Olympiad problems

2009 Korea Junior Math Olympiad, 1

For primes $a, b,c$ that satisfy the following, calculate $abc$. $\bullet$ $b + 8$ is a multiple of $a$, $\bullet$ $b^2 - 1$ is a multiple of $a$ and $c$ $\bullet$ $b + c = a^2 - 1$.

2016 IMO Shortlist, A7

Tags: functional equation , algebra

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$, \[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \]

2006 Grigore Moisil Intercounty, 1

Tags: geometry , trigonometry , similar triangles

Let $ABC$ be a triangle with $b\neq c$. Points $D$ is the midpoint of $BC$ and let $E$ be the foot of angle $A$ bisector. In the exterior of the triangle we construct the similar triangles $AMB$ and $ANC$ . Prove: a) $MN\bot AD \Longleftrightarrow MA \bot AB$ b) $MN\bot AE \Longleftrightarrow M,A,N$ are colinear.

2016 South African National Olympiad, 1

Tags: combinatorics

At the start of the Mighty Mathematicians Football Team's first game of the season, their coach noticed that the jersey numbers of the 22 players on the field were all the numbers from 1 to 22. At halftime, the coach substituted her goal-keeper, with jersey number 1, for a reserve player. No other substitutions were made by either team at or before halftime. The coach noticed that after the substitution, no two players on the field had the same jersey number and that the sums of the jersey numbers of each of the teams were exactly equal. Determine * the greatest possible jersey number of the reserve player, * the smallest possible (positive) jersey number of the reserve player.

2006 Korea National Olympiad, 7

Tags: geometry

Points $A,B,C,D,E,F$ is on the circle $O.$ A line $\ell$ is tangent to $O$ at $E$ is parallel to $AC$ and $DE>EF.$ Let $P,Q$ be the intersection of $\ell$ and $BC,CD$ ,respectively and let $R,S$ be the intersection of $\ell$ and $CF,DF$ ,respectively. Show that $PQ=RS$ if and only if $QE=ER.$

2017 CMIMC Team, 8

Tags: team

Alice and Bob have a fair coin with sides labeled $C$ and $M$, and they flip the coin repeatedly while recording the outcomes; for example, if they flip two $C$'s then an $M$, they have $CCM$ recorded. They play the following game: Alice chooses a four-character string $\mathcal A$, then Bob chooses two distinct three-character strings $\mathcal B_1$ and $\mathcal B_2$ such that neither is a substring of $\mathcal A$. Bob wins if $\mathcal A$ shows up in the running record before either $\mathcal B_1$ or $\mathcal B_2$ do, and otherwise Alice wins. Given that Alice chooses $\mathcal A = CMMC$ and Bob plays optimally, compute the probability that Bob wins.

2002 Vietnam Team Selection Test, 3

Tags: number theory

Let $m$ be a given positive integer which has a prime divisor greater than $\sqrt {2m} +1 $. Find the minimal positive integer $n$ such that there exists a finite set $S$ of distinct positive integers satisfying the following two conditions: [b]I.[/b] $m\leq x\leq n$ for all $x\in S$; [b]II.[/b] the product of all elements in $S$ is the square of an integer.

2006 Stanford Mathematics Tournament, 10

Tags:

What is the square root of the sum of the first 2006 positive odd integers?

2010 Regional Olympiad of Mexico Center Zone, 3

Tags: inequalities

Let $a$, $b$ and $c$ be real positive numbers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$ Prove that: $a^2+b^2+b^2 \ge 2a+2b+2c+9$

1982 Vietnam National Olympiad, 1

Tags: number theory

Find all positive integers $x, y, z$ such that $2^x + 2^y + 2^z = 2336$.

2021 239 Open Mathematical Olympiad, 1

Tags: angle bisector , geometry , circumcircle , angle , arc midpoint

Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$. Point $T$ lies on side $AC$. It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$?

1996 AMC 8, 25

Tags: ratio , geometry , symmetry , probability

A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region? $\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/3 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 2/3 \qquad \text{(E)}\ 3/4$

2000 All-Russian Olympiad Regional Round, 9.8

Tags: coloring , combinatorics

The cells of the $200 \times 200$ table are painted black and white so that there are $404$ more black cells than white ones. Prove that there is a $2 \times 2$ square in which the number of white cells is odd.

