Olympiad problems

2005 BAMO, 3

Let $ n\ge12$ be an integer, and let $ P_1,P_2,...P_n, Q$ be distinct points in a plane. Prove that for some $ i$, at least $ \frac{n}{6}\minus{}1$ of the distances $ P_1P_i,P_2P_i,...P_{i\minus{}1}P_i,P_{i\plus{}1}P_i,...P_nP_i$ are less than $ P_iQ$.

2018 China Team Selection Test, 4

Tags: function , algebra , functional equation , fe

Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

2001 South africa National Olympiad, 3

Tags: number theory

For a certain real number $x$, the differences between $x^{1919}$, $x^{1960}$ and $x^{2001}$ are all integers. Prove that $x$ is an integer.

1969 AMC 12/AHSME, 11

Tags: analytic geometry , graphing lines , slope

Given points $P(-1,-2)$ and $Q(4,2)$ in the $xy$-plane; point $R(1,m)$ is taken so that $PR+RQ$ is a minimum. Then $m$ equals: $\textbf{(A) }-\tfrac35\qquad \textbf{(B) }-\tfrac25\qquad \textbf{(C) }-\tfrac15\qquad \textbf{(D) }\tfrac15\qquad \textbf{(E) }\text{either }-\tfrac15\text{ or }\tfrac15$

2000 IberoAmerican, 3

Tags: algebra

Find all the solutions of the equation \[\left(x+1\right)^y-x^z=1\] For $x,y,z$ integers greater than 1.

2023 Thailand Mathematical Olympiad, 2

Tags: geometry , incenter , perpendicular bisector

Let $\triangle ABC$ which $\angle ABC$ are right angle, Let $D$ be point on $AB$ ( $D \neq A , B$ ), Let $E$ be point on line $AB$ which $B$ is the midpoint of $DE$, Let $I$ be incenter of $\triangle ACE$ and $J$ be $A$-excenter of $\triangle ACD$. Prove that perpendicular bisector of $BC$ bisects $IJ$

2007 Canada National Olympiad, 3

Tags: function , algebra , polynomial , functional inequalities

Suppose that $ f$ is a real-valued function for which \[ f(xy)+f(y-x)\geq f(y+x)\] for all real numbers $ x$ and $ y$. a) Give a non-constant polynomial that satisfies the condition. b) Prove that $ f(x)\geq 0$ for all real $ x$.

2022 Rioplatense Mathematical Olympiad, 1

Tags: combinatorics

In the blackboard there are drawn $25$ points as shown in the figure. Gastón must choose $4$ points that are vertices of a square. In how many different ways can he make this choice?$$\begin{matrix}\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \end{matrix}$$

2006 Petru Moroșan-Trident, 3

Tags: inequalities , am-gm

Let a ,b and c be positive real numbers such that $a^2+b^2+c^2=3$. Prove that for whatever positive real numbers x y and z, the inequality below holds. $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge \sqrt{xy}+\sqrt{yz}+\sqrt{zx}$ At first I noticed $\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\le \sqrt{x+y+z}\sqrt{x+y+z}=x+y+z$, so perhaps the next move is to prove $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge x+y+z$, but I don't see how to do that, the best thing that I can do with the LHS of this inequality is to prove it by AM-GM in the way that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge 3\left(\frac{xyz}{abc}\right)^{\frac{1}{3}}\ge 3(xyz)^{\frac{1}{3}}$, but this isn't going to be helpful...

LMT Team Rounds 2021+, B28

Tags: combinatorics

Maisy and Jeff are playing a game with a deck of cards with $4$ $0$’s, $4$ $1$’s, $4$ $2$’s, all the way up to $4$ $9$’s. You cannot tell apart cards of the same number. After shuffling the deck, Maisy and Jeff each take $4$ cards, make the largest $4$-digit integer they can, and then compare. The person with the larger $4$-digit integer wins. Jeff goes first and draws the cards $2,0,2,1$ from the deck. Find the number of hands Maisy can draw to beat that, if the order in which she draws the cards matters. [i]Proposed by Richard Chen[/i]

1999 Nordic, 1

Tags: functional equation , algebra

The function $f$ is defined for non-negative integers and satisfies the condition $f(n) = f(f(n + 11))$, if $n \le 1999$ and $f(n) = n - 5$, if $n > 1999$. Find all solutions of the equation $f(n) = 1999$.

