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

2020 Baltic Way, 4
Tags: functional equation , algebra
Find all functions $f:\mathbb{R} \to \mathbb{R}$ so that \[f(f(x)+x+y) = f(x+y) + y f(y)\] for all real numbers $x, y$.

2003 Alexandru Myller, 3
Tags: inequalities , real analysis
Let be a nonnegative integer $ n. $ Prove that there exists an increasing and finite sequence of positive real numbers, $ \left( a_k \right)_{0\le k\le n} , $ that satisfy the equality $$ a_0/0! +a_1/1! +a_2/2! +\cdots +a_n/n! =1/n! , $$ and the inequality $$ a_0+a_1+a_2+\cdots +a_n<\frac{3}{2^n} . $$ [i]Dorin Andrica[/i]

2018 Czech and Slovak Olympiad III A, 5
Tags: geometry
Let $ABCD$ an isosceles trapezoid with the longer base $AB$. Denote $I$ the incenter of $\Delta ABC$ and $J$ the excenter relative to the vertex $C$ of $\Delta ACD$. Show that the lines $IJ$ and $AB$ are parallel.

2023 Poland - Second Round, 5
Tags: geometry
Given is a triangle $ABC$ with $AB>AC$. Its incircle touches $AB, AC$ at $D, E$, respectively. Let $CD$ meet the incircle at $K$ and $L$ is the foot of the perpendicular from $A$ to $CK$. If $M$ is the midpoint of $DE$ and $H$ is the orthocenter of $\triangle KML$, prove that $\angle AHK=90^{o}$. [i]Proposed by Dominik Burek[/i]

2012 China Western Mathematical Olympiad, 1
Tags: modular arithmetic , prime number , number theory
Find the smallest positive integer $m$ satisfying the following condition: for all prime numbers $p$ such that $p>3$,have $105|9^{ p^2}-29^p+m.$ (September 28, 2012, Hohhot)

2020 Yasinsky Geometry Olympiad, 4
Tags: geometry , orthocenter , tangent circle
The altitudes of the acute-angled triangle $ABC$ intersect at the point $H$. On the segments $BH$ and $CH$, the points $B_1$ and $C_1$ are marked, respectively, so that $B_1C_1 \parallel BC$. It turned out that the center of the circle $\omega$ circumscribed around the triangle $B_1HC_1$ lies on the line $BC$. Prove that the circle $\Gamma$, which is circumscribed around the triangle $ABC$, is tangent to the circle $\omega$ .

2010 Junior Balkan Team Selection Tests - Romania, 3
Tags: factorial , remainder , number theory
Determine the integers $n, n \ge 2$, with the property that the numbers $1! , 2 ! , 3 ! , ..., (n- 1)!$ give different remainders when dividing by $n $.

2007 Sharygin Geometry Olympiad, 6
Tags: geometry , symmetry , axes , polygon , polyhedron
a) What can be the number of symmetry axes of a checked polygon, that is, of a polygon whose sides lie on lines of a list of checked paper? (Indicate all possible values.) b) What can be the number of symmetry axes of a checked polyhedron, that is, of a polyhedron consisting of equal cubes which border one to another by plane facets?

2008 Balkan MO Shortlist, N5
Tags:
Let $(a_n)$ be a sequence with $a_1=0$ and $a_{n+1}=2+a_n$ for odd $n$ and $a_{n+1}=2a_n$ for even $n$. Prove that for each prime $p>3$, the number \begin{align*} b=\frac{2^{2p}-1}{3} \mid a_n \end{align*} for infinitely many values of $n$

2006 Polish MO Finals, 1
Tags: algebra
Solve in reals: \begin{eqnarray*}a^2=b^3+c^3 \\ b^2=c^3+d^3 \\ c^2=d^3+e^3 \\ d^2=e^3+a^3 \\ e^2=a^3+b^3 \end{eqnarray*}

1992 Poland - First Round, 10
Tags: geometry , 3d geometry
Let $C$ be a cube and let $f: C \longrightarrow C$ be a surjection with $|PQ| \geq |f(P)f(Q)|$ for all $P,Q \in C$. Prove that $f$ is an isometry.

2019 Estonia Team Selection Test, 3
Tags: algebra , functional equation , functional
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.

