Olympiad problems

2012 Turkmenistan National Math Olympiad, 3

Tags: logarithm , inequalities

Prove that : $\frac{1}{(\log_{bc} a)^n}+\frac{1}{(\log_{ac} b)^n}+\frac{1}{(\log_{bc} a)^n}\geq 3\cdot2^{n}$ where $a,b,c>1$ and $n$ is natural number.

1984 Iran MO (2nd round), 7

Tags: geometry

Let $B$ and $C$ be two fixed point on the plane $P.$ Find the locus of the points $M$ on the plane $P$ for which $MB^2 + kMC^2 = a^2.$ ($k$ and $a$ are two given numbers and $k>0.$)

2010 China Team Selection Test, 3

Tags: pigeonhole principle , ceiling function , number theory

Let $k>1$ be an integer, set $n=2^{k+1}$. Prove that for any positive integers $a_1<a_2<\cdots<a_n$, the number $\prod_{1\leq i<j\leq n}(a_i+a_j)$ has at least $k+1$ different prime divisors.

2019 ELMO Shortlist, A1

Tags: algebra , inequality , inequalities

Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$ [i]Proposed by Milan Haiman[/i]

2025 Belarusian National Olympiad, 11.3

Tags: geometry

An arbitrary triangle $ABC$ is given. Using ruler and compass construct three pairwise tangent circles $w_A$,$w_B$, $w_C$ with equal radii such that $A \in w_A, B \in w_B, C \in w_C$. [i]Matsvei Zorka[/i]

2015 IMO Shortlist, A3

Tags: inequalities , algebra , multivariate polynomial , maximization , n-variable inequality

Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.

2017 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorics , combinatorial geometry , tiles , tiling

Two right isosceles triangles of legs equal to $1$ are glued together to form either an isosceles triangle - called [i]t-shape[/i] - of leg $\sqrt2$, or a parallelogram - called [i]p-shape[/i] - of sides $1$ and $\sqrt2$. Find all integers $m$ and $n, m, n \ge 2$, such that a rectangle $m \times n$ can be tilled with t-shapes and p-shapes.

2014 Contests, 2

Tags: geometry , 3d geometry

How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube? [i]Proposed by Evan Chen[/i]

2019 JHMT, 7

Tags: geometry

Regular hexagon $ABCDEF$ has side length $\alpha$. Line $\ell$ intersects $A$ and bisects $\overline{CD}$ (and the point of intersection is $M$), line $m$ intersects $C$ and $E$, and line $n$ intersects $B$ and $E$. Lines $n$ and $\ell$ intersect at a point $G$, and lines $m$ and $\ell$ intersect at a point $H$. $[\vartriangle CHM] : [\vartriangle GHE] : [\vartriangle ABG] = a : b : c$ where $[\vartriangle ABC]$ is the area of $\vartriangle ABC$. Find $a + b + c$.

2011 Armenian Republican Olympiads, Problem 3

Tags: number theory

Find all integers $a, m, n, k,$ such that $(a^m+1)(a^n-1)=15^k.$

2001 Cuba MO, 1

Tags: combinatorics

In each square of a $3 \times 3$ board a real number is written. The element of the $i$ -th row and the $j$ -th column is equal to abso;uteof the difference of the sum of the elements of column $j$ and the sum of the elements of row $i$. Prove that every element of the board is equal to the sum or difference of two other elements on the board.

1997 AMC 12/AHSME, 22

Tags:

Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $ \$56$. The absolute difference between the amounts Ashley and Betty had to spend was $ \$19$. The absolute difference between the amounts Betty and Carlos had was $ \$7$, between Carlos and Dick was $ \$5$, between Dick and Elgin was $ \$4$, and between Elgin and Ashley was $ \$11$. How much did Elgin have? $ \textbf{(A)}\ \$6\qquad \textbf{(B)}\ \$7\qquad \textbf{(C)}\ \$8\qquad \textbf{(D)}\ \$9\qquad \textbf{(E)}\ \$10$

1999 Yugoslav Team Selection Test, Problem 1

Tags: polynomial , algebra

For a natural number $n$, let $P(x)$ be the polynomial of $2n$−th degree such that: $P(0) = 1$ and $P(k) = 2^{k-1}$ for $k = 1, 2, . . . , 2n$. Prove that $2P(2n + 1) - P(2n + 2) = 1$. P.S. I tried to prove it by firstly expressing this polynomial using Lagrange interpolation but get bored of computations - it seems like it can be done this way, but I'd like to see more 'clever' solution. :)

2010 Kurschak Competition, 3

Tags: number theory

For what positive integers $n$ and $k$ do there exits integers $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_k$ such that the products $a_ib_j$ ($1\le i\le n,1\le j\le k$) give pairwise different residues modulo $nk$?

