Olympiad problems

2021 Caucasus Mathematical Olympiad, 1

Tags: algebra

Let $a$, $b$, $c$ be real numbers such that $a^2+b=c^2$, $b^2+c=a^2$, $c^2+a=b^2$. Find all possible values of $abc$.

1979 Chisinau City MO, 181

Tags: point , collinear , geometry , combinatorial geometry

Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.

2003 Polish MO Finals, 4

Tags: modular arithmetic , number theory

A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$

LMT Team Rounds 2010-20, A21 B23

Tags:

The LHS Math Team wants to play Among Us. There are so many people who want to play that they are going to form several games. Each game has at most 10 people. People are $\textit{happy}$ if they are in a game that has at least 8 people in it. What is the largest possible number of people who would like to play Among Us such that it is impossible to make everyone $\textit{happy}$? [i]Proposed by Sammy Charney[/i]

2014 AMC 10, 8

Tags: factorial

Which of the following numbers is a perfect square? $ \textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2 $

2013 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory , tsts

Find all integers $n \ge 2$ with the property: there is a permutation $(a_1,a2,..., a_n)$ of the set $\{1, 2,...,n\}$ so that the numbers $a_1 + a_2 +...+ a_k, k = 1, 2,...,n$ have diffferent remainders when divided by $n$

2019 CHMMC (Fall), 9

Tags: geometry

Consider a rectangle with length $6$ and height $4$. A rectangle with length $3$ and height $1$ is placed inside the larger rectangle such that it is distance $1$ from the bottom and leftmost sides of the larger rectangle. We randomly select one point from each side of the larger rectangle, and connect these $4$ points to form a quadrilateral. What is the probability that the smaller rectangle is strictly contained within that quadrilateral?

2019 Thailand Mathematical Olympiad, 10

Tags: number theory

Prove that there are infinitely many positive odd integer $n$ such that $n!+1$ is composite number.

2020 Kosovo National Mathematical Olympiad, 2

Tags: number theory , perfect square

Find all positive integers $x$, $y$ such that $2^x+5^y+2$ is a perfect square.

2015 İberoAmerican, 3

Tags: polynomial , algebra

Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$, where $q$ is a rational number larger than $2$. Let $s_1 = \alpha + \beta$, $t_1 = 1$, and for all integers $n \geq 2$: $s_n = \alpha^n + \beta^n$ $t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$ Prove that, for all odd integers $n$, $t_n$ is the square of a rational number.

2009 Balkan MO Shortlist, G6

Tags: geometry , geometric transformation , reflection , circumcircle , rotation , trapezoid , trigonometry

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

2018 Junior Balkan Team Selection Tests - Romania, 1

Tags: algebra , rational

Prove that the equation $x^2+y^2+z^2 = x+y+z+1$ has no rational solutions.

2024 IMC, 5

Tags: combinatorial geometry , probability

Let $n>d$ be positive integers. Choose $n$ independent, uniformly distributed random points $x_1,\dots,x_n$ in the unit ball $B \subset \mathbb{R}^d$ centered at the origin. For a point $p \in B$ denote by $f(p)$ the probability that the convex hull of $x_1,\dots,x_n$ contains $p$. Prove that if $p,q \in B$ and the distance of $p$ from the origin is smaller than the distance of $q$ from the origin, then $f(p) \ge f(q)$.

1998 Korea - Final Round, 2

Tags: inequalities , circumcircle , geometry

Let $D$,$E$,$F$ be points on the sides $BC$,$CA$,$AB$ respectively of a triangle $ABC$. Lines $AD$,$BE$,$CF$ intersect the circumcircle of $ABC$ again at $P$,$Q$,$R$, respectively.Show that: \[\frac{AD}{PD}+\frac{BE}{QE}+\frac{CF}{RF}\geq 9\] and find the cases of equality.

1994 AIME Problems, 10

Tags: trigonometry , calculus , integration , ratio , inequalities

In triangle $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D.$ The lengths of the sides of $\triangle ABC$ are integers, $BD=29^3,$ and $\cos B=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

MathLinks Contest 4th, 3.3

Tags: geometry

Let $ABC$ be a triangle, and let $C$ be its circumcircle. Let $T$ be the circle tangent to $AB, AC$ and $C$ internally in the points $F, E$ and $D$ respectively. Let $P, Q$ be the intersection points between the line $EF$ and the lines $DB$ and $DC$ respectively. Prove that if $DP = DQ$ then the triangle $ABC$ is isosceles.

2005 Today's Calculation Of Integral, 62

Tags: calculus , integration , limit , calculus computations , logarithm

For $a>1$, let $f(a)=\frac{1}{2}\int_0^1 |ax^n-1|dx+\frac{1}{2}\ (n=1,2,\cdots)$ and let $b_n$ be the minimum value of $f(a)$ at $a>1$. Evaluate \[\lim_{m\to\infty} b_m\cdot b_{m+1}\cdot \cdots\cdots b_{2m}\ (m=1,2,3,\cdots)\]

2011 Dutch BxMO TST, 4

Tags: number theory , inequalities , algebra

Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$. Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.

2015 Harvard-MIT Mathematics Tournament, 9

Tags: divisibility , algebra , number theory

Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$.

2002 China Team Selection Test, 1

Tags: polynomial , inequalities , algebra

Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that \begin{align*} P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right), \end{align*} where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]

2017 Putnam, A2

Tags:

Let $Q_0(x)=1$, $Q_1(x)=x,$ and \[Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}\] for all $n\ge 2.$ Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.

1987 Polish MO Finals, 5

Tags: product , number theory , min , prime

Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).

2020 USMCA, 16

Tags:

How many paths from $(0, 0)$ to $(2020, 2020)$, consisting of unit steps up and to the right, pass through at most one point with both coordinates even, other than $(0,0)$ and $(2020,2020)$?

2000 All-Russian Olympiad, 3

Tags: geometry , parallelogram , trigonometry , geometric transformation , reflection , circumcircle , angle bisector

In an acute scalene triangle $ABC$ the bisector of the acute angle between the altitudes $AA_1$ and $CC_1$ meets the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The bisector of the angle $B$ intersects the segment joining the orthocenter of $ABC$ and the midpoint of $AC$ at point $R$. Prove that $P$, $B$, $Q$, $R$ lie on a circle.

2005 Taiwan TST Round 2, 1

Tags: function , functional equation , algebra

Let $a,b$ be two constants within the open interval $(0,\frac{1}{2})$. Find all continous functions $f(x)$ such that \[f(f(x))=af(x)+bx\] holds for all real $x$. This is much harder than the problems we had in the 1st TST...