Olympiad problems

2020 Thailand TSTST, 1

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(\max \left\{ x, y \right\} + \min \left\{ f(x), f(y) \right\}) = x+y $$ for all $x,y \in \mathbb{R}$.

2007 Princeton University Math Competition, 3

Tags: geometry , analytic geometry , quadratics , calculus , integration , algebra , quadratic formula

For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?

2020 Tuymaada Olympiad, 2

Tags: Inequality , inequalities , n-variable inequality

Given positive real numbers $a_1, a_2, \dots, a_n$. Let \[ m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right). \] Prove the inequality \[ \sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m. \]

MBMT Guts Rounds, 2015.16

Tags:

Your math teacher asks you to rationalize the denominator of the expression $\frac{a}{b + \sqrt{c}}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. You find that $\frac{a}{b + \sqrt{c}}$ is equal to $\frac{30 - 5\sqrt{14}}{11}$. Compute the triple $(a,b,c)$.

2015 India Regional MathematicaI Olympiad, 3

Tags: algebra , polynomial , algebra proposed , Integer Polynomial , Divisibility

Let $P(x)$ be a polynomial whose coefficients are positive integers. If $P(n)$ divides $P(P(n)-2015)$ for every natural number $n$, prove that $P(-2015)=0$. [hide]One additional condition must be given that $P$ is non-constant, which even though is understood.[/hide]

1991 Tournament Of Towns, (294) 4

Tags: combinatorial geometry , geometry , 3D geometry

(a) Is it possible to place five wooden cubes in space so that each of them has a part of its face touching each of the others? (b) Answer the same question, but with $6$ cubes.

1966 Miklós Schweitzer, 3

Tags: limit , advanced fields , advanced fields unsolved

Let $ f(n)$ denote the maximum possible number of right triangles determined by $ n$ coplanar points. Show that \[ \lim_{n\rightarrow \infty} \frac{f(n)}{n^2}\equal{}\infty \;\textrm{and}\ \lim_{n\rightarrow \infty}\frac{f(n)}{n^3}\equal{}0 .\] [i]P. Erdos[/i]

2008 Chile National Olympiad, 2

Tags: right triangle , isosceles , Triangle , geometry

Let $ABC$ be right isosceles triangle with right angle in $A$. Given a point $P$ inside the triangle, denote by $a, b$ and $c$ the lengths of $PA, PB$ and $PC$, respectively. Prove that there is a triangle whose sides have a length of $a\sqrt2 , b$ and $c$.

Russian TST 2016, P1

Tags: number theory , Diophantine equation

Find all $ x, y, z\in\mathbb{Z}^+ $ such that \[ (x-y)(y-z)(z-x)=x+y+z \]

1996 All-Russian Olympiad Regional Round, 9.2

Tags: geometry , isosceles , paralell

In triangle $ABC$, in which $AB = BC$, on side $AB$ is selected point $D$, and the ciscumcircles of triangles $ADC$ and $BDC$ , $S1$ and $S2$ respectively. The tangent drawn to $S_1$ at point $D$ intersects $S_2$ for second time at point $M$. Prove that $BM \parallel AC$.

2014 Cono Sur Olympiad, 3

Tags: geometry , rectangle

Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$. Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other. Note: the pieces can be rotated and flipped over.

2004 Croatia National Olympiad, Problem 3

Tags: number theory

The sequence $(p_n)_{n\in\mathbb N}$ is defined by $p_1=2$ and, for $n\ge2$, $p_n$ is the largest prime factor of $p_1p_2\cdots p_{n-1}+1$. Show that $p_n\ne5$ for all $n$.

2022 VIASM Summer Challenge, Problem 4

Tags: geometry

Given a triangle $ABC$ inscribed in $(O)$. Choose points $M,N,P$ on the sides $AB,BC,CA$ such that $AMNP$ is a parallelogram. The segment $CM$ intersects $NP$ at $E$; the segment $BP$ intersects $NM$ at $F$; and the segment $BE$ intersects $CF$ at $D.$ a) Prove that: $A,D,N$ are collinear. b) Let $I,J$ be the circumcenters of $\triangle MBF, \triangle PCE,$ respectively. Prove that: $OD$ passes through the midpoint of $IJ.$

1984 IMO Longlists, 14

Tags: algebra , Sequence , recurrence relation , Linear Recurrences , IMO Shortlist

Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows: \[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\] Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$

2023 Indonesia TST, 2

Tags: geometry , IMO Shortlist

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

2018 Estonia Team Selection Test, 1

Tags: Coloring , combinatorics , combinatorial geometry

There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.

2013 Sharygin Geometry Olympiad, 8

Tags: geometry , circumcircle , Asymptote , geometry proposed

Let $X$ be an arbitrary point inside the circumcircle of a triangle $ABC$. The lines $BX$ and $CX$ meet the circumcircle in points $K$ and $L$ respectively. The line $LK$ intersects $BA$ and $AC$ at points $E$ and $F$ respectively. Find the locus of points $X$ such that the circumcircles of triangles $AFK$ and $AEL$ touch.

2023 Assam Mathematics Olympiad, 14

Tags:

Find all possible triples of integers $a, b, c$ satisfying $a+b-c = 1$ and $a^2+b^2-c^2 =-1$.

2021 Iran Team Selection Test, 6

Tags: geometry , Euler Line

Point $D$ is chosen on the Euler line of triangle $ABC$ and it is inside of the triangle. Points $E,F$ are were the line $BD,CD$ intersect with $AC,AB$ respectively. Point $X$ is on the line $AD$ such that $\angle EXF =180 - \angle A$, also $A,X$ are on the same side of $EF$. If $P$ is the second intersection of circumcircles of $CXF,BXE$ then prove the lines $XP,EF$ meet on the altitude of $A$ Proposed by [i]Alireza Danaie[/i]