integers $0\leq n < 2013$, is $n^4+2n^3-20n^2+2n-21$ divisible by $2013$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ \text{None of above} $

2014 AMC 12/AHSME, 3

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Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip? $ \textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7} $

2022 Purple Comet Problems, 8

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Find the number of divisors of $20^{22}$ that are perfect squares.

2015 District Olympiad, 4

[b]a)[/b] Show that the three last digits of $ 1038^2 $ are equal with $ 4. $ [b]b)[/b] Show that there are infinitely many perfect squares whose last three digits are equal with $ 4. $ [b]c)[/b] Prove that there is no perfect square whose last four digits are equal to $ 4. $