Olympiad problems

2014 Romania National Olympiad, 2

Tags: function

Let be a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ satisfying $ \text{(i)} f(1)=1 $ $ \text{(ii)} f(p)=1+f(p-1), $ for any prime $ p $ $ \text{(iii)} f(p_1p_2\cdots p_u)=f(p_1)+f(p_2)+\cdots f(p_u), $ for any natural number $ u $ and any primes $ p_1,p_2,\ldots ,p_u. $ Show that $ 2^{f(n)}\le n^3\le 3^{f(n)}, $ for any natural $ n\ge 2. $

1966 Miklós Schweitzer, 7

Tags: function , search , real analysis

Does there exist a function $ f(x,y)$ of two real variables that takes natural numbers as its values and for which $ f(x,y)\equal{}f(y,z)$ implies $ x\equal{}y\equal{}z?$ [i]A. Hajnal[/i]

2013 Argentina Cono Sur TST, 6

Tags:

Let $m\geq 4$ and $n\geq 4$. An integer is written on each cell of a $m \times n$ board. If each cell has a number equal to the arithmetic mean of some pair of numbers written on its neighbouring cells, determine the maximum amount of distinct numbers that the board may have. Note: two neighbouring cells share a common side.

2003 Brazil National Olympiad, 3

Tags: geometry , rhombus , trigonometry , projective geometry

$ABCD$ is a rhombus. Take points $E$, $F$, $G$, $H$ on sides $AB$, $BC$, $CD$, $DA$ respectively so that $EF$ and $GH$ are tangent to the incircle of $ABCD$. Show that $EH$ and $FG$ are parallel.

2005 MOP Homework, 7

Tags: inequalities

Let $n$ be a positive integer with $n>1$, and let $a_1$, $a_2$, ..., $a_n$ be positive integers such that $a_1<a_2<...<a_n$ and $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n} \le 1$. Prove that $(\frac{1}{a_1^2+x^2}+\frac{1}{a_2^2+x^2}+...+\frac{1}{a_n^2+x^2})^2 \le \frac{1}{2} \cdot \frac{1}{a_1(a_1-1)+x^2}$ for all real numbers $x$.

2024 ELMO Shortlist, G5

Tags: geometry

Let $ABC$ be a triangle with circumcenter $O$ and circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be points on the circumcircles of triangles $AOB$ and $AOC$, respectively, such that $A$, $P$, and $Q$ are collinear. Prove that if the circumcircle of triangle $OPQ$ is tangent to $\omega$ at $T$, then $\angle BTD=\angle CAP$. [i]Tiger Zhang[/i]

1960 Polish MO Finals, 3

Tags: geometry , hexagon , cyclic

On the circle 6 distinct points $ A $, $ B $, $ C $, $ D $, $ E $, $ F $ are chosen in such a way that $ AB $ is parallel to $ DE $, and $ DC $ is parallel to $ AF $. Prove that $ BC $ is parallel to $ EF $

2025 PErA, P4

Tags: geometry

Let $ ABC $ be an acute-angled scalene triangle. Let $ B_1 $ and $ B_2 $ be points on the rays $ BC $ and $ BA $, respectively, such that $ BB_1 = BB_2 = AC $. Similarly, let $ C_1 $ and $ C_2 $ be points on the rays $ CB $ and $ CA $, respectively, such that $ CC_1 = CC_2 = AB $. Prove that if $ B_1B_2 $ and $ C_1C_2 $ intersect at $ K $, then $ AK $ is parallel to $ BC $.

2022 JHMT HS, 7

Tags: geometry , optimization , slope

Two rays emanate from the origin $O$ and form a $45^\circ$ angle in the first quadrant of the Cartesian coordinate plane. For some positive numbers $X$, $Y$, and $S$, the ray with the larger slope passes through point $A = (X, S)$, and the ray with the smaller slope passes through point $B = (S, Y)$. If $6X + 6Y + 5S = 600$, then determine the maximum possible area of $\triangle OAB$.

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.

2018 CMIMC Algebra, 3

Tags: algebra

Let $P(x)=x^2+4x+1$. What is the product of all real solutions to the equation $P(P(x))=0$?

2009 Tournament Of Towns, 5

Tags: combinatorics

A country has two capitals and several towns. Some of them are connected by roads. Some of the roads are toll roads where a fee is charged for driving along them. It is known that any route from the south capital to the north capital contains at least ten toll roads. Prove that all toll roads can be distributed among ten companies so that anybody driving from the south capital to the north capital must pay each of these companies. [i](5 points)[/i]

2024 ELMO Shortlist, G1

Tags: geometry , water balloon , power of a point

In convex quadrilateral $ABCD$, let diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Let the circumcircles of $ADE$ and $BCE$ intersect $\overline{AB}$ again at $P \neq A$ and $Q \neq B$, respectively. Let the circumcircle of $ACP$ intersect $\overline{AD}$ again at $R \neq A$, and let the circumcircle of $BDQ$ intersect $\overline{BC}$ again at $S \neq B$. Prove that $A$, $B$, $R$, and $S$ are concyclic. [i]Tiger Zhang[/i]

2019 Finnish National High School Mathematics Comp, 5

Tags: combinatorics

A teacher is known to have $2^k$ apples for some $k \in \mathbb{N}$. He ets one of the apples and distributes the rest of the apples to his students $A$ and $B$. The students do not see how many apples the other gets, and they do not know the number $k$. However, they have pre-selected a discreet way to reveal one another something about the number of apples: each of the students scratches their head either by their right, left or both hands, depending on the number of apples they have received. To the teacher's surprise, the students will always know which one of the students got more apples, or that the teacher ate the only apple by herself. How is this possible?

