Olympiad problems

1996 IMO Shortlist, 4

Determine whether or nor there exist two disjoint infinite sets $ A$ and $ B$ of points in the plane satisfying the following conditions: a.) No three points in $ A \cup B$ are collinear, and the distance between any two points in $ A \cup B$ is at least 1. b.) There is a point of $ A$ in any triangle whose vertices are in $ B,$ and there is a point of $ B$ in any triangle whose vertices are in $ A.$

1999 Brazil Team Selection Test, Problem 3

Tags: algebra , sequence , recurrence relation

A sequence $a_n$ is defined by $$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.

2017 Hong Kong TST, 3

Tags: algebra , polynomial

Let $f(x)$ be a monic cubic polynomial with $f(0)=-64$, and all roots of $f(x)$ are non-negative real numbers. What is the largest possible value of $f(-1)$? (A polynomial is monic if its leading coefficient is 1.)

LMT Team Rounds 2021+, 7

Tags: number theory

How many $2$-digit factors does $555555$ have?

2021 All-Russian Olympiad, 8

Tags: combinatorics

Each girl among $100$ girls has $100$ balls; there are in total $10000$ balls in $100$ colors, from each color there are $100$ balls. On a move, two girls can exchange a ball (the first gives the second one of her balls, and vice versa). The operations can be made in such a way, that in the end, each girl has $100$ balls, colored in the $100$ distinct colors. Prove that there is a sequence of operations, in which each ball is exchanged no more than 1 time, and at the end, each girl has $100$ balls, colored in the $100$ colors.

1973 Miklós Schweitzer, 8

Tags: advanced fields

What is the radius of the largest disc that can be covered by a finite number of closed discs of radius $ 1$ in such a way that each disc intersects at most three others? [i]L. Fejes-Toth[/i]

2021 Junior Balkan Team Selection Tests - Moldova, 7

Tags: diophantine , number theory

Determine all pairs of integer numbers $(a, b)$ that satisfy the relation: $$(a + b)(a + b + 6) = 34 \cdot 3^{|2a-b|}- 7$$

2014 AMC 12/AHSME, 7

Tags: ratio , geometric sequence

The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term? $\textbf{(A) }1\qquad \textbf{(B) }\sqrt[7]3\qquad \textbf{(C) }\sqrt[8]3\qquad \textbf{(D) }\sqrt[9]3\qquad \textbf{(E) }\sqrt[10]3\qquad$

1991 AMC 8, 24

Tags: 3d geometry , geometry

A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$ $\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20$

2006 Kyiv Mathematical Festival, 5

Tags: number theory

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $a, b, c, d$ be positive integers and $p$ be prime number such that $a^2+b^2=p$ and $c^2+d^2$ is divisible by $p.$ Prove that there exist positive integers $e$ and $f$ such that $e^2+f^2=\frac{c^2+d^2}{p}.$

2014 Math Prize For Girls Problems, 16

Tags: trigonometry , vector , geometry , complex number

If $\sin x + \sin y = \frac{96}{65}$ and $\cos x + \cos y = \frac{72}{65}$, then what is the value of $\tan x + \tan y$?

2011 Princeton University Math Competition, A6

Tags: number theory

Let $a$ and $b$ be positive integers such that $a + bz = x^3 + y^4$ has no solutions for any integers $x, y, z$, with $b$ as small as possible, and $a$ as small as possible for the minimum $b$. Find $ab$.

1988 Tournament Of Towns, (200) 3

Tags: permutation , combinatorics

The integers $1 , 2,..., n$ are rearranged in such a way that if the integer $k, 1 \le k\le n$, is not the first term, then one of the integers $k + 1$ or $k-1$ occurs to the left of $k$ . How many arrangements of the integers $1 , 2,..., n$ satisfy this condition? (A. Andjans, Riga)

2024 Vietnam National Olympiad, 5

Tags: algebra , polynomial

For each polynomial $P(x)$, define $$P_1(x)=P(x), \forall x \in \mathbb{R},$$ $$P_2(x)=P(P_1(x)), \forall x \in \mathbb{R},$$ $$...$$ $$P_{2024}(x)=P(P_{2023}(x)), \forall x \in \mathbb{R}.$$ Let $a>2$ be a real number. Is there a polynomial $P$ with real coefficients such that for all $t \in (-a, a)$, the equation $P_{2024}(x)=t$ has $2^{2024}$ distinct real roots?

