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

2004 Indonesia MO, 3
Tags:
In how many ways can we change the sign $ *$ with $ \plus{}$ or $ \minus{}$, such that the following equation is true? \[ 1 *2*3*4*5*6*7*8*9*10\equal{}29\]

1993 Austrian-Polish Competition, 8
Tags: polynomial , functional , functional equation , algebra
Determine all real polynomials $P(z)$ for which there exists a unique real polynomial $Q(x)$ satisfying the conditions $Q(0)= 0$, $x + Q(y + P(x))= y + Q(x + P(y))$ for all $x,y \in R$.

2020 Purple Comet Problems, 15
Tags: geometry
Daniel had a string that formed the perimeter of a square with area $98$. Daniel cut the string into two pieces. With one piece he formed the perimeter of a rectangle whose width and length are in the ratio $2 : 3$. With the other piece he formed the perimeter of a rectangle whose width and length are in the ratio $3 : 8$. The two rectangles that Daniel formed have the same area, and each of those areas is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

1964 AMC 12/AHSME, 10
Tags: geometry
Given a square side of length $s$. On a diagonal as base a triangle with three unequal sides is constructed so that its area equals that of the square. The length of the altitude drawn to the base is: ${{ \textbf{(A)}\ s\sqrt{2} \qquad\textbf{(B)}\ s/\sqrt{2} \qquad\textbf{(C)}\ 2s \qquad\textbf{(D)}\ 2\sqrt{s} }\qquad\textbf{(E)}\ 2/ \sqrt{s} } $

1989 AMC 8, 17
Tags:
The number $\text{N}$ is between $9$ and $17$. The average of $6$, $10$, and $\text{N}$ could be $\text{(A)}\ 8 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$

2022 JHMT HS, 8
Tags: algebra , complex number
Let $\omega$ be a complex number satisfying $\omega^{2048} = 1$ and $\omega^{1024} \neq 1$. Find the unique ordered pair of nonnegative integers $(p, q)$ satisfying \[ 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}. \]

2024 TASIMO, 6
Tags: abstract algebra , number theory , complete residue system
We call a positive integer $n\ge 4$[i] beautiful[/i] if there exists some permutation $$\{x_1,x_2,\dots ,x_{n-1}\}$$ of $\{1,2,\dots ,n-1\}$ such that $\{x^1_1,\ x^2_2,\ \dots,x^{n-1}_{n-1}\}$ gives all the residues $\{1,2,\dots, n-1\}$ modulo $n$. Prove that if $n$ is beautiful then $n=2p,$ for some prime number $p.$

1999 Harvard-MIT Mathematics Tournament, 5
Tags:
For any finite set $S$, let $f(S)$ be the sum of the elements of $S$ (if $S$ is empty then $f(S)=0$). Find the sum over all subsets $E$ of $S$ of $\dfrac{f(E)}{f(S)}$ for $S=\{1,2,\cdots,1999\}$.

2012 Pre-Preparation Course Examination, 4
Tags: linear algebra , matrix , vector
Prove that these two statements are equivalent for an $n$ dimensional vector space $V$: [b]$\cdot$[/b] For the linear transformation $T:V\longrightarrow V$ there exists a base for $V$ such that the representation of $T$ in that base is an upper triangular matrix. [b]$\cdot$[/b] There exist subspaces $\{0\}\subsetneq V_1 \subsetneq ...\subsetneq V_{n-1}\subsetneq V$ such that for all $i$, $T(V_i)\subseteq V_i$.

Brazil L2 Finals (OBM) - geometry, 2011.2
Tags: geometry
Let $ ABCD $ be a convex quadrilateral such that $ AD = DC, AC = AB $ and $ \angle ADC = \angle CAB $. If $ M $ and $ N $ are midpoints of the $ AD $ and $ AB $ sides, prove that the $ MNC $ triangle is isosceles.

2007 Stanford Mathematics Tournament, 8
Tags:
If $r+s+t=3$, $r^2+s^2+t^2=1$, and $r^3+s^3+t^3=3$, compute $rst$.

2021 Israel TST, 2
Tags: functional equation , functional equation in z , algebra
Find all unbounded functions $f:\mathbb Z \rightarrow \mathbb Z$ , such that $f(f(x)-y)|x-f(y)$ holds for any integers $x,y$.

