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

2013 BMT Spring, 1
Tags: algebra , system of equations
Find the value of $a$ satisfying \begin{align*} a+b&=3\\ b+c&=11\\ c+a&=61 \end{align*}

2006 AMC 10, 12
Tags:
The lines $ x \equal{} \frac 14y \plus{} a$ and $ y \equal{} \frac 14x \plus{} b$ intersect at the point $ (1,2)$. What is $ a \plus{} b$? $ \textbf{(A) } 0 \qquad \textbf{(B) } \frac 34 \qquad \textbf{(C) } 1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } \frac 94$

2024-25 IOQM India, 17
Tags: geometry , circumcircle
Consider an isosceles triangle $ABC$ with sides $BC = 30, CA = AB = 20$. Let $D$ be the foot of the perpendicular from $A$ to $BC$, and let $M$ be the midpoint of $AD$. Let $PQ$ be a chord of the circumcircle of triangle $ABC$, such that $M$ lies on $PQ$ and $PQ$ is parallel to $BC$. The length of $PQ$ is:

2007 Thailand Mathematical Olympiad, 6
Tags: geometry , geometric inequalities
A triangle has perimeter $2s$, inradius $r$, and incenter $I$. If $s_a, s_b$ and $s_c$ are the distances from $I$ to the three vertices, then show that $$\frac34 +\frac{r}{s_a}+\frac{r}{s_b}+\frac{r}{s_c} \le \frac{s^2}{12r^2}$$

2017 Harvard-MIT Mathematics Tournament, 10
Tags: geometry
Let $ABC$ be a triangle in the plane with $AB = 13$, $BC = 14$, $AC = 15$. Let $M_n$ denote the smallest possible value of $(AP^n + BP^n + CP^n)^{\frac{1}{n}}$ over all points $P$ in the plane. Find $\lim_{n \to \infty} M_n$.

2022 Junior Macedonian Mathematical Olympiad, P3
Tags: geometry , geometric transformation , reflection
Let $\triangle ABC$ be an acute triangle with orthocenter $H$. The circle $\Gamma$ with center $H$ and radius $AH$ meets the lines $AB$ and $AC$ at the points $E$ and $F$ respectively. Let $E'$, $F'$ and $H'$ be the reflections of the points $E$, $F$ and $H$ with respect to the line $BC$, respectively. Prove that the points $A$, $E'$, $F'$ and $H'$ lie on a circle. [i]Proposed by Jasna Ilieva[/i]

2000 239 Open Mathematical Olympiad, 2
Tags: combinatorics , tournament
100 volleyball teams played a one-round tournament. No two matches held at the same time. It turned out that before each match the teams playing against each other had the same number of wins. Find all possible number of wins for the winner of this tournament.

1965 Poland - Second Round, 6
Tags: geometry , 3d geometry , polyhedron
Prove that there is no polyhedron whose every plane section is a triangle.

2005 Today's Calculation Of Integral, 81
Tags: calculus , integration , trigonometry , inequalities , derivative , function , logarithm
Prove the following inequality. \[\frac{1}{12}(\pi -6+2\sqrt{3})\leq \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \ln (1+\cos 2x) dx\leq \frac{1}{4}(2-\sqrt{3})\]

1987 Czech and Slovak Olympiad III A, 1
Tags: geometry , constructive geometry , quadrilateral , cyclic quadrilateral , trapezoid , circumcircle
Given a trapezoid, divide it by a line into two quadrilaterals in such a way that both of them are cyclic with the same circumradius. Discuss conditions of solvability.

1995 AMC 8, 5
Tags:
Find the smallest whole number that is larger than the sum \[2\dfrac{1}{2}+3\dfrac{1}{3}+4\dfrac{1}{4}+5\dfrac{1}{5}.\] $\text{(A)}\ 14 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 17 \qquad \text{(E)}\ 18$

1978 IMO Shortlist, 12
Tags: geometry , inradius , circumcircle , incenter , triangle
In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

2009 Sharygin Geometry Olympiad, 7
Tags: geometry , construction , circles , tangent circle
Given two intersecting circles with centers $O_1, O_2$. Construct the circle touching one of them externally and the second one internally such that the distance from its center to $O_1O_2$ is maximal. (M.Volchkevich)

2017 VJIMC, 2
Tags: function , real analysis
Prove or disprove the following statement. If $g:(0,1) \to (0,1)$ is an increasing function and satisfies $g(x) > x$ for all $x \in (0,1)$, then there exists a continuous function $f:(0,1) \to \mathbb{R}$ satisfying $f(x) < f(g(x)) $ for all $x \in (0,1)$, but $f$ is not an increasing function.