2014 EGMO, 3

Tags: quadratic , number theory , combinatorial number theory , divisor

We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.

1982 Vietnam National Olympiad, 3

Tags: 3d geometry , geometry

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ and $A'B'C'D'$ are faces and $AA',BB',CC',DD'$ are edges). Consider the four lines $AA', BC, D'C'$ and the line joining the midpoints of $BB'$ and $DD'$. Show that there is no line which cuts all the four lines.

2017 ELMO Shortlist, 2

Tags: number theory

An integer $n>2$ is called [i]tasty[/i] if for every ordered pair of positive integers $(a,b)$ with $a+b=n,$ at least one of $\frac{a}{b}$ and $\frac{b}{a}$ is a terminating decimal. Do there exist infinitely many tasty integers? [i]Proposed by Vincent Huang[/i]

2017 Harvard-MIT Mathematics Tournament, 16

Tags: algebra , complex number

Let $a$ and $b$ be complex numbers satisfying the two equations \begin{align*} a^3 - 3ab^2 & = 36 \\ b^3 - 3ba^2 & = 28i. \end{align*} Let $M$ be the maximum possible magnitude of $a$. Find all $a$ such that $|a| = M$.

Kyiv City MO Juniors Round2 2010+ geometry, 2021.8.2

Tags: geometry , incenter , right triangle

In a triangle $ABC$, $\angle B=90^o$ and $\angle A=60^o$, $I$ is the point of intersection of its angle bisectors. A line passing through the point $I$ parallel to the line $AC$, intersects the sides $AB$ and $BC$ at the points $P$ and $T$ respectively. Prove that $3PI+IT=AC$ . (Anton Trygub)

2012 Tournament of Towns, 2

Tags: number theory , prime , divisor

Let $C(n)$ be the number of prime divisors of a positive integer n. (For example, $C(10) = 2,C(11) = 1, C(12) = 2$). Consider set S of all pairs of positive integers $(a, b)$ such that $a\ne b$ and $C(a + b) = C(a) + C(b)$. Is set $S$ finite or infinite?

2012 NIMO Problems, 3

Tags:

Compute the sum of the distinct prime factors of $10101$. [i]Proposed by Lewis Chen[/i]

2014 Purple Comet Problems, 12

Tags: trigonometry , trapezoid , rectangle , geometry

The vertices of hexagon $ABCDEF$ lie on a circle. Sides $AB = CD = EF = 6$, and sides $BC = DE = F A = 10$. The area of the hexagon is $m\sqrt3$. Find $m$.

2011 May Olympiad, 4

Tags: combinatorics

Given $n$ points in a circle, Arnaldo write 0 or 1 in all the points. Bernado can do a operation, he can chosse some point and change its number and the numbers of the points on the right and left side of it. Arnaldo wins if Bernado can´t change all the numbers in the circle to 0, and Bernado wins if he can a) Show that Bernado can win if $n=101$ b) Show that Arnaldo wins if $n=102$

2007 Miklós Schweitzer, 4

Tags: abstract algebra , combinatorics

Let $p$ be a prime number and $a_1, \ldots, a_{p-1}$ be not necessarily distinct nonzero elements of the $p$-element $\mathbb Z_p \pmod{p}$ group. Prove that each element of $\mathbb Z_p$ equals a sum of some of the $a_i$'s (the empty sum is $0$). (translated by Miklós Maróti)

2015 Online Math Open Problems, 19

Tags:

Let $ABC$ be a triangle with $AB = 80, BC = 100, AC = 60$. Let $D, E, F$ lie on $BC, AC, AB$ such that $CD = 10, AE = 45, BF = 60$. Let $P$ be a point in the plane of triangle $ABC$. The minimum possible value of $AP+BP+CP+DP+EP+FP$ can be expressed in the form $\sqrt{x}+\sqrt{y}+\sqrt{z}$ for integers $x, y, z$. Find $x+y+z$. [i]Proposed by Yang Liu[/i]