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

1992 India Regional Mathematical Olympiad, 7
Tags: parameterization
Solve the system \begin{eqnarray*} \\ (x+y)(x+y+z) &=& 18 \\ (y+z)(x+y+z) &=& 30 \\ (x+z)(x+y+z) &=& 2A \end{eqnarray*} in terms of the parameter $A$.

1995 Poland - First Round, 5
Tags: inequalities , trigonometry
Given triangle $ABC$ in the plane such that $\angle CAB = a > \pi/2$. Let $PQ$ be a segment whose midpoint is the point $A$. Prove that $(BP+CQ) \tan a/2 \geq BC$.

Kvant 2019, M2586
Tags: combinatorics
A polygon is given in which any two adjacent sides are perpendicular. We call its two vertices non-friendly if the bisectors of the polygon emerging from these vertices are perpendicular. Prove that for any vertex the number of vertices that are not friends with it is even.

2014 Sharygin Geometry Olympiad, 20
Tags: concurrency , circumcircle , circles , geometry
A quadrilateral $KLMN$ is given. A circle with center $O$ meets its side $KL$ at points $A$ and $A_1$, side $LM$ at points $B$ and $B_1$, etc. Prove that if the circumcircles of triangles $KDA, LAB, MBC$ and $NCD$ concur at point $P$, then a) the circumcircles of triangles $KD_1A_1, LA_1B_1, MB_1C_1$ and $NC1D1$ also concur at some point $Q$; b) point $O$ lies on the perpendicular bisector to $PQ$.

1986 AIME Problems, 7
Tags:
The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\text{th}}$ term of this sequence.

2010 Indonesia TST, 3
Tags: combinatorics
In a party, each person knew exactly $ 22$ other persons. For each two persons $ X$ and $ Y$, if $ X$ and $ Y$ knew each other, there is no other person who knew both of them, and if $ X$ and $ Y$ did not know each other, there are exactly $ 6$ persons who knew both of them. Assume that $ X$ knew $ Y$ iff $ Y$ knew $ X$. How many people did attend the party? [i]Yudi Satria, Jakarta[/i]

2006 Harvard-MIT Mathematics Tournament, 7
Tags: calculus
Find all positive real numbers $c$ such that the graph of $f\text{ : }\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^3-cx$ has the property that the circle of curvature at any local extremum is centered at a point on the $x$-axis.

1997 AMC 8, 4
Tags:
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech? $\textbf{(A)}\ 2250 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 4200 \qquad \textbf{(D)}\ 4350 \qquad \textbf{(E)}\ 5650$

2012 Sharygin Geometry Olympiad, 18
Tags: geometry , incenter
A triangle and two points inside it are marked. It is known that one of the triangle’s angles is equal to $58^{\circ}$, one of two remaining angles is equal to $59^{\circ}$, one of two given points is the incenter of the triangle and the second one is its circumcenter. Using only the ruler without partitions determine where is each of the angles and where is each of the centers.

2013 Kazakhstan National Olympiad, 1
Tags: geometry , circumcircle , geometric transformation , incenter
Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.

2006 Peru MO (ONEM), 4
Tags: combinatorics
In each of the squares of an $n \times n$ board, with $n \ge 3$, a positive integer is written in such a way that the absolute value of the difference of the numbers written in any two neighboring cells is less than or equal to $2$ (two neighboring cells are those that have a common side). a) Show a $5 \times 5$ board on which $15$ integers have been written different following the indicated rule. b) Find, as a function of $n$, the maximum number of different numbers that can have the board of $n \times n$ squares.

2016 Taiwan TST Round 2, 1
Tags: triangle , geometry
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

2023 Thailand October Camp, 6
Tags: geometry
Let $ABC$ be an acute triangle with orthocenter $H$. Let $D$ and $E$ be feet of the altitudes from $B$ and $C$ respectively. Let $M$ be the midpoint of segment $AH$ and $F$ be the intersection point of $AH$ and $DE$. Furthermore, let $P$ and $Q$ be the points inside triangle $ADE$ so that $P$ is an intersection of $CM$ and the circumcircle of $DFH$, and $Q$ is an intersection of $BM$ and the circumcircle of $EFH$. Prove that the intersection of lines $DQ$ and $EP$ lies on segment $AH$.

