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

2018 CHMMC (Fall), 3
Tags: sum , algebra
Compute $$\sum^{\infty}_{n=1} \left( \frac{1}{n^2 + 3n} - \frac{1}{n^2 + 3n + 2}\right)$$

2005 AMC 10, 14
Tags: calculus , integration
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? $ \textbf{(A)}\ 41\qquad \textbf{(B)}\ 42\qquad \textbf{(C)}\ 43\qquad \textbf{(D)}\ 44\qquad \textbf{(E)}\ 45$

2020 Polish Junior MO Second Round, 2.
Tags: geometry , parallelogram , perpendicular bisector
Let $ABCD$ be the parallelogram, such that angle at vertex $A$ is acute. Perpendicular bisector of the segment $AB$ intersects the segment $CD$ in the point $X$. Let $E$ be the intersection point of the diagonals of the parallelogram $ABCD$. Prove that $XE = \frac{1}{2}AD$.

2007 Belarusian National Olympiad, 2
Tags: geometry
Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$, respectively, pass through the centers of each other. Let $A$ be one of their intersection points. Two points $M_1$ and $M_2$ begin to move simultaneously starting from $A$. Point $M_1$ moves along $S_1$ and point $M_2$ moves along $S_2$. Both points move in clockwise direction and have the same linear velocity $v$. (a) Prove that all triangles $AM_1M_2$ are equilateral. (b) Determine the trajectory of the movement of the center of the triangle $AM_1M_2$ and find its linear velocity.

2013 Stars Of Mathematics, 2
Tags: geometry , rectangle
Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. i) Prove at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle. ii) Assuming the three points inside the rectangle are three corners of it, prove at least two of the three concave quadrilaterals formed by these four points have perimeters lesser than that of the rectangle. [i](Dan Schwarz)[/i]

1961 Putnam, B3
Tags: geometry , circumcircle
Consider four points in the plane, no three of which are collinear, and such that the circle through three of them does not pass through the fourth. Prove that one of the four points can be selected having the property that it lies inside the circle determined by the other three.

2017 Azerbaijan EGMO TST, 4
Tags: calculus , integration , induction , modular arithmetic , number theory
Find all natural numbers a, b such that $ a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1}$ where c and n are naturals.

2003 Vietnam National Olympiad, 2
Tags: geometry , geometric transformation , homothety , invariant , trigonometry , ratio
The circles $ C_{1}$ and $ C_{2}$ touch externally at $ M$ and the radius of $ C_{2}$ is larger than that of $ C_{1}$. $ A$ is any point on $ C_{2}$ which does not lie on the line joining the centers of the circles. $ B$ and $ C$ are points on $ C_{1}$ such that $ AB$ and $ AC$ are tangent to $ C_{1}$. The lines $ BM$, $ CM$ intersect $ C_{2}$ again at $ E$, $ F$ respectively. $ D$ is the intersection of the tangent at $ A$ and the line $ EF$. Show that the locus of $ D$ as $ A$ varies is a straight line.

2018 JHMT, 9
Tags: geometry
In a trapezoid $ABCD$, $AD \parallel BC$ and $\angle A = 60^o$. Let $E$ be a point on $AB$, and let $O_1$ and $O_2$ be circumcenters of $\vartriangle AED$ and $\vartriangle BEC$, respectively. Let $\frac{\overline{O_1O_2}}{\overline{DC}}$ be $x$. $x^2$ is in the form $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $p + q$.

1985 IMO Longlists, 82
Tags: diophantine equation , algebra , polynomial
Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.

2020 Regional Olympiad of Mexico Center Zone, 2
Tags: algebra , inequalities
Let $a$, $b$ and $c$ be positive real numbers, prove that \[\frac{2a^2 b^2}{a^5+b^5}+\frac{2b^2 c^2}{b^5+c^5}+\frac{2c^2 a^2}{c^5+a^5}\le\frac{a+b}{2ab}+\frac{b+c}{2bc}+\frac{c+a}{2ca}\]

1996 All-Russian Olympiad Regional Round, 11.6
Tags: algebra , number theory
Find all natural $n$ such that for some different natural $a, b, c$ and $d$ among numbers $$\frac{(a-c)(b-d)}{(b-c)(a-d)} , \frac{(b-c)(a-d)}{(a-c)(b-d)} , \frac{(a-b)(d-c)}{(a-d)(b-c)} , \frac{(a-c)(b-d)}{(a-b)(c-d)} ,$$ there are at least two numbers equal to $n$.

