Olympiad problems

1987 Polish MO Finals, 1

There are $n \ge 2$ points in a square side $1$. Show that one can label the points $P_1, P_2, ... , P_n$ such that $\sum_{i=1}^n |P_{i-1} - P_i|^2 \le 4$, where we use cyclic subscripts, so that $P_0$ means $P_n$.

2017 Online Math Open Problems, 2

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The numbers $a,b,c,d$ are $1,2,2,3$ in some order. What is the greatest possible value of $a^{b^{c^d}}$? [i]Proposed by Yannick Yao and James Lin[/i]

2023 Iran MO (3rd Round), 2

Tags: geometry

In triangle $\triangle ABC$ , $M$ is the midpoint of arc $(BAC)$ and $N$ is the antipode of $A$ in $(ABC)$. The line through $B$ perpendicular to $AM$ , intersects $AM , (ABC)$ at $D,P$ respectively and a line through $D$ perpendicular to $AC$ , intersects $BC,AC$ at $F,E$ respectively. Prove that $PE,MF,ND$ are concurrent.

2021 BMT, 12

Tags: algebra

Let $a$, $b$, and $c$ be the solutions of the equation $$x^3 - 3 \cdot 2021^2x = 2 \cdot 20213.$$ Compute $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$

1985 ITAMO, 7

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Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$.

Revenge EL(S)MO 2024, 5

Tags: hyperbola , ellipse , geometry , conic

In triangle $ABC$ let the $A$-foot be $E$ and the $B$-excenter be $L$. Suppose the incircle of $ABC$ is tangent to $AC$ at $I$. Construct a hyperbola $\mathcal H$ through $A$ with $B$ and $C$ as the foci such that $A$ lies on the branch of the $\mathcal H$ closer to $C$. Construct an ellipse $\mathcal E$ passing through $I$ with $B$ and $C$ as the foci. Suppose $\mathcal E$ meets $\overline{AB}$ again at point $H$. Let $\overline{CH}$ and $\overline{BI}$ intersect the $C$-branch of $\mathcal H$ at points $M$ and $O$ respectively. Prove $E$, $L$, $M$, $O$ are concyclic. Proposed by [i]Alex Wang[/i]

2010 Postal Coaching, 4

Tags: relatively prime , number theory

For each $n\in \mathbb{N}$, let $S(n)$ be the sum of all numbers in the set $\{ 1, 2, 3, \cdots , n \}$ which are relatively prime to $n$. $(a)$ Show that $2 \cdot S(n)$ is not a perfect square for any $n$. $(b)$ Given positive integers $m, n$, with odd $n$, show that the equation $2 \cdot S(x) = y^n$ has at least one solution $(x, y)$ among positive integers such that $m|x$.

2007 India IMO Training Camp, 3

Tags: function , ratio , algebra

Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation \[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\] For all $x,y\in\mathbb R.$

1969 IMO, 3

Tags: geometry , 3d geometry , tetrahedron , triangle inequality

For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.

2021 Middle European Mathematical Olympiad, 7

Tags: number theory , diophantine equation , easy number theorem

Find all pairs $(n, p)$ of positive integers such that $p$ is prime and \[ 1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2). \]

1951 AMC 12/AHSME, 49

Tags: pythagorean theorem , geometry

The medians of a right triangle which are drawn from the vertices of the acute angles are $ 5$ and $ \sqrt {40}$. The value of the hypotenuse is: $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 2\sqrt {40} \qquad\textbf{(C)}\ \sqrt {13} \qquad\textbf{(D)}\ 2\sqrt {13} \qquad\textbf{(E)}\ \text{none of these}$

1995 AIME Problems, 13

Tags: floor function , inequalities

Let $f(n)$ be the integer closest to $\sqrt[4]{n}.$ Find $\sum_{k=1}^{1995}\frac 1{f(k)}.$

1955 Moscow Mathematical Olympiad, 311

Tags: number theory , floor function

Find all numbers $a$ such that (1) all numbers $[a], [2a], . . . , [Na]$ are distinct and (2) all numbers $\left[ \frac{1}{a}\right], \left[ \frac{2}{a}\right], ..., \left[ \frac{M}{a}\right]$ are distinct.

2016 German National Olympiad, 6

Tags: cyclic function , function , inequality , maximum , algebra

Let \[ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 \] be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$. Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.

2017 Online Math Open Problems, 23

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Call a nonempty set $V$ of nonzero integers \emph{victorious} if there exists a polynomial $P(x)$ with integer coefficients such that $P(0)=330$ and that $P(v)=2|v|$ holds for all elements $v\in V$. Find the number of victorious sets. [i]Proposed by Yannick Yao[/i]

2004 Germany Team Selection Test, 2

Tags: geometry , fixed point

Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.

2003 Austria Beginners' Competition, 3

Tags: number theory , consecutive , divide , divisible

a) Show that the product of $5$ consecutive even integers is divisible by $15$. b) Determine the largest integer $D$ such that the product of $5$ consecutive even integers is always divisible by $D$.

2010 Junior Balkan Team Selection Tests - Moldova, 5

Tags: inequalities

For any strictly positive numbers $a$ and $b$ , prove the inequality $$\frac{a}{a+b} \cdot \frac{a+2b}{a+3b} < \sqrt{ \frac{a}{a+4b}}.$$

2011 Costa Rica - Final Round, 6

Tags: geometric transformation , reflection , incenter , parallelogram , geometry

Let $ABC$ be a triangle. The incircle of $ABC$ touches $BC,AC,AB$ at $D,E,F$, respectively. Each pair of the incircles of triangles $AEF, BDF,CED$ has two pair of common external tangents, one of them being one of the sides of $ABC$. Show that the other three tangents divide triangle $DEF$ into three triangles and three parallelograms.

2015 Bosnia and Herzegovina Junior BMO TST, 2

Tags: number theory , solve , triplets

Find all triplets of positive integers $a$, $b$ and $c$ such that $a \geq b \geq c$ and $\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right)=2$

2015 Online Math Open Problems, 9

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Let $s_1, s_2, \dots$ be an arithmetic progression of positive integers. Suppose that \[ s_{s_1} = x+2, \quad s_{s_2} = x^2+18, \quad\text{and}\quad s_{s_3} = 2x^2+18. \] Determine the value of $x$. [i] Proposed by Evan Chen [/i]

1986 Swedish Mathematical Competition, 1

Tags: algebra , polynomial

Show that the polynomial $x^6 -x^5 +x^4 -x^3 +x^2 -x+\frac34$ has no real zeroes.

2018 Korea Winter Program Practice Test, 2

Tags: geometric inequality , geometry , circumcircle , inequalities , area

Let $\Delta ABC$ be a triangle and $P$ be a point in its interior. Prove that \[ \frac{[BPC]}{PA^2}+\frac{[CPA]}{PB^2}+\frac{[APB]}{PC^2} \ge \frac{[ABC]}{R^2} \] where $R$ is the radius of the circumcircle of $\Delta ABC$, and $[XYZ]$ is the area of $\Delta XYZ$.

JOM 2024, 3

Tags: algebra , functional equation

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that for all $x, y\in\mathbb{R}^+$, \[ \frac{f(x)}{y^2} - \frac{f(y)}{x^2} \le \left(\frac{1}{x}-\frac{1}{y}\right)^2\] ($\mathbb{R}^+$ denotes the set of positive real numbers.) [i](Proposed by Ivan Chan Guan Yu)[/i]

2019 South Africa National Olympiad, 1

Tags: number theory , exponent

Determine all positive integers $a$ for which $a^a$ is divisible by $20^{19}$.