2016 Japan MO Preliminary, 7
Tags: algebra , system of equations
Let $a, b, c, d$ be real numbers satisfying the system of equation $\[(a+b)(c+d)=2 \\ (a+c)(b+d)=3 \\ (a+d)(b+c)=4\]$ Find the minimum value of $a^2+b^2+c^2+d^2$.

2017 Canadian Mathematical Olympiad Qualification, 6
Tags: number theory
Let $N$ be a positive integer. There are $N$ tasks, numbered $1, 2, 3, \ldots, N$, to be completed. Each task takes one minute to complete and the tasks must be completed subjected to the following conditions: [list] [*] Any number of tasks can be performed at the same time. [*] For any positive integer $k$, task $k$ begins immediately after all tasks whose numbers are divisors of $k$, not including $k$ itself, are completed. [*] Task 1 is the first task to begin, and it begins by itself. [/list] Suppose $N = 2017$. How many minutes does it take for all of the tasks to complete? Which tasks are the last ones to complete?

2025 Al-Khwarizmi IJMO, 6
Tags: algebra
Let $a,b,c$ be real numbers such that \[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\] Find the smallest possible value of $a^2 + b^2 + c^2$. [i]Binh Luan and Nhan Xet, Vietnam[/i]

2024 Sharygin Geometry Olympiad, 7
Tags: geometry , bicentric quadrilateral
Restore a bicentral quadrilateral if two opposite vertices and the incenter are given.

2011 China National Olympiad, 1
Tags: floor function , function , inequalities
Let $a_1,a_2,\ldots,a_n$ are real numbers, prove that; \[\sum_{i=1}^na_i^2-\sum_{i=1}^n a_i a_{i+1} \le \left\lfloor \frac{n}{2}\right\rfloor(M-m)^2.\] where $a_{n+1}=a_1,M=\max_{1\le i\le n} a_i,m=\min_{1\le i\le n} a_i$.

2017 Iran Team Selection Test, 4
Tags: number theory , prime number , modular arithmetic
We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$. Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$. Is there always a number $x$ that satisfies all the equations? [i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]

2005 China Team Selection Test, 2
Tags: inequalities
Let $a$, $b$, $c$ be nonnegative reals such that $ab+bc+ca = \frac{1}{3}$. Prove that \[\frac{1}{a^{2}-bc+1}+\frac{1}{b^{2}-ca+1}+\frac{1}{c^{2}-ab+1}\leq 3 \]

2012 ITAMO, 5
Tags: circumcircle , geometric transformation , reflection , symmetry , geometry
$ABCD$ is a square. Describe the locus of points $P$, different from $A, B, C, D$, on that plane for which \[\widehat{APB}+\widehat{CPD}=180^\circ\]

2023 Princeton University Math Competition, B2
Tags: algebra , polynomial
Let $f$ be a polynomial with degree at most $n-1$. Show that $$ \sum_{k=0}^n\left(\begin{array}{l} n \\ k \end{array}\right)(-1)^k f(k)=0 $$

2023 Princeton University Math Competition, A4 / B6
Tags: geometry
Let $\vartriangle ABC$ be a triangle with $AB = 4$, $BC = 6$, and $CA = 5$. Let the angle bisector of $\angle BAC$ intersect $BC$ at the point $D$ and the circumcircle of $\vartriangle ABC$ again at the point $M\ne A$. The perpendicular bisector of segment $DM$ intersects the circle centered at $M$ passing through $B$ at two points, $X$ and $Y$ . Compute $AX \cdot AY$.

2013 Saudi Arabia Pre-TST, 3.1
Tags: algebra , functional , functional equation
Let $f : R \to R$ be a function satisfying $f(f(x)) = 4x + 1$ for all real number $x$. Prove that the equation $f(x) = x$ has a unique solution.