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

1963 German National Olympiad, 6
Tags: geometry , 3d geometry , sphere , tetrahedron
Consider a pyramid $ABCD$ whose base $ABC$ is a triangle. Through a point $M$ of the edge $DA$, the lines $MN$ and $MP$ on the plane of the surfaces $DAB$ and $DAC$ are drawn respectively, such that $N$ is on $DB$ and $P$ is on $DC$ and $ABNM$ , $ACPM$ are cyclic quadrilaterals. a) Prove that $BCPN$ is also a cyclic quadrilateral. b) Prove that the points $A,B,C,M,N, P$ lie on a sphere.

2010 Nordic, 2
Tags: geometry
Three circles $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$ share a common point of intersection $O$. The other common point of $\Gamma_A$ and $\Gamma_B$ is $C$, that of $\Gamma_A$ and $\Gamma_C$ is $B$, and that of $\Gamma_C$ and $\Gamma_B$ is $A$. The line $AO$ intersects the circle $\Gamma_A$ in the point $X \ne O$. Similarly, the line $BO$ intersects the circle $\Gamma_B$ in the point $Y \ne O$, and the line $CO$ intersects the circle $\Gamma_C$ in the point $Z \ne O$. Show that \[\frac{|AY |\cdot|BZ|\cdot|CX|}{|AZ|\cdot|BX|\cdot|CY |}= 1.\]

2000 Moldova National Olympiad, Problem 2
Tags: number theory , algebra
Thirty numbers are arranged on a circle in such a way that each number equals the absolute difference of its two neighbors. Given that the sum of the numbers is $2000$, determine the numbers.

2001 Moldova National Olympiad, Problem 3
Tags: geometry
For an arbitrary point $D$ on side $BC$ of an acute-angled triangle $ABC$, let $O_1$ and $O_2$ be the circumcenters of the triangles $ABD$ and $ACD$, and $O$ be the circumcenter of the triangle $AO_1O_2$. Find the locus of $O$ when $D$ moves across $BC$.

2010 AMC 10, 1
Tags:
What is $ 100(100\minus{}3) \minus{} (100 \cdot 100 \minus{} 3)$? $ \textbf{(A)}\ \minus{}20,000 \qquad \textbf{(B)}\ \minus{}10,000 \qquad \textbf{(C)}\ \minus{}297 \qquad \textbf{(D)}\ \minus{}6 \qquad \textbf{(E)}\ 0$

1984 IMO Longlists, 43
Tags: number theory , equation , divisibility , power of 2
Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

2007 Harvard-MIT Mathematics Tournament, 9
Tags: geometry
$\triangle ABC$ is right angled at $A$. $D$ is a point on $AB$ such that $CD=1$. $AE$ is the altitude from $A$ to $BC$. If $BD=BE=1$, what is the length of $AD$?

2011 AIME Problems, 9
Tags: number theory , relatively prime
Let $x_1,x_2,\dots ,x_6$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5+x_6=1$, and $x_1x_3x_5+x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $x_1x_2x_3+x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2$. Find $p+q$.

2019 Junior Balkan Team Selection Tests - Romania, 3
Tags: geometry , circumcircle , orthocenter , parallel , concyclic , cyclic quadrilateral
Let $ABC$ be a triangle in which $AB < AC, D$ is the foot of the altitude from $A, H$ is the orthocenter, $O$ is the circumcenter, $M$ is the midpoint of the side $BC, A'$ is the reflection of $A$ across $O$, and $S$ is the intersection of the tangents at $B$ and $C$ to the circumcircle. The tangent at $A'$ to the circumcircle intersects $SC$ and $SB$ at $X$ and $Y$ , respectively. If $M,S,X,Y$ are concyclic, prove that lines $OD$ and $SA'$ are parallel.

2021 Girls in Math at Yale, 2
Tags: college
A box of strawberries, containing $12$ strawberries total, costs $\$ 2$. A box of blueberries, containing $48$ blueberries total, costs $ \$ 3$. Suppose that for $\$ 12$, Sareen can either buy $m$ strawberries total or $n$ blueberries total. Find $n - m$. [i]Proposed by Andrew Wu[/i]

2000 Estonia National Olympiad, 3
Tags: number theory , diophantine , diophantine equation
Are there any (not necessarily positive) integers $m$ and $n$ such that a) $\frac{1}{m}-\frac{1}{n}=\frac{1}{m-n}$ ? b) $\frac{1}{m}-\frac{1}{n}=\frac{1}{n-m}$

1995 Kurschak Competition, 2
Tags: polynomial , algebra
Consider a polynomial in $n$ variables with real coefficients. We know that if every variable is $\pm1$, the value of the polynomial is positive, or negative if the number of $-1$'s is even, or odd, respectively. Prove that the degree of this polynomial is at least $n$.

