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

2000 All-Russian Olympiad, 5
Tags: trigonometry , inequalities
Prove the inequality \[ \sin^n (2x) + \left( \sin^n x - \cos^n x \right)^2 \le 1. \]

2017 Azerbaijan EGMO TST, 3
Tags: geometry
In $\bigtriangleup$$ABC$ $BL$ is bisector. Arbitrary point $M$ on segment $CL$ is chosen. Tangent to $\odot$$(ABC)$ at $B$ intersects $CA$ at $P$. Tangents to $\odot$$BLM$ at $B$ and $M$ intersect at point $Q$. Prove that $PQ$$\parallel$$BL$.

2020 ITAMO, 4
Tags: geometry , circumcircle
Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.

2024 Middle European Mathematical Olympiad, 8
Tags: sequence , divisibility , constant , number theory
Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that \[a_ia_{i+1} \mid k-a_i^2\] for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all integers $n \ge M$.

2015 India Regional MathematicaI Olympiad, 4
Tags:
Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

2013 All-Russian Olympiad, 4
Tags: geometry , rectangle , combinatorics
A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly $ k $ rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly $\ell$ rectangles. How much the number of rectangles can be? [i]I. Bogdanov, D. Fon-Der-Flaass[/i]

2019 Online Math Open Problems, 1
Tags:
Daniel chooses some distinct subsets of $\{1, \dots, 2019\}$ such that any two distinct subsets chosen are disjoint. Compute the maximum possible number of subsets he can choose. [i]Proposed by Ankan Bhattacharya[/i]

2021 Nordic, 1
Tags: number theory , prime
On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.

2018 German National Olympiad, 1
Tags: algebra , system of equations
Find all real numbers $x,y,z$ satisfying the following system of equations: \begin{align*} xy+z&=-30\\ yz+x &= 30\\ zx+y &=-18 \end{align*}

LMT Speed Rounds, 2010.11
Tags:
Compute the number of positive integers $n$ less than $100$ for which $1+2+\dots+n$ is not divisible by $n.$

1996 Estonia National Olympiad, 5
Tags: triangle , combinatorial geometry , combinatorics
Suppose that $n$ triangles are given in the plane such that any three of them have a common vertex, but no four of them do. Find the greatest possible $n$.

2023 AMC 10, 21
Tags: probability
Each of $2023$ balls is placed in on of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? $\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$

2009 Today's Calculation Of Integral, 417
Tags: calculus , integration , function , geometry , algebra , polynomial , vieta
The functions $ f(x) ,\ g(x)$ satify that $ f(x) \equal{} \frac {x^3}{2} \plus{} 1 \minus{} x\int_0^x g(t)\ dt,\ g(x) \equal{} x \minus{} \int_0^1 f(t)\ dt$. Let $ l_1,\ l_2$ be the tangent lines of the curve $ y \equal{} f(x)$, which pass through the point $ (a,\ g(a))$ on the curve $ y \equal{} g(x)$. Find the minimum area of the figure bounded by the tangent tlines $ l_1,\ l_2$ and the curve $ y \equal{} f(x)$ .

2023 VN Math Olympiad For High School Students, Problem 8
Tags: algebra , polynomial
Prove that: for all positive integers $n\ge 2,$ the polynomial$$(x^2-1)^2(x^2-1)^2...(x^2-2023)^2+1$$ is irreducible in $\mathbb{Q}[x].$

KoMaL A Problems 2022/2023, A. 848
Tags: combinatorics , graph theory
Let $G$ be a planar graph, which is also bipartite. Is it always possible to assign a vertex to each face of the graph such that no two faces have the same vertex assigned to them? [i]Submitted by Dávid Matolcsi, Budapest[/i]

2008 Tuymaada Olympiad, 1
Tags: combinatorics
Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $ 5\over 4$ times greater than Vasya's result. What minimum number of portraits can be on the wall? [i]Author: V. Frank[/i]

2022 Girls in Mathematics Tournament, 1
Tags: geometry
Let $ABC$ be a triangle with $BA=BC$ and $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the midpoints of $CA$ and $BA$ respectively. The point $F$ is inside of $\triangle ABC$ such that $\triangle DEF$ is equilateral. Let $X=BF\cap AC$ and $Y=AF\cap DB$. Prove that $DX=YD$.

JOM 2015 Shortlist, G6
Tags: geometry
Let $ABC$ be a triangle. Let $\omega_1$ be circle tangent to $BC$ at $B$ and passes through $A$. Let $\omega_2$ be circle tangent to $BC$ at $C$ and passes through $A$. Let $\omega_1$ and $\omega_2$ intersect again at $P \neq A$. Let $\omega_1$ intersect $AC$ again at $E\neq A$, and let $\omega_2$ intersect $AB$ again at $F\neq A$. Let $R$ be the reflection of $A$ about $BC$, Prove that lines $BE, CF, PR$ are concurrent.

2023 Moldova Team Selection Test, 9
Tags: inequalities
Let $ n $ $(n\geq2)$ be an integer. Find the greatest possible value of the expression $$E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}$$ if the positive real numbers $a_1,a_2,\ldots,a_n$ satisfy $a_1+a_2+\ldots+a_n=\frac{n}{2}.$ What are the values of $a_1,a_2,\ldots,a_n$ when the greatest value is achieved?

1999 Estonia National Olympiad, 2
Tags: calculus , integration , integral
Find the value of the integral $\int_{-1}^{1} ln \left(x +\sqrt{1 + x^2}\right) dx$.

1979 Chisinau City MO, 172
Tags: geometry , right triangle , angle bisector
Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.

2018 Online Math Open Problems, 27
Tags:
Let $p=2^{16}+1$ be a prime. Let $N$ be the number of ordered tuples $(A,B,C,D,E,F)$ of integers between $0$ and $p-1$, inclusive, such that there exist integers $x,y,z$ not all divisible by $p$ with $p$ dividing all three of $Ax+Ez+Fy, By+Dz+Fx, Cz+Dy+Ex$. Compute the remainder when $N$ is divided by $10^6$. [i]Proposed by Vincent Huang[/i]

1954 Moscow Mathematical Olympiad, 259
Tags: geometry , pentagon , convex polygon
A regular star-shaped hexagon is split into $4$ parts. Construct from them a convex polygon. Note: A regular six-pointed star is a figure that is obtained by combining a regular triangle and a triangle symmetrical to it relative to its center

1981 Tournament Of Towns, (010) 4
Tags: minimum , combinatorics
Each of $K$ friends simultaneously learns one different item of news. They begin to phone one another to tell them their news. Each conversation lasts exactly one hour, during which time it is possible for two friends to tell each other all of their news. What is the minimum number of hours needed in order for all of the friends to know all of the news? Consider in this problem: (a) $K = 64$. (b) $K = 55$. (c) $K = 100$. (A Andjans, Riga) PS. (a) was the junior problem, (a),(b),(c) the senior one

2019 India Regional Mathematical Olympiad, 6
Tags: algebra , geometry , coordinate geometry , analytic geometry , parabola , conic
Let $k$ be a positive real number. In the $X-Y$ coordinate plane, let $S$ be the set of all points of the form $(x,x^2+k)$ where $x\in\mathbb{R}$. Let $C$ be the set of all circles whose center lies in $S$, and which are tangent to $X$-axis. Find the minimum value of $k$ such that any two circles in $C$ have at least one point of intersection.