2004 Gheorghe Vranceanu, 3
Tags: function , equation , inversability , algebra
Consider the function $ f:(-\infty,1]\longrightarrow\mathbb{R} $ defined as $$ f(x)=\left\{ \begin{matrix} \frac{5}{2} +2^x-\frac{1}{2^x} ,& \quad x<-1 \\ 3^{\sqrt{1-x^2}} ,& \quad x\in [-1,1] \end{matrix} \right. . $$ [b]a)[/b] For a fixed parameter, find the roots of $ f-m. $ [b]b)[/b] Study the inversability of the restrictions of $ f $ to $ (-\infty,-1] $ and $ [-1,1] $ and find the inverses of these that admit them. [i]D. Zaharia[/i]

2010 Math Prize For Girls Problems, 8
Tags: number theory , prime factorization
When Meena turned 16 years old, her parents gave her a cake with $n$ candles, where $n$ has exactly 16 different positive integer divisors. What is the smallest possible value of $n$?

Swiss NMO - geometry, 2015.4
Tags: geometry , construction , circles
Given a circle $k$ and two points $A$ and $B$ outside the circle. Specify how to can construct a circle with a compass and ruler, so that $A$ and $B$ lie on that circle and that circle is tangent to $k$.

2007 ITest, 20
Tags: modular arithmetic
Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$. $\textbf{(A) }1\hspace{14em}\textbf{(B) }2\hspace{14em}\textbf{(C) }3$ $\textbf{(D) }4\hspace{14em}\textbf{(E) }5\hspace{14em}\textbf{(F) }6$ $\textbf{(G) }7\hspace{14em}\textbf{(H) }8\hspace{14em}\textbf{(I) }9$ $\textbf{(J) }10\hspace{13.7em}\textbf{(K) }11\hspace{13.5em}\textbf{(L) }12$ $\textbf{(M) }13\hspace{13.3em}\textbf{(N) }14\hspace{13.4em}\textbf{(O) }15$ $\textbf{(P) }16\hspace{13.6em}\textbf{(Q) }55\hspace{13.4em}\textbf{(R) }63$ $\textbf{(S) }64\hspace{13.7em}\textbf{(T) }2007$

2020 Romania EGMO TST, P2
Tags: function , algebra
Suppose a function $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x+y)|\geqslant|f(x)+f(y)|$ for all real numbers $x$ and $y$. Prove that equality always holds. Is the conclusion valid if the sign of the inequality is reversed?

1958 AMC 12/AHSME, 16
Tags: geometry
The area of a circle inscribed in a regular hexagon is $ 100\pi$. The area of hexagon is: $ \textbf{(A)}\ 600\qquad \textbf{(B)}\ 300\qquad \textbf{(C)}\ 200\sqrt{2}\qquad \textbf{(D)}\ 200\sqrt{3}\qquad \textbf{(E)}\ 120\sqrt{5}$

1975 AMC 12/AHSME, 11
Tags:
Let $ P$ be an interior point of circle $ K$ other than the center of $ K$. Form all chords of $ K$ which pass through $ P$, and determine their midpoints. The locus of these midpoints is $ \textbf{(A)}\ \text{a circle with one point deleted} \qquad$ $ \textbf{(B)}\ \text{a circle if the distance from } P \text{ to the center of } K \text{ is less than}$ $ \text{one half the radius of } K \text{; otherwise a circular arc of less than}$ $ 360^{\circ}\qquad$ $ \textbf{(C)}\ \text{a semicircle with one point deleted} \qquad$ $ \textbf{(D)}\ \text{a semicircle} \qquad$ $ \textbf{(E)}\ \text{a circle}$

2000 Moldova National Olympiad, Problem 8
Tags: geometry , 3d geometry
A rectangular parallelepiped has dimensions $a,b,c$ that satisfy the relation $3a+4b+10c=500$, and the length of the main diagonal $20\sqrt5$. Find the volume and the total area of the surface of the parallelepiped.

2018 Harvard-MIT Mathematics Tournament, 9
Tags:
Evan has a simple graph with $v$ vertices and $e$ edges. Show that he can delete at least $\frac{e-v+1}{2}$ edges so that each vertex still has at least half of its original degree.

2002 IMO Shortlist, 2
Tags: combinatorics , tiling , dissection
For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

2010 Germany Team Selection Test, 3
Tags: algebra , polynomial , number theory , permutation
Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

2015 ASDAN Math Tournament, 3
Tags:
For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tournament can generate.

2014 Brazil Team Selection Test, 2
Tags: number theory , prime divisor , polynomial
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.