Olympiad problems

2010 Tuymaada Olympiad, 2

In acute triangle $ABC$, let $H$ denote its orthocenter and let $D$ be a point on side $BC$. Let $P$ be the point so that $ADPH$ is a parallelogram. Prove that $\angle DCP<\angle BHP$.

2024 LMT Fall, 11

Tags: team

Let $\phi=\tfrac{1+\sqrt 5}{2}$. Find \[\left(4+\phi^{\frac12}\right)\left(4-\phi^{\frac12}\right)\left(4+i\phi^{-\frac12}\right)\left(4-i\phi^{-\frac12}\right).\]

1985 Federal Competition For Advanced Students, P2, 5

Tags: number theory

A sequence $ (a_n)$ of positive integers satisfies: $ a_n\equal{}\sqrt{\frac{a_{n\minus{}1}^2\plus{}a_{n\plus{}1}^2}{2}}$ for all $ n \ge 1$. Prove that this sequence is constant.

2006 China Team Selection Test, 3

Tags: geometry

$\triangle{ABC}$ can cover a convex polygon $M$.Prove that there exsit a triangle which is congruent to $\triangle{ABC}$ such that it can also cover $M$ and has one side line paralel to or superpose one side line of $M$.

2010 Contests, 3

Tags: trigonometry , angle bisector , geometry

Let $ABC$ be an isosceles triangle with apex at $C.$ Let $D$ and $E$ be two points on the sides $AC$ and $BC$ such that the angle bisectors $\angle DEB$ and $\angle ADE$ meet at $F,$ which lies on segment $AB.$ Prove that $F$ is the midpoint of $AB.$

2021 AMC 10 Spring, 15

Tags:

The real number $x$ satisfies the equation $x+\frac{1}{x}=\sqrt{5}$. What is the value of $x^{11}-7x^7+x^3$? $\textbf{(A)}\ -1 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 8$

2006 Cono Sur Olympiad, 3

Tags: inequalities , number theory

Let $n$ be a natural number. The finite sequence $\alpha$ of positive integer terms, there are $n$ different numbers ($\alpha$ can have repeated terms). Moreover, if from one from its terms any we subtract 1, we obtain a sequence which has, between its terms, at least $n$ different positive numbers. What's the minimum value of the sum of all the terms of $\alpha$?

PEN M Problems, 9

Tags: floor function , recursive sequences

An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=2, \; a_{n+1}=\left\lfloor \frac{3}{2}a_{n}\right\rfloor.\] Show that it has infinitely many even and infinitely many odd integers.

2021 MMATHS, 5

Tags:

Suppose that $a_1 = 1$, and that for all $n \ge 2$, $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$ Suppose furthermore that $b_n = a_1 + a_2 + \ldots + a_n$ for all $n$. If $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$ for some $k$, find $k$. [i]Proposed by Andrew Wu[/i]

2005 Serbia Team Selection Test, 5

Tags: inequalities

Let $a,b,c$ be positive reals such that $abc=1$ .Prove the inequality $\frac{a}{a^2+2}+\frac{b}{b^2+2}+\frac{c}{c^2+2}\leq 1$

1971 IMO Longlists, 9

Tags: 3d geometry , prism , trigonometry , geometry

The base of an inclined prism is a triangle $ABC$. The perpendicular projection of $B_1$, one of the top vertices, is the midpoint of $BC$. The dihedral angle between the lateral faces through $BC$ and $AB$ is $\alpha$, and the lateral edges of the prism make an angle $\beta$ with the base. If $r_1, r_2, r_3$ are exradii of a perpendicular section of the prism, assuming that in $ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1, \angle A < \angle B < \angle C,$ and $BC = a$, calculate $r_1r_2 + r_1r_3 + r_2r_3.$

2022 Grosman Mathematical Olympiad, P4

Tags: combinatorics , inequality

Along a circle-shaped path are $100$ boys and $100$ girls. The distance between two points on the path is defined as the length of the smaller arc through which it is possible to get from one point to the other. Prove that the sum of distances between pairs of the same gender is always less than or equal to the sum of distances between all pairs of a boy and a girl.

