Olympiad problems

2004 Romania National Olympiad, 3

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 $x^2+ y^2+ z^2 - xy - yz - zx = 3$

2016 Switzerland Team Selection Test, Problem 1

Tags: number theory , greatest common divisor , set

Let $n$ be a natural number. Two numbers are called "unsociable" if their greatest common divisor is $1$. The numbers $\{1,2,...,2n\}$ are partitioned into $n$ pairs. What is the minimum number of "unsociable" pairs that are formed?

2014 Harvard-MIT Mathematics Tournament, 20

Tags:

A deck of $8056$ cards has $2014$ ranks numbered $1$–$2014$. Each rank has four suits - hearts, diamonds, clubs, and spades. Each card has a rank and a suit, and no two cards have the same rank and the same suit. How many subsets of the set of cards in this deck have cards from an odd number of distinct ranks?