Olympiad problems

2013 Bangladesh Mathematical Olympiad, 5

Tags: number theory

Higher Secondary P5 Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$.

2001 IMO Shortlist, 4

Tags: combinatorics , combinatorial number theory , partition , coloring , greedy algorithm

A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called [i]historic[/i] if $\{z-y,y-x\} = \{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.

2010 Indonesia TST, 1

Tags: inequalities

Let $ a$, $ b$, and $ c$ be non-negative real numbers and let $ x$, $ y$, and $ z$ be positive real numbers such that $ a\plus{}b\plus{}c\equal{}x\plus{}y\plus{}z$. Prove that \[ \dfrac{a^3}{x^2}\plus{}\dfrac{b^3}{y^2}\plus{}\dfrac{c^3}{z^2} \ge a\plus{}b\plus{}c.\] [i]Hery Susanto, Malang[/i]

2013 Harvard-MIT Mathematics Tournament, 24

Tags: hmmt , function , geometry

Given a point $p$ and a line segment $l$, let $d(p,l)$ be the distance between them. Let $A$, $B$, and $C$ be points in the plane such that $AB=6$, $BC=8$, $AC=10$. What is the area of the region in the $(x,y)$-plane formed by the ordered pairs $(x,y)$ such that there exists a point $P$ inside triangle $ABC$ with $d(P,AB)+x=d(P,BC)+y=d(P,AC)?$

2014 China Western Mathematical Olympiad, 6

Tags: ceiling function , inequalities

Let $n\ge 2$ is a given integer , $x_1,x_2,\ldots,x_n $ be real numbers such that $(1) x_1+x_2+\ldots+x_n=0 $, $(2) |x_i|\le 1$ $(i=1,2,\cdots,n)$. Find the maximum of Min$\{|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|\}$.

2022 Princeton University Math Competition, 12

Tags: analytic geometry

Observe the set $S =\{(x, y) \in Z^2 : |x| \le 5$ and $-10 \le y\le 0\}$. Find the number of points $P$ in $S$ such that there exists a tangent line from $P$ to the parabola $y = x^2 + 1$ that can be written in the form $y = mx + b$, where $m$ and $b$ are integers.

2002 Pan African, 2

Tags: geometry , circumcircle , perpendicular bisector

$\triangle{AOB}$ is a right triangle with $\angle{AOB}=90^{o}$. $C$ and $D$ are moving on $AO$ and $BO$ respectively such that $AC=BD$. Show that there is a fixed point $P$ through which the perpendicular bisector of $CD$ always passes.

2016 Iran MO (3rd Round), 1

Tags: number theory , algebra , polynomial

Let $F$ be a subset of the set of positive integers with at least two elements and $P(x)$ be a polynomial with integer coefficients such that for any two distinct elements of $F$ like $a$ and $b$, the following two conditions hold [list] [*] $a+b \in F$, and [*] $\gcd(P(a),P(b))=1$. [/list] Prove that $P(x)$ is a constant polynomial.

2010 Today's Calculation Of Integral, 643

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^{\pi} \frac{x}{\sqrt{1+\sin ^ 3 x}}\{(3\pi \cos x+4\sin x)\sin ^ 2 x+4\}dx.\] Own

2024 Irish Math Olympiad, P8

Tags: inequalities

Let $a,b,c$ be positive real numbers with $a \leq c$ and $b \leq c$. Prove that $$ (a +10b)(b +22c)(c +7a) \geq 2024 abc.$$

IV Soros Olympiad 1997 - 98 (Russia), 11.9

Tags: combinatorics , geometry , 3d geometry , combinatorial geometry , pyramid

Cut pyramid $ABCD$ into $8$ equal and similar pyramids, if: a) $AB = BC = CD$, $\angle ABC =\angle BCD = 90^o$, dihedral angle at edge $BC$ is right b) all plane angles at vertex $B$ are right and $AB = BC = BD\sqrt2$. Note. Whether there are other types of triangular pyramids that can be cut into any number similar to the original pyramids (their number is not necessarily $8$ and the pyramids are not necessarily equal to each other) is currently unknown

1954 Moscow Mathematical Olympiad, 284

Tags: geometry , 3d geometry , symmetry , pyramid

How many planes of symmetry can a triangular pyramid have?