1994 Miklós Schweitzer, 4
Tags: real analysis , irrational number
For a given irrational number $\alpha$ , $y_{1,\alpha} = \alpha$. If $y_{n-1, \alpha}$ is given, let $y_{n, \alpha}$ be the first member of the sequence $\big (\{k \alpha \} \big) ^ \infty_{k = 1}$ to fall in the interval $(0, y_{n-1,\alpha})$ ({ x } denotes the fraction of the number x ). Show that there exists an open set $G\subset (0,1)$ , which has a limit point 0 and for all irrational $\alpha$ , infinitely many members of the $(y_{n,\alpha})$ sequence do not belong to G.

2006 Turkey Team Selection Test, 3
Tags: combinatorics
Each one of 2006 students makes a list with 12 schools among 2006. If we take any 6 students, there are two schools which at least one of them is included in each of 6 lists. A list which includes at least one school from all lists is a good list. a) Prove that we can always find a good list with 12 elements, whatever the lists are; b) Prove that students can make lists such that no shorter list is good.

2021 Latvia TST, 2.3
Tags: polynomial , linear combination , algebra
Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

2020 JBMO Shortlist, 4
Tags: prime number , shortlist , number theory
Find all prime numbers $p$ such that $(x + y)^{19} - x^{19} - y^{19}$ is a multiple of $p$ for any positive integers $x$, $y$.

2006 Moldova Team Selection Test, 1
Tags: number theory
Let $(a_n)$ be the Lucas sequence: $a_0=2,a_1=1, a_{n+1}=a_n+a_{n-1}$ for $n\geq 1$. Show that $a_{59}$ divides $(a_{30})^{59}-1$.

2010 Contests, 3
Tags: combinatorics
A total of $2010$ coins are distributed in $5$ boxes. At the beginning the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that she can do the following transformation finitely many times: from a box with at least 4 coins she can transfer one coin to each of the other boxes. What is the maximum number of coins that Martha can take away?

2022 Flanders Math Olympiad, 4
Tags: algebra , polynomial
Determine all real polynomials $P$ of degree at most $22$ for which $$kP (k + 1) - (k + 1)P (k) = k^2 + k + 1$$ for all $k \in \{1, 2, 3, . . . , 21, 22\}$.

1986 Traian Lălescu, 1.2
Tags: function , algebra
Prove that there exists a surjective function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ having the property that for all natural numbers $ n\ge 2, $ there exists an infinite set $ A_n $ such that $ f(x)=n, $ for all $ x\in A_n. $

2010 Princeton University Math Competition, 5
Tags:
Given that $x$, $y$ are positive integers with $x(x+1)|y(y+1)$, but neither $x$ nor $x+1$ divides either of $y$ or $y+1$, and $x^2 + y^2$ as small as possible, find $x^2 + y^2$.

2002 All-Russian Olympiad, 2
Tags: geometric transformation , geometry , circumcircle , cyclic quadrilateral
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $O$. The circumcircles of triangles $AOB$ and $COD$ intersect again at $K$. Point $L$ is such that the triangles $BLC$ and $AKD$ are similar and equally oriented. Prove that if the quadrilateral $BLCK$ is convex, then it is tangent [has an incircle].

2015 Baltic Way, 13
Tags: trigonometry , geometry
Let $D$ be the footpoint of the altitude from $B$ in the triangle $ABC$ , where $AB=1$ . The incircle of triangle $BCD$ coincides with the centroid of triangle $ABC$. Find the lengths of $AC$ and $BC$.

2002 BAMO, 3
Tags: combinatorics , game theory
A game is played with two players and an initial stack of $n$ pennies $(n \geq 3)$. The players take turns choosing one of the stacks of pennies on the table and splitting it into two stacks. The winner is the player who makes a move that causes all stacks to be of height $1$ or $2.$ For which starting values of n does the player who goes first win, assuming best play by both players?