1991 Irish Math Olympiad, 4

Tags: functional equation , functional equation in q , algebra

Let $\mathbb{P}$ be the set of positive rational numbers and let $f:\mathbb{P}\to\mathbb{P}$ be such that $$f(x)+f\left(\frac{1}{x}\right)=1$$ and $$f(2x)=2f(f(x))$$ for all $x\in\mathbb{P}$. Find, with proof, an explicit expression for $f(x)$ for all $x\in \mathbb{P}$.

2014 Saudi Arabia IMO TST, 2

Tags: analytic geometry , graph theory , combinatorics

Define a [i]domino[/i] to be an ordered pair of [i]distinct[/i] positive integers. A [i]proper sequence[/i] of dominoes is a list of distinct dominoes in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i, j)$ and $(j, i)$ do not [i]both[/i] appear for any $i$ and $j$. Let $D_n$ be the set of all dominoes whose coordinates are no larger than $n$. Find the length of the longest proper sequence of dominoes that can be formed using the dominoes of $D_n$.

1966 Czech and Slovak Olympiad III A, 2

Tags: combinatorial geometry , circles , geometry

Into how many regions do $n$ circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?

2021 AMC 10 Fall, 11

Tags: emily

Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster tha the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship? $\textbf{(A) }70\qquad\textbf{(B) }84\qquad\textbf{(C) }98\qquad\textbf{(D) }105\qquad\textbf{(E) }126$

2012 NIMO Problems, 2

Tags: algebra , polynomial , vieta

If $r_1$, $r_2$, and $r_3$ are the solutions to the equation $x^3 - 5x^2 + 6x - 1 = 0$, then what is the value of $r_1^2 + r_2^2 + r_3^2$? [i]Proposed by Eugene Chen[/i]

2024/2025 TOURNAMENT OF TOWNS, P3

Tags: geometry

In an acute-angled triangle ${ABC}$ , its incenter $I$ and circumcenter $O$ are marked. The lines ${AI}$ and ${CI}$ have second intersections with the circumcircle of ${ABC}$ at points $N$ and $M$ respectively. The segments ${MN}$ and ${BO}$ intersect at the point $X$ . Prove that the lines ${XI}$ and ${AC}$ are perpendicular. Fedor Ivlev

2024 Indonesia TST, A

Tags: algebra , inequalities , absolute value

Given real numbers $x,y,z$ which satisfies $$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$ Show that $max\{ |x|,|y|,|z|\} \le 1$.

1983 Polish MO Finals, 2

Tags: inequalities , algebra , irrational

Let be given an irrational number $a$ in the interval $(0,1)$ and a positive integer $N$. Prove that there exist positive integers $p,q,r,s$ such that $\frac{p}{q} < a <\frac{r}{s}, \frac{r}{s} -\frac{p}{q}<\frac{1}{N}$, and $rq- ps = 1$.

1999 VJIMC, Problem 2

Tags: limit , real analysis , function

Let $a,b\in\mathbb R$, $a\le b$. Assume that $f:[a,b]\to[a,b]$ satisfies $f(x)-f(y)\le|x-y|$ for every $x,y\in[a,b]$. Choose an $x_1\in[a,b]$ and define $$x_{n+1}=\frac{x_n+f(x_n)}2,\qquad n=1,2,3,\ldots.$$Show that $\{x_n\}^\infty_{n=1}$ converges to some fixed point of $f$.

2016 Thailand Mathematical Olympiad, 7

Tags: polynomial , absolute value , algebra

Given $P(x)=a_{2016}x^{2016}+a_{2015}x^{2015}+...+a_1x+a_0$ be a polynomial with real coefficients and $a_{2016} \neq 0$ satisfies $|a_1+a_3+...+a_{2015}| > |a_0+a_2+...+a_{2016}|$ Prove that $P(x)$ has an odd number of complex roots with absolute value less than $1$ (count multiple roots also) edited: complex roots

2024 Al-Khwarizmi IJMO, 7

Tags: geometry

Two circles with centers $O_{1}$ and $O_{2}$ intersect at $P$ and $Q$. Let $\omega$ be the circumcircle of the triangle $P O_{1} O_{2}$; the circle $\omega$ intersect the circles centered at $O_{1}$ and $O_{2}$ at points $A$ and $B$, respectively. The point $Q$ is inside triangle $P A B$ and $P Q$ intersects $\omega$ at $M$. The point $E$ on $\omega$ is such that $P Q=Q E$. Let $M E$ and $A B$ meet at $L$, prove that $\angle Q L A=\angle M L A$. [i]Proposed by Amir Parsa Hoseini Nayeri, Iran[/i]