2005 Junior Balkan MO, 1

Tags: quadratic

Find all positive integers $x,y$ satisfying the equation \[ 9(x^2+y^2+1) + 2(3xy+2) = 2005 . \]

2022 Bolivia IMO TST, P1

Tags: algebra , gauss identity

Find all possible values of $\frac{1}{x}+\frac{1}{y}$, if $x,y$ are real numbers not equal to $0$ that satisfy $$x^3+y^3+3x^2y^2=x^3y^3$$

2021 Korea Winter Program Practice Test, 8

Tags: number theory , polynomial , algebra

$P$ is an monic integer coefficient polynomial which has no integer roots. deg$P=n$ and define $A$ $:=${$v_2(P(m))|m\in Z, v_2(P(m)) \ge 1$}. If $|A|=n$, show that all of the elements of $A$ is smaller than $\frac{3}{2}n^2$.

Estonia Open Senior - geometry, 2012.1.3

Tags: right triangle , median , circumcenter , geometry

Let $ABC$ be a triangle with median AK. Let $O$ be the circumcenter of the triangle $ABK$. a) Prove that if $O$ lies on a midline of the triangle $ABC$, but does not coincide with its endpoints, then $ABC$ is a right triangle. b) Is the statement still true if $O$ can coincide with an endpoint of the midsegment?

2005 Croatia National Olympiad, 1

Tags: algebra , equation , polynomial

Let $a \not = 0, b, c$ be real numbers. If $x_{1}$ is a root of the equation $ax^{2}+bx+c = 0$ and $x_{2}$ a root of $-ax^{2}+bx+c = 0$, show that there is a root $x_{3}$ of $\frac{a}{2}\cdot x^{2}+bx+c = 0$ between $x_{1}$ and $x_{2}$.

2005 Turkey MO (2nd round), 6

Tags: number theory

Suppose that a sequence $(a_n)_{n=1}^{\infty}$ of integers has the following property: For all $n$ large enough (i.e. $n \ge N$ for some $N$ ), $a_n$ equals the number of indices $i$, $1 \le i < n$, such that $a_i + i \ge n$. Find the maximum possible number of integers which occur infinitely many times in the sequence.

1992 China National Olympiad, 1

Tags: inequalities , triangle inequality , algebra

Let equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0$ with real coefficients satisfy $0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1$. Suppose that $\lambda$ ($|\lambda|>1$) is a complex root of the equation, prove that $\lambda^{n+1}=1$.

2016 NIMO Problems, 1

Tags: arithmetic sequence

Suppose $a_1$, $a_2$, $a_3$, $\dots$ is an arithmetic sequence such that \[a_1+a_2+a_3+\cdots+a_{48}+a_{49}=1421.\] Find the value of $a_1+a_4+a_7+a_{10}+\cdots+a_{49}$. [i]Proposed by Tony Kim[/i]

LMT Accuracy Rounds, 2022 S3

Tags: number theory

Find the difference between the greatest and least values of $lcm (a,b,c)$, where $a$, $b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive.

2023 Bulgaria JBMO TST, 3

Tags: geometry , incircle , excircle , similarity , circumcircle , incenter

Let $ABC$ be a non-isosceles triangle with circumcircle $k$, incenter $I$ and $C$-excenter $I_C$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of arc $\widehat{ACB}$ on $k$. Prove that $\angle IMI_C + \angle INI_C = 180^{\circ}$.

2016 Switzerland - Final Round, 3

Tags: algebra , number theory

Find all primes $p, q$ and natural numbers $n$ such that: $p(p+1)+q(q+1)=n(n+1)$

2005 Turkey Team Selection Test, 1

Tags: search , function , number theory

Show that for any integer $n\geq2$ and all integers $a_{1},a_{2},...,a_{n}$ the product $\prod_{i<j}{(a_{j}-a_{i})}$ is divisible by $\prod_{i<j}{(j-i)}$ .

2007 IMO Shortlist, 3

Tags: combinatorics , modular arithmetic , counting , triplets

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]