Olympiad problems

1983 AIME Problems, 2

Let $f(x) = |x - p| + |x - 15| + |x - p - 15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \le x \le 15$.

2022 Taiwan TST Round 2, 3

Tags: geometry , circumcircle , excircle , ddit

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.

2024 Belarus Team Selection Test, 4.3

Tags: geometry

An isosceles triangle $ABC$ is given($AB=BC$). Point $D$ lies inside of it such that $\angle ADC=150$, $E$ lies on $CD$ such that $AE=AB$. It turned out that $\angle EBC+\angle BAE=60$. Prove that $\angle BDC+\angle CAE=90$ [i]D. Vasilyev[/i]

1996 IMO Shortlist, 8

Tags: inequalities , trigonometry , geometry , circumcircle

Let $ ABCD$ be a convex quadrilateral, and let $ R_A, R_B, R_C, R_D$ denote the circumradii of the triangles $ DAB, ABC, BCD, CDA,$ respectively. Prove that $ R_A \plus{} R_C > R_B \plus{} R_D$ if and only if $ \angle A \plus{} \angle C > \angle B \plus{} \angle D.$

2008 Hungary-Israel Binational, 2

Tags: induction , algebra

The sequence $ a_n$ is defined as follows: $ a_0\equal{}1, a_1\equal{}1, a_{n\plus{}1}\equal{}\frac{1\plus{}a_{n}^2}{a_{n\minus{}1}}$. Prove that all the terms of the sequence are integers.

2016 Harvard-MIT Mathematics Tournament, 7

Tags: hmmt

Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon.

2023 Austrian Junior Regional Competition, 2

Tags: geometry , collinear , hexagon

Let $ABCDEF$ be a regular hexagon with sidelength s. The points $P$ and $Q$ are on the diagonals $BD$ and $DF$, respectively, such that $BP = DQ = s$. Prove that the three points $C$, $P$ and $Q$ are on a line. [i](Walther Janous)[/i]

2024 Al-Khwarizmi IJMO, 5

Tags: combinatorics

At a party, every guest is a friend of exactly fourteen other guests (not including him or her). Every two friends have exactly six other attending friends in common, whereas every pair of non-friends has only two friends in common. How many guests are at the party? Please explain your answer with proof. [i]Proposed by Alexander Slavik, Czech Republic[/i]

2013 Princeton University Math Competition, 2

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Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish.

1996 Spain Mathematical Olympiad, 1

Tags: number theory , greatest common divisor , integer

The natural numbers $a$ and $b$ are such that $ \frac{a+1}{b}+ \frac{b+1}{a}$ is an integer. Show that the greatest common divisor of a and b is not greater than $\sqrt{a+b}$.

2008 ITest, 57

Tags: trigonometry , number theory , relatively prime

Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \dfrac{193}{137}.\] If $a+b=m/n$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.

2013 Math Prize For Girls Problems, 20

Tags: inequalities , trigonometry

Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and \[ a_{n} = 2 a_{n-1}^2 - 1 \] for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality \[ a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}. \] What is the value of $100c$, rounded to the nearest integer?

2014 JBMO TST - Turkey, 4

Tags: combinatorics

Alice and Bob play a game on a complete graph $G$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses a positive integer number $m,$ $1 \le m \le 1000$ and after that directs $m$ undirected edges of $G$. The game ends when all edges are directed. If there is some directed cycle in $G$ Alice wins. Determine whether Alice has a winning strategy.

LMT Speed Rounds, 2016.18

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Let $\triangle ABC$ be a triangle with $AB=5, BC=6, CA=7$. Suppose $P$ is a point inside $\triangle ABC$ such that $\triangle BPA\sim \triangle APC$. If $AP$ intersects $BC$ at $X$, find $\frac{BX}{CX}$. [i]Proposed by Nathan Ramesh

1950 AMC 12/AHSME, 6

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The values of y which will satisfy the equations $ 2x^2\plus{}6x\plus{}5y\plus{}1\equal{}0, 2x\plus{}y\plus{}3\equal{}0$ may be found by solving: $\textbf{(A)}\ y^2+14y-7=0 \qquad \textbf{(B)}\ y^2+8y+1=0 \qquad \textbf{(C)}\ y^2+10y-7=0 \qquad \textbf{(D)}\ y^2+y-12=0 \qquad \textbf{(E)}\ \text{None of these equations}$

2024 Belarusian National Olympiad, 8.1

Tags: number theory

Numbers $7^2$,$8^2,\ldots,2023^2$,$2024^2$ are written on the board. Is it possible to add to one of them $7$, to some other one $8$, $\ldots$, to the remaining $2024$ such that all numbers became prime [i]M. Zorka[/i]

2009 Ukraine National Mathematical Olympiad, 2

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There is convex $2009$-gon on the plane. [b]a)[/b] Find the greatest number of vertices of $2009$-gon such that no two forms the side of the polygon. [b]b)[/b] Find the greatest number of vertices of $2009$-gon such that among any three of them there is one that is not connected with other two by side.

2018 lberoAmerican, 6

Tags: geometry , perpendicular bisector

Let $ABC$ be an acute triangle with $AC > AB > BC$. The perpendicular bisectors of $AC$ and $AB$ cut line $BC$ at $D$ and $E$ respectively. Let $P$ and $Q$ be points on lines $AC$ and $AB$ respectively, both different from $A$, such that $AB = BP$ and $AC = CQ$, and let $K$ be the intersection of lines $EP$ and $DQ$. Let $M$ be the midpoint of $BC$. Show that $\angle DKA = \angle EKM$.

2023 India IMO Training Camp, 3

Tags: algebra

Prove that for all integers $k>2$, there exists $k$ distinct positive integers $a_1, \dots, a_k$ such that $$\sum_{1 \le i<j \le k} \frac{1}{a_ia_j} =1.$$ [i]Proposed by Anant Mudgal[/i]

2009 Kurschak Competition, 1

Tags: combinatorics

Let $n,k$ be arbitrary positive integers. We fill the entries of an $n\times k$ array with integers such that all the $n$ rows contain the integers $1,2,\dots,k$ in some order. Add up the numbers in all $k$ columns – let $S$ be the largest of these sums. What is the minimal value of $S$?

2005 Argentina National Olympiad, 2

Tags: combinatorics

On Babba Island they use a two-letter alphabet, $a$ and $b$, and every (finite) sequence of letters is a word. For each set $P$ of six words of $4$ letters each, we denote $N_P$ to the set of all words that do not contain any of the words of $P$ as a syllable (subword). Prove that if $N_P$ is finite, then all its words are of length less than or equal to $10$, and find a set $P$ such that $N_P$ is finite and contains at least one word of length $10$.

1996 AIME Problems, 8

Tags: sfft

The harmonic mean of two positive numbers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}.$

1995 All-Russian Olympiad, 3

Tags: quadratic , polynomial , symmetry , function , complex number , algebra

Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$? [i]S. Tokarev[/i]

2014 Moldova Team Selection Test, 1

Tags: quadratic , number theory

Find all pairs of non-negative integers $(x,y)$ such that \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]

2015 NZMOC Camp Selection Problems, 7

Tags: circumcircle , cyclic quadrilateral , circumcenter , geometry

Let $ABC$ be an acute-angled scalene triangle. Let $P$ be a point on the extension of $AB$ past $B$, and $Q$ a point on the extension of $AC$ past $C$ such that $BPQC$ is a cyclic quadrilateral. Let $N$ be the foot of the perpendicular from $A$ to $BC$. If $NP = NQ$ then prove that $N$ is also the centre of the circumcircle of $APQ$.