Olympiad problems

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 $B'C'$.

1998 National High School Mathematics League, 1

Tags: logarithm

If $a>1,b>1,\lg(a+b)=\lg a+\lg b$, then the value of $\lg(a-1)+\lg(b-1)$ is $\text{(A)}\lg2\qquad\text{(B)}1\qquad\text{(C)}0\qquad\text{(D)}$ not sure

2015 Postal Coaching, 4

Tags: sequence , induction , recurrence , number theory

The sequence $<a_n>$ is defined as follows, $a_1=a_2=1$, $a_3=2$, $$a_{n+3}=\frac{a_{n+2}a_{n+1}+n!}{a_n},$$ $n \ge 1$. Prove that all the terms in the sequence are integers.

1999 India Regional Mathematical Olympiad, 6

Tags: quadratic

Find all solutions in integers $m,n$ of the equation \[ (m-n)^2 = \frac{4mn}{ m+n-1}. \]

2019 BMT Spring, Tie 2

Tags: algebra

If $P$ is a function such that $P(2x) = 2^{-3}P(x) + 1$, find $P(0)$.

1963 Putnam, B3

Tags: function , algebra , functional equation

Find every twice-differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the functional equation $$ f(x)^2 -f(y)^2 =f(x+y)f(x-y)$$ for all $x,y \in \mathbb{R}. $

2015 AIME Problems, 8

Tags:

For positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Find the smallest positive integer $n$ satisfying $s(n)=s(n+864)=20$.

2019 Adygea Teachers' Geometry Olympiad, 2

Tags: triangle , geometry

Inside the triangle $T$ there are three other triangles that do not have common points. Is it true that one can choose such a point inside $T$ and draw three rays from it so that the triangle breaks into three parts, in each of which there will be one triangle?

2018 Regional Olympiad of Mexico Northeast, 4

Tags: algebra , sequence

We have an infinite sequence of integers $\{x_n\}$, such that $x_1 = 1$, and, for all $n \ge 1$, it holds that $x_n < x_{n+1} \le 2n$. Prove that there are two terms of the sequence,$ x_r$ and $x_s$, such that $x_r - x_s = 2018$.

2016 NIMO Summer Contest, 6

Tags:

A positive integer $n$ is lucky if $2n+1$, $3n+1$, and $4n+1$ are all composite numbers. Compute the smallest lucky number. [i]Proposed by Michael Tang[/i]

2013 Harvard-MIT Mathematics Tournament, 12

Tags: hmmt , inequalities

For how many integers $1\leq k\leq 2013$ does the decimal representation of $k^k$ end with a $1$?

2013 Harvard-MIT Mathematics Tournament, 8

Tags: geometry , power of a point , radical axis

Let points $A$ and $B$ be on circle $\omega$ centered at $O$. Suppose that $\omega_A$ and $\omega_B$ are circles not containing $O$ which are internally tangent to $\omega$ at $A$ and $B$, respectively. Let $\omega_A$ and $\omega_B$ intersect at $C$ and $D$ such that $D$ is inside triangle $ABC$. Suppose that line $BC$ meets $\omega$ again at $E$ and let line $EA$ intersect $\omega_A$ at $F$. If $ FC \perp CD $, prove that $O$, $C$, and $D$ are collinear.

2020 Princeton University Math Competition, A8

Tags: number theory

What is the smallest integer $a_0$ such that, for every positive integer $n$, there exists a sequence of positive integers $a_0, a_1, ..., a_{n-1}, a_n$ such that the first $n-1$ are all distinct, $a_0 = a_n$, and for $0 \le i \le n -1$, $a_i^{a_{i+1}}$ ends in the digits $\overline{0a_i}$ when expressed without leading zeros in base $10$.

1989 AMC 12/AHSME, 1

Tags:

$(-1)^{5^2} + 1^{2^5} =$ $\textbf{(A)}\ -7 \qquad \textbf{(B)}\ -2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 57$

1995 Moldova Team Selection Test, 9

Tags: number theory

For every nonempty set $M{}$ of integers denote $S(M)$ the sum of all its elements. Let $A=\{a_1,a_2,\ldots,a_{11}\}$ be a set of positive integers with the properties: 1) $a_1<a_2<\ldots<a_{11};$ 2) for every positive integer $n\leq 1500$ there is a subset $M{}$ of $A{}$ for which $S(M)=n.$ Find the smallest possible value of $a_{10}.$

2023 Bundeswettbewerb Mathematik, 2

Tags: algebra , math competition

Determine all triples $(x, y, z)$ of integers that satisfy the equation $x^2+ y^2+ z^2 - xy - yz - zx = 3$