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

1996 Austrian-Polish Competition, 8
Tags: algebra , polynomial , functional equation , functional
Show that there is no polynomial $P(x)$ of degree $998$ with real coefficients which satisfies $P(x^2 + 1) = P(x)^2 - 1$ for all $x$.

2017 ASDAN Math Tournament, 1
Tags: geometry test
What is the surface area of a cube with volume $64$?

2016 Denmark MO - Mohr Contest, 3
Tags: geometry , right angle , equal segments , area
Prove that all quadrilaterals $ABCD$ where $\angle B = \angle D = 90^o$, $|AB| = |BC|$ and $|AD| + |DC| = 1$, have the same area. [img]https://1.bp.blogspot.com/-55lHuAKYEtI/XzRzDdRGDPI/AAAAAAAAMUk/n8lYt3fzFaAB410PQI4nMEz7cSSrfHEgQCLcBGAsYHQ/s0/2016%2Bmohr%2Bp3.png[/img]

1976 AMC 12/AHSME, 22
Tags: analytic geometry
Given an equilateral triangle with side of length $s$, consider the locus of all points $\mathit{P}$ in the plane of the triangle such that the sum of the squares of the distances from $\mathit{P}$ to the vertices of the triangle is a fixed number $a$. This locus $\textbf{(A) }\text{is a circle if }a>s^2\qquad$ $\textbf{(B) }\text{contains only three points if }a=2s^2\text{ and is a circle if }a>2s^2\qquad$ $\textbf{(C) }\text{is a circle with positive radius only if }s^2<a<2s^2\qquad$ $\textbf{(D) }\text{contains only a finite number of points for any value of }a\qquad $ $\textbf{(E) }\text{is none of these}$

2005 Slovenia National Olympiad, Problem 2
Tags: sequence , algebra
Let $(a_n)$ be a geometrical progression with positive terms. Define $S_n=\log a_1+\log a_2+\ldots+\log a_n$. Prove that if $S_n=S_m$ for some $m\ne n$, then $S_{n+m}=0$.

2016 Mathematical Talent Reward Programme, MCQ: P 5
Tags: geometry , complex geometry , complex plane
$ABCD$ is a quadrilateral on complex plane whose four vertices satisfy $z^4+z^3+z^2+z+1=0$. Then $ABCD$ is a [list=1] [*] Rectangle [*] Rhombus [*] Isosceles Trapezium [*] Square [/list]

1996 Estonia National Olympiad, 2
Tags: geometry , trapezoid , equal segments , angle
Three sides of a trapezoid are equal, and a circle with the longer base as a diameter halves the two non-parallel sides. Find the angles of the trapezoid.

2009 Sharygin Geometry Olympiad, 18
Tags: incenter , inradius , rectangle , geometry
Given three parallel lines on the plane. Find the locus of incenters of triangles with vertices lying on these lines (a single vertex on each line).

1980 IMO Longlists, 21
Tags: symmetry , geometry
Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$

1988 IMO Shortlist, 9
Tags: number theory , divisibility , diophantine equation , perfect square , vieta jumping
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

1996 Tuymaada Olympiad, 6
Tags: algebra , sequence , limit of sequence , limit , trigonometric
Given the sequence $f_1(a)=sin(0,5\pi a)$ $f_2(a)=sin(0,5\pi (sin(0,5\pi a)))$ $...$ $f_n(a)=sin(0,5\pi (sin(...(sin(0,5\pi a))...)))$ , where $a$ is any real number. What limit aspire the members of this sequence as $n \to \infty$?

1988 Czech And Slovak Olympiad IIIA, 4
Tags: combinatorics , coloring , number theory , combinatorial number theory
Prove that each of the numbers $1, 2, 3, ..., 2^n$ can be written in one of two colors (red and blue) such that no non-constant $2n$-term arithmetic sequence chosen from these numbers is monochromatic .

1955 AMC 12/AHSME, 49
Tags: quadratic
The graphs of $ y\equal{}\frac{x^2\minus{}4}{x\minus{}2}$ and $ y\equal{}2x$ intersect in: $ \textbf{(A)}\ \text{1 point whose abscissa is 2} \qquad \textbf{(B)}\ \text{1 point whose abscissa is 0}\\ \textbf{(C)}\ \text{no points} \qquad \textbf{(D)}\ \text{two distinct points} \qquad \textbf{(E)}\ \text{two identical points}$

2017 Balkan MO Shortlist, N1
Tags: algebra , equation
Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$

1989 USAMO, 5
Tags: function , geometric series , algebra
Let $u$ and $v$ be real numbers such that \[ (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. \] Determine, with proof, which of the two numbers, $u$ or $v$, is larger.

2021 Canada National Olympiad, 5
Tags: combinatorics
Nina and Tadashi play the following game. Initially, a triple $(a, b, c)$ of nonnegative integers with $a+b+c=2021$ is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer $k$ and one of the three entries on the board; then the player increases the chosen entry by $k$ and decreases the other two entries by $k$. A player loses if, on their turn, some entry on the board becomes negative. Find the number of initial triples $(a, b, c)$ for which Tadashi has a winning strategy.

2018 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: algebra , equation , floor function , frac function
Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number

2020 LIMIT Category 2, 18
Tags: trigonometry , sum , limit , calculus
Evaluate the following sum: $n \choose 1$ $\sin (a) +$ $n \choose 2$ $\sin (2a) +...+$ $n \choose n$ $\sin (na)$ (A) $2^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (B) $2^n \sin^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$ (C) $2^n \sin^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (D) $2^n \cos^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$

2015 India IMO Training Camp, 3
Tags: trigonometry , inequalities
Prove that for any triangle $ABC$, the inequality $\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2)$ holds.

2017 Iran Team Selection Test, 5
Tags: circumcircle , geometry
In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively. Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other. [i]Proposed by Iman Maghsoudi[/i]

2006 ITAMO, 6
Tags: combinatorics
Alberto and Barbara play the following game. Initially, there are some piles of coins on a table. Each player in turn, starting with Albert, performs one of the two following ways: 1) take a coin from an arbitrary pile; 2) select a pile and divide it into two non-empty piles. The winner is the player who removes the last coin on the table. Determine which player has a winning strategy with respect to the initial state.

2011 IMO Shortlist, 4
Tags: function , algebra , functional equation
Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$. [i]Proposed by Bojan Bašić, Serbia[/i]

2020 JBMO Shortlist, 6
Tags: shortlist , number theory , diophantine
Are there any positive integers $m$ and $n$ satisfying the equation $m^3 = 9n^4 + 170n^2 + 289$ ?

2018 Switzerland - Final Round, 8
Tags: inequalities
Let $a,b,c,d,e$ be positive real numbers. Find the largest possible value for the expression $$\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}.$$

1983 Putnam, A5
Tags: number theory , algebra
Prove or disprove that there exists a positive real $u$ such that $\lfloor u^n\rfloor-n$ is an even integer for all positive integers $n$.