Olympiad problems

2006 IMO Shortlist, 4

Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]

2018 India IMO Training Camp, 2

Tags: function , algebra

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

2016 Azerbaijan JBMO TST, 1

Tags: inequalities

Let $a,b,c \ge 0$ and $a+b+c=3$. Prove that: $2(ab+bc+ca)-3abc\ge \sum_{cyc}^{}a\sqrt{\frac{b^2+c^2}{2}}$

2008 Sharygin Geometry Olympiad, 3

Tags: inequalities , inradius , trigonometry , geometry

(R.Pirkuliev) Prove the inequality \[ \frac1{\sqrt {2\sin A}} \plus{} \frac1{\sqrt {2\sin B}} \plus{} \frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}}, \] where $ p$ and $ r$ are the semiperimeter and the inradius of triangle $ ABC$.

2010 Romania National Olympiad, 4

Tags: limit , logarithm , real analysis

Let $a\in \mathbb{R}_+$ and define the sequence of real numbers $(x_n)_n$ by $x_1=a$ and $x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1$. Prove that the sequence is convergent and find it's limit.

2021 Brazil Team Selection Test, 1

Tags: number theory

Let $p>10$ be a prime. Prove that there is positive integers $m,n$ with $m+n<p$ such that $p$ divides $5^m7^n -1$

2003 Nordic, 3

Tags: right triangle , equilateral triangle , geometry

The point ${D}$ inside the equilateral triangle ${\triangle ABC}$ satisfies ${\angle ADC = 150^o}$. Prove that a triangle with side lengths ${|AD|, |BD|, |CD|}$ is necessarily a right-angled triangle.

2011 F = Ma, 6

Tags:

A child is sliding out of control with velocity $v_\text{c}$ across a frozen lake. He runs head-on into another child, initially at rest, with $3$ times the mass of the first child, who holds on so that the two now slide together. What is the velocity of the couple after the collision? (A) $2v_\text{c}$ (B) $v_\text{c}$ (C) $v_\text{c}/2$ (D) $v_\text{c}/3$ (E) $v_\text{c}/4$

2017 AMC 12/AHSME, 10

Tags: ratio

At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it? $\textbf{(A) } 10\%\qquad \textbf{(B) } 12\%\qquad \textbf{(C) } 20\%\qquad \textbf{(D) } 25\%\qquad \textbf{(E) } 33\frac{1}{3}\%$

2024 China Team Selection Test, 19

Tags: combinatorics , trapezoid

$n$ is a positive integer. An equilateral triangle of side length $3n$ is split into $9n^2$ unit equilateral triangles, each colored one of red, yellow, blue, such that each color appears $3n^2$ times. We call a trapezoid formed by three unit equilateral triangles as a "standard trapezoid". If a "standard trapezoid" contains all three colors, we call it a "colorful trapezoid". Find the maximum possible number of "colorful trapezoids".

1998 AMC 12/AHSME, 11

Tags: geometry , rectangle

Let R be a rectangle. How many circles in the plane of R have a diameter both of whose endpoints are vertices of R? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

2025 Vietnam National Olympiad, 1

Tags: algebra , polynomial , sequence , converges

Let $P(x) = x^4-x^3+x$. a) Prove that for all positive real numbers $a$, the polynomial $P(x) - a$ has a unique positive zero. b) A sequence $(a_n)$ is defined by $a_1 = \dfrac{1}{3}$ and for all $n \geq 1$, $a_{n+1}$ is the positive zero of the polynomial $P(x) - a_n$. Prove that the sequence $(a_n)$ converges, and find the limit of the sequence.

2017 IFYM, Sozopol, 1

Tags: algebra , inequalities

Let $x,y,z\in \mathbb{R}^+$ be such that $xy+yz+zx=x+y+z$. Prove the following inequality: $\frac{1}{x^2+y+1}+\frac{1}{y^2+z+1}+\frac{1}{z^2+x+1}\leq 1$.