May Olympiad L2 - geometry, 2019.3
Tags: geometry , equal angles
On the sides $AB, BC$ and $CA$ of a triangle $ABC$ are located the points $P, Q$ and $R$ respectively, such that $BQ = 2QC, CR = 2RA$ and $\angle PRQ = 90^o$. Show that $\angle APR =\angle RPQ$.

PEN O Problems, 52
Tags: induction
Is there a set $S$ of positive integers such that a number is in $S$ if and only if it is the sum of two distinct members of $S$ or a sum of two distinct positive integers not in $S$?

2023 Swedish Mathematical Competition, 3
Tags: algebra , inequalities
Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be different real numbers, placed one after the other in any order. We say we have a [i]local minimum[/i] in one of the numbers if this is less than both of their neighbors. Which is the average number of local minima over all possible ways of ordering the numbers each other?

2016 CCA Math Bonanza, L1.3
Tags:
If the GCD of $a$ and $b$ is $12$ and the LCM of $a$ and $b$ is $168$, what is the value of $a\times b$? [i]2016 CCA Math Bonanza L1.3[/i]

Ukraine Correspondence MO - geometry, 2007.7
Tags: midpoint , geometry , isosceles
Let $ABC$ be an isosceles triangle ($AB = AC$), $D$ be the midpoint of $BC$, and $M$ be the midpoint of $AD$. On the segment $BM$ take a point $N$ such that $\angle BND = 90^o$. Find the angle $ANC$.

1998 Iran MO (3rd Round), 3
Tags: algebra , functional equation
Find all functions $f : \mathbb R \to \mathbb R$ such that for all $x, y,$ \[f(f(x) + y) = f(x^2 - y) + 4f(x)y.\]

2003 Iran MO (3rd Round), 19
Tags: number theory
An integer $ n$ is called a good number if and only if $ |n|$ is not square of another intger. Find all integers $ m$ such that they can be written in infinitely many ways as sum of three different good numbers and product of these three numbers is square of an odd number.

2023 ELMO Shortlist, C5
Tags: combinatorics
Define the [i]mexth[/i] of \(k\) sets as the \(k\)th smallest positive integer that none of them contain, if it exists. Does there exist a family \(\mathcal F\) of sets of positive integers such that [list] [*]for any nonempty finite subset \(\mathcal G\) of \(\mathcal F\), the mexth of \(\mathcal G\) exists, and [*]for any positive integer \(n\), there is exactly one nonempty finite subset \(\mathcal G\) of \(\mathcal F\) such that \(n\) is the mexth of \(\mathcal G\). [/list] [i]Proposed by Espen Slettnes[/i]

1976 Euclid, 2
Tags: geometric sequence , arithmetic sequence
Source: 1976 Euclid Part B Problem 2 ----- Given that $x$, $y$, and $2$ are in geometric progression, and that $x^{-1}$, $y^{-1}$, and $9x^{-2}$ are in are in arithmetic progression, then find the numerical value of $xy$.

1994 Putnam, 5
Tags: limit
Let $(r_n)_{n\ge 0}$ be a sequence of positive real numbers such that $\lim_{n\to \infty} r_n = 0$. Let $S$ be the set of numbers representable as a sum \[ r_{i_1} + r_{i_2} +\cdots + r_{i_{1994}} ,\] with $i_1 < i_2 < \cdots< i_{1994}.$ Show that every nonempty interval $(a, b)$ contains a nonempty subinterval $(c, d)$ that does not intersect $S$.

1974 IMO Longlists, 25
Tags: trigonometry , limit , algebra
Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$ \[x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).\] Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$

2007 Princeton University Math Competition, 7
Tags: geometry
$A, B, C$, and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, what is the area of $ABCD$?

2024 Israel TST, P3
Tags: combinatorics
Let $0<c<1$ and $n$ a positive integer. Alice and Bob are playing a game. Bob writes $n$ integers on the board, not all equal. On a player's turn, they erase two numbers from the board and write their arithmetic mean instead. Alice starts and performs at most $cn$ moves. After her, Bob makes moves until there are only two numbers left on the board. Alice wins if these two numbers are different, and otherwise, Bob wins. For which values of $c$ does Alice win for all large enough $n$?