2013 Costa Rica - Final Round, F2
Tags: functional equation , functional , algebra
Find all functions $f:R -\{0,2\} \to R$ that satisfy for all $x \ne 0,2$ $$f(x) \cdot \left(f\left(\sqrt[3]{\frac{2+x}{2-x}}\right) \right)^2=\frac{x^3}{4}$$

1972 IMO, 2
Tags: geometry , trapezoid , dissection , cyclic quadrilateral
Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

2022 Princeton University Math Competition, A4 / B6
Tags: geometry
Let $\vartriangle ABC$ be an equilateral triangle. Points $D,E, F$ are drawn on sides $AB$,$BC$, and $CA$ respectively such that $[ADF] = [BED] + [CEF]$ and $\vartriangle ADF \sim \vartriangle BED \sim \vartriangle CEF$. The ratio $\frac{[ABC]}{[DEF]}$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$. (Here $[P]$ denotes the area of polygon $P$.)

1997 Moldova Team Selection Test, 2
Tags: ratio , pentagon , parallelogram , geometry
In a convex pentagon every diagonal is parallel to one side. Show that the ratios between the lengths of diagonals and the sides parallel to them are equal and find their value.

2012 Czech-Polish-Slovak Match, 1
Tags: ratio , circumcircle , area of a triangle , geometry
Let $ABC$ be a right angled triangle with hypotenuse $AB$ and $P$ be a point on the shorter arc $AC$ of the circumcircle of triangle $ABC$. The line, perpendicuar to $CP$ and passing through $C$, intersects $AP$, $BP$ at points $K$ and $L$ respectively. Prove that the ratio of area of triangles $BKL$ and $ACP$ is independent of the position of point $P$.

2001 Junior Balkan MO, 4
Tags: geometry , perimeter , trigonometry , combinatorics
Let $N$ be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of $N$ which form a triangle of area smaller than 1.

2015 Silk Road, 1 (original)
Tags: inequalities , algebra
Given positive real numbers $a,b,c,d$ such that $ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6 \quad \text{and} \quad \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}=36.$ Prove the inequality ${{a}^{2}}+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}>ab+ac+ad+bc+bd+cd.$

1998 Flanders Math Olympiad, 3
Tags: linear algebra , matrix , geometry , geometric transformation , rotation
a magical $3\times3$ square is a $3\times3$ matrix containing all number from 1 to 9, and of which the sum of every row, every column, every diagonal, are all equal. Determine all magical $3\times3$ square

2010 Balkan MO Shortlist, G6
Tags: geometry , circumcircle
In a triangle $ABC$ the excircle at the side $BC$ touches $BC$ in point $D$ and the lines $AB$ and $AC$ in points $E$ and $F$ respectively. Let $P$ be the projection of $D$ on $EF$. Prove that the circumcircle $k$ of the triangle $ABC$ passes through $P$ if and only if $k$ passes through the midpoint $M$ of the segment $EF$.

2013 China Team Selection Test, 1
Tags: modular arithmetic , geometry , geometric transformation , real analysis , number theory
Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]

2013 BMT Spring, 2
Tags: algebra
Find the sum of all positive integers $N$ such that $s =\sqrt[3]{2 + \sqrt{N}} + \sqrt[3]{2 - \sqrt{N}}$ is also a positive integer

VI Soros Olympiad 1999 - 2000 (Russia), 10.2
Tags: geometry , angle
In the triangle $ABC$, the point $X$ is the projection of the touchpoint of the inscribed circle to the side $BC$ on the middle line parallel to $BC$. It is known that $\angle BAC \ge 60^o$. Prove that the angle $BXC$ is obtuse.

1999 All-Russian Olympiad Regional Round, 11.5
Tags: algebra , inequalities
Are there real numbers $a, b$ and $c$ such that for all real $x$ and $y$ the following inequality holds: $$|x + a| + |x + y + b| + |y + c| > |x| + |x + y| + |y|?$$