1963 All Russian Mathematical Olympiad, 030

Tags: number theory , greatest common divisor , relatively prime

Natural numbers $a$ and $b$ are relatively prime. Prove that the greatest common divisor of $(a+b)$ and $(a^2+b^2)$ is either $1$ or $2$.

2017 BMT Spring, 5

Tags: algebra , number theory

How many pairs of positive integers $(a, b)$ satisfy the equation $log_a 16 = b$?

1982 IMO, 2

Tags: geometry , hexagon , collinearity , golden ratio

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine $r$ if $B,M$ and $N$ are collinear.

2023 JBMO TST - Turkey, 1

Tags: number theory

Let $n,k$ are integers and $p$ is a prime number. Find all $(n,k,p)$ such that $|6n^2-17n-39|=p^k$

2002 VJIMC, Problem 4

Tags: inequalities

The numbers $1,2,\ldots,n$ are assigned to the vertices of a regular $n$-gon in an arbitrary order. For each edge, compute the product of the two numbers at the endpoints and sum up these products. What is the smallest possible value of this sum?

1998 IberoAmerican Olympiad For University Students, 2

Tags: ellipse , geometry , conic

In a plane there is an ellipse $E$ with semiaxis $a,b$. Consider all the triangles inscribed in $E$ such that at least one of its sides is parallel to one of the axis of $E$. Find both the locus of the centroid of all such triangles and its area.

2005 Taiwan TST Round 1, 1

Tags: diophantine equation , number theory

Prove that there exists infinitely many positive integers $n$ such that $n, n+1$, and $n+2$ can be written as the sum of two perfect squares.

1991 Federal Competition For Advanced Students, 4

Tags: geometry

Let $ AB$ be a chord of a circle $ k$ of radius $ r$, with $ AB\equal{}c$. $ (a)$ Construct the triangle $ ABC$ with $ C$ on $ k$ in which a median from $ A$ or $ B$ is of a given length $ d.$ $ (b)$ For which $ c$ and $ d$ is this triangle unique?

2015 Princeton University Math Competition, A7

Tags: geometry

Triangle $ABC$ has $\overline{AB} = \overline{AC} = 20$ and $\overline{BC} = 15$. Let $D$ be the point in $\triangle ABC$ such that $\triangle ADB \sim \triangle BDC$. Let $l$ be a line through $A$ and let $BD$ and $CD$ intersect $l$ at $P$ and $Q$, respectively. Let the circumcircles of $\triangle BDQ$ and $\triangle CDP$ intersect at $X$. The area of the locus of $X$ as $l$ varies can be expressed in the form $\tfrac{p}{q}\pi$ for positive coprime integers $p$ and $q$. What is $p + q$?

2002 Moldova National Olympiad, 1

Tags: trigonometry

Solve in $ \mathbb R$ the equation $ \sqrt{1\minus{}x}\equal{}2x^2\minus{}1\plus{}2x\sqrt{1\minus{}x^2}$.

2017 Korea Winter Program Practice Test, 2

Tags: function , number theory

Find all functions $f : \mathbb{N} \to \mathbb{N}$ satisfying the following conditions: [list] [*]For every $n \in \mathbb{N}$, $f^{(n)}(n) = n$. (Here $f^{(1)} = f$ and $f^{(k)} = f^{(k-1)} \circ f$.) [*]For every $m, n \in \mathbb{N}$, $\lvert f(mn) - f(m) f(n) \rvert < 2017$. [/list]

1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 4

Tags: geometry , similar triangles

Given three squares as in the figure (where the vertex of B is touching square A --- the diagram had an error), where the largest square has area 1, and the area $ A$ is known. What is the area $ B$ of the smallest square? [img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1995Number4.jpg[/img] A. $ A/8$ B. $ \frac {A^2}{2}$ C. $ \frac {A^4}{4}$ D. $ A(1 \minus{} A)$ E. $ \frac {(1 \minus{} A)^2}{4}$