2012 India IMO Training Camp, 3

Tags: function , algebra

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a function such that $f(x+y+xy)=f(x)+f(y)+f(xy)$ for all $x, y\in\mathbb{R}$. Prove that $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x, y\in\mathbb{R}$.

Kyiv City MO Juniors Round2 2010+ geometry, 2012.7.3

Tags: geometry , median , equal segments

In the triangle $ABC $ the median $BD$ is drawn, which is divided into three equal parts by the points $E $ and $F$ ($BE = EF = FD$). It is known that $AD = AF$ and $AB = 1$. Find the length of the segment $CE$.

2016 Thailand Mathematical Olympiad, 6

Tags: prime , power of 2 , number theory

Let $m$ and $n$ be positive integers. Prove that if $m^{4^n+1} - 1$ is a prime number, then there exists an integer $t \ge 0$ such that $n = 2^t$.

2023 Junior Balkan Mathematical Olympiad, 2

Tags: inequality

Prove that for all non-negative real numbers $x,y,z$, not all equal to $0$, the following inequality holds $\displaystyle \dfrac{2x^2-x+y+z}{x+y^2+z^2}+\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\dfrac{2z^2+x+y-z}{x^2+y^2+z}\geq 3.$ Determine all the triples $(x,y,z)$ for which the equality holds. [i]Milan Mitreski, Serbia[/i]

1969 Vietnam National Olympiad, 4

Tags: equal , geometry , circles , minimum , ratio

Two circles centers $O$ and $O'$, radii $R$ and $R'$, meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$. Find the locus of $H$ and $H'$. If $OO'^2 < R^2 + R'^2$, find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value. Show that this value does not depend on the position of $L$. Comment on the case $OO'^2 > R^2 + R'^2$.

2021 MOAA, 17

Tags: team

Compute the remainder when $10^{2021}$ is divided by $10101$. [i]Proposed by Nathan Xiong[/i]

Kvant 2019, M2575

Tags: algebra , polynomial

Let $t\in (1,2)$. Show that there exists a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ with the coefficients in $\{1,-1\}$ such that $\left|P(t)-2019\right| \leqslant 1.$ [i]Proposed by N. Safaei (Iran)[/i]

2015 FYROM JBMO Team Selection Test, 1

Tags: number theory , diophantine equation , integer

Solve the equation $x^2+y^4+1=6^z$ in the set of integers.

2008 District Round (Round II), 2

Tags:

Two circles $U,V$ have distinct radii,tangent to each other externally at $T$.$A,B$ are points on $U,V$ respectively,both distinct from $T$,such that $\angle ATB=90$. (1)Prove that line $AB$ passes through a fixed point; (2)Find the locus of the midpoint of $AB$.

2020 Azerbaijan National Olympiad, 1

Tags: combinatorics

$13$ fractions are corrected by using each of the numbers $1,2,...,26$ once.[b]Example:[/b]$\frac{12}{5},\frac{18}{26}.... $ What is the maximum number of fractions which are integers?

1994 National High School Mathematics League, 4

Tags:

$0<b<1,0<a<\frac{\pi}{4}$,$x=(\sin a)^{\log_{b}\sin a},y=(\cos a)^{\log_{b}\cos a},z=(\sin a)^{\log_{b}\cos a}$. Then the order of $x,y,z$ is $\text{(A)}x<z<y\qquad\text{(B)}y<z<x\qquad\text{(C)}z<x<y\qquad\text{(D)}x<y<z$

1950 Poland - Second Round, 2

Tags: algebra , inequalities

Prove that if $a > 0$, $b > 0$, $abc=1$, then $$a+b+c \ge 3$$

2008 Baltic Way, 18

Tags: trigonometry , circumcircle , radical axis , geometry

Let $ AB$ be a diameter of a circle $ S$, and let $ L$ be the tangent at $ A$. Furthermore, let $ c$ be a fixed, positive real, and consider all pairs of points $ X$ and $ Y$ lying on $ L$, on opposite sides of $ A$, such that $ |AX|\cdot |AY| \equal{} c$. The lines $ BX$ and $ BY$ intersect $ S$ at points $ P$ and $ Q$, respectively. Show that all the lines $ PQ$ pass through a common point.