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

1980 Poland - Second Round, 4
Tags: polynomial , algebra
Prove that if $ a $ and $ b $ are real numbers and the polynomial $ ax^3 - ax^2 + 9bx - b $ has three positive roots, then they are equal.

2014 BMT Spring, 2
Tags: geometry
Regular hexagon $ABCDEF$ has side length $2$ and center $O$. The point $P$ is defined as the intersection of $AC$ and $OB$. Find the area of quadrilateral $OPCD$.

India EGMO 2025 TST, 4
Tags:
For a positive integer $m$, let $f(m)$ denote the smallest power of $2024$ not less than $m$ (e.g. $f(1)=1, f(2023)=f(2024)=2024,$ and $f(2025)=2024^2$). Find all positive real numbers $c$ for which there exists a sequence $x_1,x_2,\cdots$ of real numbers in $[0,1]$ such that $$|x_m-x_n|\geq\frac{c}{f(m)}$$ for all positive integers $m>n\geq1$. Proposed by Shantanu Nene

2017 China Second Round Olympiad, 10
Tags: inequalities
Let $x_1,x_2,x_3\geq 0$ and $x_1+x_2+x_3=1$. Find the minimum value and the maximum value of $(x_1+3x_2+5x_3)\left(x_1+\frac{x_2}{3}+\frac{x_3}{5}\right).$

2016 PAMO, 1
Tags: common tangent , geometry
Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect each other at two distinct points $M$ and $N$. A common tangent lines touches $\mathcal{C}_1$ at $P$ and $\mathcal{C}_2$ at $Q$, the line being closer to $N$ than to $M$. The line $PN$ meets the circle $\mathcal{C}_2$ again at the point $R$. Prove that the line $MQ$ is a bisector of the angle $\angle{PMR}$.

2019 Kurschak Competition, 2
Tags: combinatorics
Find all family $\mathcal{F}$ of subsets of $[n]$ such that for any nonempty subset $X\subseteq [n]$, exactly half of the elements $A\in \mathcal{F}$ satisfies that $|A\cap X|$ is even.

1952 AMC 12/AHSME, 34
Tags:
The price of an article was increased $ p\%$. Later the new price was decreased $ p\%$. If the last price was one dollar, the original price was: $ \textbf{(A)}\ \frac {1 \minus{} p^2}{200} \qquad\textbf{(B)}\ \frac {\sqrt {1 \minus{} p^2}}{100} \qquad\textbf{(C)}\ \text{one dollar} \qquad\textbf{(D)}\ 1 \minus{} \frac {p^2}{10000 \minus{} p^2}$ $ \textbf{(E)}\ \frac {10000}{10000 \minus{} p^2}$

1984 Bundeswettbewerb Mathematik, 1
Tags: number theory , game , winning strategy , combinatorics
Let $n$ be a positive integer and $M = \{1, 2, 3, 4, 5, 6\}$. Two persons $A$ and $B$ play in the following Way: $A$ writes down a digit from $M$, $B$ appends a digit from $M$, and so it becomes alternately one digit from $M$ is appended until the $2n$-digit decimal representation of a number has been created. If this number is divisible by $9$, $B$ wins, otherwise $A$ wins. For which $n$ can $A$ and for which $n$ can $B$ force the win?

2009 Indonesia TST, 3
Tags: calculus , geometry , function , derivative , 3d geometry , inequalities
Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.

2012 Belarus Team Selection Test, 3
Tags: subset , coloring , combinatorics
Define $M_n = \{1,2,...,n\}$, for any $n\in N$. A collection of $3$-element subsets of $M_n$ is said to be [i]fine [/i] if for any coloring of elements of $M_n$ in two colors there is a subset of the collection all three elements of which are of the same color. For any $n\ge 5$ find the minimal possible number of the $3$-element subsets of $M_n$ in the fine collection. (E. Barabanov, S. Mazanik, I. Voronovich)

PEN H Problems, 75
Tags: diophantine equation
Let $a,b$, and $x$ be positive integers such that $x^{a+b}=a^b{b}$. Prove that $a=x$ and $b=x^{x}$.

2008 Kurschak Competition, 3
Tags: combinatorics
In a far-away country, travel between cities is only possible by bus or by train. One can travel by train or by bus between only certain cities, and there are not necessarily rides in both directions. We know that for any two cities $A$ and $B$, one can reach $B$ from $A$, [i]or[/i] $A$ from $B$ using only bus, or only train rides. Prove that there exists a city such that any other city can be reached using only one type of vehicle (but different cities may be reached with different vehicles).