2013 Gheorghe Vranceanu, 1

Tags: modular arithmetic , number theory

Find all natural numbers $ a,b $ such that $ a^3+b^3 $ a power of $3.$

2021 Bangladeshi National Mathematical Olympiad, 2

Tags: number theory , greatest common divisor , least common multiple

Let $x$ and $y$ be positive integers such that $2(x+y)=gcd(x,y)+lcm(x,y)$. Find $\frac{lcm(x,y)}{gcd(x,y)}$.

1958 AMC 12/AHSME, 22

Tags: parabola , analytic geometry , conic

A particle is placed on the parabola $ y \equal{} x^2 \minus{} x \minus{} 6$ at a point $ P$ whose $ y$-coordinate is $ 6$. It is allowed to roll along the parabola until it reaches the nearest point $ Q$ whose $ y$-coordinate is $ \minus{}6$. The horizontal distance traveled by the particle (the numerical value of the difference in the $ x$-coordinates of $ P$ and $ Q$) is: $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 1$

2013 May Olympiad, 5

Tags: combinatorics

An $8\times 8$ square is drawn on the board divided into $64$ $1\times1$ squares by lines parallel to the sides. Gustavo erases some segments of length $ 1$ so that every $1\times 1$ square he erases $0, 1$ or $2$ sides. Gustavo states that he erased $6$ segments of length $1$ from the edge of the $8\times 8$ square and that the amount of $1\times 1$ squares that have exactly $ 1$ side erased is equal to $5$. Decide if what Gustavo said it may be true.

2024 Saint Petersburg Mathematical Olympiad, 5

Tags: number theory , combinatorics

$2 \ 000 \ 000$ points with integer coordinates are marked on the numeric axis. Segments of lengths $97$, $100$ and $103$ with ends at these points are considered. What is the largest number of such segments?

2009 China Team Selection Test, 1

Tags: inequalities , number theory

Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$

2011 National Olympiad First Round, 18

Tags:

How many positive integer divides the expression $n(n^2-1)(n^2+3)(n^2+5)$ for every possible value of positive integer $n$? $\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$

2021 CCA Math Bonanza, T8

Tags:

Let $ABC$ be a triangle with $AB = 9$ and $AC=12$. Point $B'$ is chosen on line $AC$ such that the midpoint of $B$ and $B'$ is equidistant from $A$ and $C$. Point $C'$ is chosen similarly. Given that the circumcircle of $AB'C'$ is tangent to $BC$, compute $BC^2$. [i]2021 CCA Math Bonanza Team Round #8[/i]

2018 JBMO Shortlist, A5

Tags: algebra , inequalities

Let a$,b,c,d$ and $x,y,z,t$ be real numbers such that $0\le a,b,c,d \le 1$ , $x,y,z,t \ge 1$ and $a+b+c+d +x+y+z+t=8$. Prove that $a^2+b^2+c^2+d^2+x^2+y^2+z^2+t^2\le 28$

2024 Harvard-MIT Mathematics Tournament, 3

Tags: number theory

Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base $10$) positive integers $\underline{a}\, \underline{b} \, \underline{c}$, if $\underline{a} \, \underline{b} \, \underline{c}$ is a multiple of $x$, then the three-digit (base $10$) number $\underline{b} \, \underline{c} \, \underline{a}$ is also a multiple of $x$.

2012 Brazil Team Selection Test, 3

Tags: geometry , circumcircle , symmetry , homothety , radical axis , marvio

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]