2006 District Olympiad, 3
Tags: probability , real analysis
Let $\{x_n\}_{n\geq 0}$ be a sequence of real numbers which satisfy \[ (x_{n+1} - x_n)(x_{n+1}+x_n+1) \leq 0, \quad n\geq 0. \] a) Prove that the sequence is bounded; b) Is it possible that the sequence is not convergent?

1996 Polish MO Finals, 2
Tags: circumcircle , geometry
Let $P$ be a point inside a triangle $ABC$ such that $\angle PBC = \angle PCA < \angle PAB$. The line $PB$ meets the circumcircle of triangle $ABC$ at a point $E$ (apart from $B$). The line $CE$ meets the circumcircle of triangle $APE$ at a point $F$ (apart from $E$). Show that the ratio $\frac{\left|APEF\right|}{\left|ABP\right|}$ does not depend on the point $P$, where the notation $\left|P_1P_2...P_n\right|$ stands for the area of an arbitrary polygon $P_1P_2...P_n$.

2025 Kyiv City MO Round 1, Problem 5
Tags: number theory
Find all quadruples of positive integers \( (a, p, q, r) \), where \( p, q, r \) are prime numbers, such that the following equation holds: \[ p^2q^2 + q^2r^2 + r^2p^2 + 3 = 4 \cdot 13^a. \] [i]Proposed by Oleksii Masalitin[/i]

2004 Purple Comet Problems, 12
Tags: function
If $f(x, y) = xy + 2x + y + 1$, find $f(f(2, f(3, 4)), 5)$.

2023 Romania Team Selection Test, P2
Tags: number theory
Find all positive integers, such that there exist positive integers $a, b, c$, satisfying $\gcd(a, b, c)=1$ and $n=\gcd(ab+c, ac-b)=a+b+c$.

PEN S Problems, 23
Tags:
Observe that \[\frac{1}{1}+\frac{1}{3}=\frac{4}{3}, \;\; 4^{2}+3^{2}=5^{2},\] \[\frac{1}{3}+\frac{1}{5}=\frac{8}{15}, \;\; 8^{2}+{15}^{2}={17}^{2},\] \[\frac{1}{5}+\frac{1}{7}=\frac{12}{35}, \;\;{12}^{2}+{35}^{2}={37}^{2}.\] State and prove a generalization suggested by these examples.

2004 Romania Team Selection Test, 9
Tags: pigeonhole principle , linear algebra , matrix , inequalities , combinatorics
Let $n\geq 2$ be a positive integer, and $X$ a set with $n$ elements. Let $A_{1},A_{2},\ldots,A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $\frac{50}{51}n$ elements. Prove that among these $101$ subsets there exist $3$ subsets such that any two of them have a common element.

2015 China Second Round Olympiad, 1
Tags: inequalities , probabilistic method
Let $a_1, a_2, \ldots, a_n$ be real numbers.Prove that you can select $\varepsilon _1, \varepsilon _2, \ldots, \varepsilon _n\in\{-1,1\}$ such that$$\left( \sum_{i=1}^{n}a_{i}\right)^2 +\left( \sum_{i=1}^{n}\varepsilon _ia_{i}\right)^2 \leq(n+1)\left( \sum_{i=1}^{n}a^2_{i}\right).$$

2013 Online Math Open Problems, 34
Tags: number theory , relatively prime
For positive integers $n$, let $s(n)$ denote the sum of the squares of the positive integers less than or equal to $n$ that are relatively prime to $n$. Find the greatest integer less than or equal to \[ \sum_{n\mid 2013} \frac{s(n)}{n^2}, \] where the summation runs over all positive integers $n$ dividing $2013$. [i]Ray Li[/i]

2014 Balkan MO Shortlist, N4
Tags: number theory
A [i]special number[/i] is a positive integer $n$ for which there exists positive integers $a$, $b$, $c$, and $d$ with \[ n = \frac {a^3 + 2b^3} {c^3 + 2d^3}. \] Prove that i) there are infinitely many special numbers; ii) $2014$ is not a special number. [i]Romania[/i]

2004 Purple Comet Problems, 13
Tags:
How many three digit numbers are made up of three distinct digits?

2012 Flanders Math Olympiad, 4
Tags: geometry , angle bisector , angle chasing , angle
In $\vartriangle ABC, \angle A = 66^o$ and $| AB | <| AC |$. The outer bisector in $A$ intersects $BC$ in $D$ and $| BD | = | AB | + | AC |$. Determine the angles of $\vartriangle ABC$.

2019 Regional Competition For Advanced Students, 2
Tags: cyclic , parallel , equal segments , geometry
The convex pentagon $ABCDE$ is cyclic and $AB = BD$. Let point $P$ be the intersection of the diagonals $AC$ and $BE$. Let the straight lines $BC$ and $DE$ intersect at point $Q$. Prove that the straight line $PQ$ is parallel to the diagonal $AD$.