1957 Czech and Slovak Olympiad III A, 2

Tags: geometry , computational geometry , constructive geometry , 3d geometry , pyramid

Consider a (right) square pyramid $ABCDV$ with the apex $V$ and the base (square) $ABCD$. Denote $d=AB/2$ and $\varphi$ the dihedral angle between planes $VAD$ and $ABC$. (1) Consider a line $XY$ connecting the skew lines $VA$ and $BC$, where $X$ lies on line $VA$ and $Y$ lies on line $BC$. Describe a construction of line $XY$ such that the segment $XY$ is of the smallest possible length. Compute the length of segment $XY$ in terms of $d,\varphi$. (2) Compute the distance $v$ between points $V$ and $X$ in terms of $d,\varphi.$

2015 Princeton University Math Competition, A6

Tags: number theory

For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that \[\sum_{t \mid n} d(t)^3\]is divisible by $35$?

1991 Greece National Olympiad, 3

Tags: combinatorics

In how many ways can we construct a square with dimensions $4\times 4$ using $4$ white, $4$ green , $4$ red and 4 $blue$ squares of dimensions $1\times 1$, such that in every horizontal and in every certical line, squares have different colours .

2022 Francophone Mathematical Olympiad, 3

Tags: parallel , geometry , circles

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Denote $\Delta$ the tangent at $A$ to the circle $\Gamma$. $\Gamma_1$ is a circle tangent to the lines $\Delta$, $(AB)$ and $(BC)$, and $E$ its touchpoint with the line $(AB)$. Let $\Gamma_2$ be a circle tangent to the lines $\Delta$, $(AC)$ and $(BC)$, and $F$ its touchpoint with the line $(AC)$. We suppose that $E$ and $F$ belong respectively to the segments $[AB]$ and $[AC]$, and that the two circles $\Gamma_1$ and $\Gamma_2$ lie outside triangle $ABC$. Show that the lines $(BC)$ and $(EF)$ are parallel.

1991 French Mathematical Olympiad, Problem 5

Tags: polynomial , geometry , algebra , roots of unity , complex number

(a) For given complex numbers $a_1,a_2,a_3,a_4$, we define a function $P:\mathbb C\to\mathbb C$ by $P(z)=z^5+a_4z^4+a_3z^3+a_2z^2+a_1z$. Let $w_k=e^{2ki\pi/5}$, where $k=0,\ldots,4$. Prove that $$P(w_0)+P(w_1)+P(w_2)+P(w_3)+P(w_4)=5.$$(b) Let $A_1,A_2,A_3,A_4,A_5$ be five points in the plane. A pentagon is inscribed in the circle with center $A_1$ and radius $R$. Prove that there is a vertex $S$ of the pentagon for which $$SA_1\cdot SA_2\cdot SA_3\cdot SA_4\cdot SA_5\ge R^5.$$

1998 Slovenia National Olympiad, Problem 2

Tags: function , algebra

find all functions $f(x)$ satisfying: $(\forall x\in R) f(x)+xf(1-x)=x^2+1$

2006 Harvard-MIT Mathematics Tournament, 6

Tags: vieta

Let $a,b,c$ be the roots of $x^3-9x^2+11x-1=0$, and let $s=\sqrt{a}+\sqrt{b}+\sqrt{c}$. Find $s^4-18s^2-8s$.

2022/2023 Tournament of Towns, P3

Tags: number theory , decimal representation

Let us call a positive integer [i]pedestrian[/i] if all its decimal digits are equal to 0 or 1. Suppose that the product of some two pedestrian integers also is pedestrian. Is it necessary in this case that the sum of digits of the product equals the product of the sums of digits of the factors? [i]Viktor Kleptsyn, Konstantin Knop[/i]

2006 Cezar Ivănescu, 1

Tags: geometry , quadrilateral

Let be two quadrilaterals $ ABCD,A'B'C'D' $ with $ AB,BC,CD,AC,BD $ being perpendicular to $ A'B',B'C',C'D',A'C',B'D', $ respectively. Show that $ AD $ is perpendicular to $ A'D'. $

2010 Princeton University Math Competition, 1

Tags:

Let the operation $\bigstar$ be defined by $x\bigstar y=y^x-xy$. Calculate $(3\bigstar4)-(4\bigstar3)$.

2003 Bosnia and Herzegovina Junior BMO TST, 4

Tags: geometry , trapezoid , geometric inequality

In the trapezoid $ABCD$ ($AB \parallel DC$) the bases have lengths $a$ and $c$ ($c < a$), while the other sides have lengths $b$ and $d$. The diagonals are of lengths $m$ and $n$. It is known that $m^2 + n^2 = (a + c)^2$. a) Find the angle between the diagonals of the trapezoid. b) Prove that $a + c < b + d$. c) Prove that $ac < bd$.

2023 Purple Comet Problems, 19

Tags: geometry

A trapezoid has side lengths $24$, $25$, $26$, and $27$ in some order. Find its area.