2008 Germany Team Selection Test, 1

Tags: induction , algebra

A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and \[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}. \] Determine $ S_{1024}.$

2019 Thailand TSTST, 2

Tags: inequalities

Let $a,b,c\in(0,\frac{4}{3})$ and $a + b + c = 3$. Prove that $$\frac{4abc}{(a+b)(a+c)}+\frac{(a+b)^2+(a+c)^2}{(a+b)+(a+c)}\leq\sum_{cyc}\frac{1}{a^2(3b+3c-5)}.$$

2019 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt , algebra , function

Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \to \mathbb{N}$ be a function satisfying [list] [*] $f(1) = 1$, [*] for $n \in \mathbb{N}$, $f(2n) = 2f(n)$ and $f(2n+1) = 2f(n) - 1$. [/list] Determine the sum of all positive integer solutions to $f(x) = 19$ that do not exceed 2019.

2004 Romania National Olympiad, 3

Tags: 3d geometry , pyramid , trigonometry , geometry

Let $ABCD A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a truncated regular pyramid in which $BC^{\prime}$ and $DA^{\prime}$ are perpendicular. (a) Prove that $\measuredangle \left( AB^{\prime},DA^{\prime} \right) = 60^{\circ}$; (b) If the projection of $B^{\prime}$ on $(ABC)$ is the center of the incircle of $ABC$, then prove that $d \left( CB^{\prime},AD^{\prime} \right) = \frac12 BC^{\prime}$. [i]Mircea Fianu[/i]

2018 Turkey MO (2nd Round), 1

Tags: inequalities , algebra , system of equations

Find all pairs $(x,y)$ of real numbers that satisfy, \begin{align*} x^2+y^2+x+y &= xy(x+y)-\frac{10}{27}\\ |xy| & \leq \frac{25}{9}. \end{align*}

1987 ITAMO, 6

Tags: coloring , probability , combinatorics

There are three balls of distinct colors in a bag. We repeatedly draw out the balls one by one, the balls are put back into the bag after each drawing. What is the probability that, after $n$ drawings, (a) exactly one color occured? (b) exactly two oclors occured? (c) all three colors occured?

1993 Mexico National Olympiad, 5

Tags: chord , geometry , collinear

$OA, OB, OC$ are three chords of a circle. The circles with diameters $OA, OB$ meet again at $Z$, the circles with diameters $OB, OC$ meet again at $X$, and the circles with diameters $OC, OA$ meet again at $Y$. Show that $X, Y, Z$ are collinear.

2008 Mathcenter Contest, 9

Tags: polynomial , algebra

Set $P$ as a polynomial function by $p_n(x)=\sum_{k=0}^{n-1} x^k$. a) Prove that for $m,n\in N$, when dividing $p_n(x)$ by $p_m(x)$, the remainder is $$p_i(x),\forall i=0,1,...,m-1.$$ b) Find all the positive integers $i,j,k$ that make $$p_i(x)+p_j(x^2)+p_k(x^4)=p_{100}(x).$$ [i](square1zoa)[/i]

2020 USOMO, 3

Tags: number theory

Let $p$ be an odd prime. An integer $x$ is called a [i]quadratic non-residue[/i] if $p$ does not divide $x-t^2$ for any integer $t$. Denote by $A$ the set of all integers $a$ such that $1\le a<p$, and both $a$ and $4-a$ are quadratic non-residues. Calculate the remainder when the product of the elements of $A$ is divided by $p$. [i]Proposed by Richard Stong and Toni Bluher[/i]

2020 Centroamerican and Caribbean Math Olympiad, 4

Tags: geometry , incenter , circles

Consider a triangle $ABC$ with $BC>AC$. The circle with center $C$ and radius $AC$ intersects the segment $BC$ in $D$. Let $I$ be the incenter of triangle $ABC$ and $\Gamma$ be the circle that passes through $I$ and is tangent to the line $CA$ at $A$. The line $AB$ and $\Gamma$ intersect at a point $F$ with $F \neq A$. Prove that $BF=BD$.

2012 HMNT, 10

Tags: geometry

Triangle $ABC$ has $AB = 4$, $BC = 5$, and $CA = 6$. Points $A'$, $B'$, $C'$ are such that $B'C'$ is tangent to the circumcircle of $ABC$ at $A$, $C'A'$ is tangent to the circumcircle at $B$, and $A'B'$ is tangent to the circumcircle at $C$. Find the length $B'C'$.