Olympiad problems

1992 Tournament Of Towns, (340) 2

On each side of a parallelogram an arbitrary point is chosen. Each pair of chosen points on neighbouring sides (i.e. sides with a common vertex) are connected by a line segment. Prove that the centres of the circumscribed circles of the four triangles so created are themselves vertices of a parallelogram. (ED Kulanin)

2011 Germany Team Selection Test, 2

Tags: floor function , number theory

Let $n$ be a positive integer prove that $$6\nmid \lfloor (\sqrt[3]{28}-3)^{-n} \rfloor.$$

2017 Iran MO (3rd round), 2

Tags: algebra , polynomial , complex number

Let $P(z)=a_d z^d+\dots+ a_1z+a_0$ be a polynomial with complex coefficients. The $reverse$ of $P$ is defined by $$P^*(z)=\overline{a_0}z^d+\overline{a_1}z^{d-1}+\dots+\overline{a_d}$$ (a) Prove that $$P^*(z)=z^d \overline{ P\left( \frac{1}{\overline{z}} \right) } $$ (b) Let $m$ be a positive integer and let $q(z)$ be a monic nonconstant polynomial with complex coefficients. Suppose that all roots of $q(z)$ lie inside or on the unit circle. Prove that all roots of the polynomial $$Q(z)=z^m q(z)+ q^*(z)$$ lie on the unit circle.

2004 China Team Selection Test, 3

Tags: inequalities

Let $k \geq 2, 1 < n_1 < n_2 < \ldots < n_k$ are positive integers, $a,b \in \mathbb{Z}^+$ satisfy \[ \prod^k_{i=1} \left( 1 - \frac{1}{n_i} \right) \leq \frac{a}{b} < \prod^{k-1}_{i=1} \left( 1 - \frac{1}{n_i} \right) \] Prove that: \[ \prod^k_{i=1} n_i \geq (4 \cdot a)^{2^k - 1}. \]

2024 Serbia Team Selection Test, 3

Tags: geometry

Let $ABC$ be a triangle with circumcenter $O$, angle bisector $AD$ with $D \in BC$ and altitude $AE$ with $E \in BC$. The lines $AO$ and $BC$ meet at $I$. The circumcircle of $\triangle ADE$ meets $AB, AC$ at $F, G$ and $FG$ meets $BC$ at $H$. The circumcircles of triangles $AHI$ and $ABC$ meet at $J$. Show that $AJ$ is a symmedian in $\triangle ABC$

2000 Brazil Team Selection Test, Problem 2

Tags: number theory

For a positive integer $n$, let $A_n$ be the set of all positive numbers greater than $1$ and less than $n$ which are coprime to $n$. Find all $n$ such that all the elements of $A_n$ are prime numbers.

2010 Puerto Rico Team Selection Test, 4

Tags: algebra , inequalities , min

Find the largest possible value in the real numbers of the term $$\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}$$ with $x^2 + y^2 \ne 0$.

2009 QEDMO 6th, 8

Tags: algebra , sum

Given $n$ integers $a_1, a_2, ..., a_n$, which $a_1 = 1$ and $a_i \le a_{i + 1} \le 2a_i$ for each $i \in \{1,2,...,n-1\}$ . Prove that if $a_1 + a_2 +... + a_n$ is even, you do select some of the numbers so that their sum equals $\frac{a_1 + a_2 +... + a_n}{2}$ .

1984 Brazil National Olympiad, 4

Tags: right triangle , minimum , projection , segment , geometry

$ABC$ is a triangle with $\angle A = 90^o$. For a point $D$ on the side $BC$, the feet of the perpendiculars to $AB$ and $AC$ are $E$ and$ F$. For which point $D$ is $ EF$ a minimum?

2016 Mathematical Talent Reward Programme, MCQ: P 1

Tags: algebra , polynomial , sum of roots , trigonometric equation

Sum of the roots in the range $\left(-\frac{\pi}{2},\frac{\pi}{2} \right)$ of the equation $\sin x\tan x=x^2$ is [list=1] [*] $\frac{\pi}{2}$ [*] 0 [*] 1 [*] None of these [/list]