2020 Peru EGMO TST, 5
Tags: geometry , angle bisector
Let $AD$ be the diameter of a circle $\omega$ and $BC$ is a chord of $\omega$ which is perpendicular to $AD$. Let $M,N,P$ be points on the segments $AB,AC,BC$ respectively, such that $MP\parallel AC$ and $PN\parallel AB$. The line $MN$ cuts the line $PD$ in the point $Q$ and the angle bisector of $\angle MPN$ in the point $R$. Prove that the points $B,R,Q,C$ are concyclic.

2007 JBMO Shortlist, 3
Tags: chessboard , combinatorics
The nonnegative integer $n$ and $ (2n + 1) \times (2n + 1)$ chessboard with squares colored alternatively black and white are given. For every natural number $m$ with $1 < m < 2n+1$, an $m \times m$ square of the given chessboard that has more than half of its area colored in black, is called a $B$-square. If the given chessboard is a $B$-square, fi nd in terms of $n$ the total number of $B$-squares of this chessboard.

IV Soros Olympiad 1997 - 98 (Russia), 9.5
Tags: rotation , geometry , locus , geometric transformation
Given triangle $ABC$. Find the locus of points $M$ such that there is a rotation with center at $M$ that takes $C$ to a certain point on side $AB$.

2014 JBMO TST - Turkey, 1
Tags: ratio , geometry , angle bisector
In a triangle $ABC$, the external bisector of $\angle BAC$ intersects the ray $BC$ at $D$. The feet of the perpendiculars from $B$ and $C$ to line $AD$ are $E$ and $F$, respectively and the foot of the perpendicular from $D$ to $AC$ is $G$. Show that $\angle DGE + \angle DGF = 180^{\circ}$.

2018 Cyprus IMO TST, 3
Tags: inequalities
Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$ obtains its minimum value.

1989 Putnam, A2
Tags: integration
Evaluate $\int^{a}_{0}{\int^{b}_{0}{e^{max(b^{2}x^{2},a^{2}y^{2})}dy dx}}$

2019 CCA Math Bonanza, L4.1
Tags:
The Garfield Super Winners play $100$ games of foosball, in which teams score a non-negative integer number of points and the team with more points after ten minutes wins (if both teams have the same number of points, it is a draw). Suppose that the Garfield Super Winners score an average of $7$ points per game but allow an average of $8$ points per game. Given that the Garfield Super Winners never won or lost by more than $10$, what is the largest possible number of games that they could win? [i]2019 CCA Math Bonanza Lightning Round #4.1[/i]

1995 Turkey Team Selection Test, 3
Tags: circumcircle , geometry
Let $D$ be a point on the small arc $AC$ of the circumcircle of an equilateral triangle $ABC$, different from $A$ and $C$. Let $E$ and $F$ be the projections of $D$ onto $BC$ and $AC$ respectively. Find the locus of the intersection point of $EF$ and $OD$, where $O$ is the center of $ABC$.

1974 IMO Longlists, 43
Tags: rectangle , matrix , pigeonhole principle , linear algebra , combinatorics , geometry
An $(n^2+n+1) \times (n^2+n+1)$ matrix of zeros and ones is given. If no four ones are vertices of a rectangle, prove that the number of ones does not exceed $(n + 1)(n^2 + n + 1).$

2001 Romania National Olympiad, 2
Tags: superior algebra
Let $A$ be a finite ring. Show that there exists two natural numbers $m,p$ where $m> p\ge 1$, such that $a^m=a^p$ for all $a\in A$.

2017 NIMO Problems, 3
Tags:
Compute the number of ordered quadruples of complex numbers $(a,b,c,d)$ such that \[ (ax+by)^3 + (cx+dy)^3 = x^3 + y^3 \] holds for all complex numbers $x, y$. [i]Proposed by Evan Chen[/i]

Russian TST 2018, P1
Tags: number theory
The natural numbers $a > b$ are such that $a-b=5b^2-4a^2$. Prove that the number $8b + 1$ is composite.

2007 China Western Mathematical Olympiad, 2
Tags: number theory
Find all natural numbers $n$ such that there exist $ x_1,x_2,\ldots,x_n,y\in\mathbb{Z}$ where $x_1,x_2,\ldots,x_n,y\neq 0$ satisfying: \[x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n \equal{} 0\] \[ny^2 \equal{} x_1^2 \plus{} x_2^2 \plus{} \ldots \plus{} x_n^2\]