Olympiad problems

2022 Nigerian Senior MO Round 2, Problem 6

Let $k, l, m, n$ be positive integers. Given that $k+l+m+n=km=ln$, find all possible values of $k+l+m+n$.

1987 Kurschak Competition, 2

Tags: geometry

Is there a set of points in space whose intersection with any plane is a finite but nonempty set of points?

2003 All-Russian Olympiad Regional Round, 8.7

Tags: geometry , equal segments , equal angles

In triangle $ABC$, angle $C$ is a right angle. Found on the side $AC$ point $D$, and on the segment $BD$, point $K$ such that $\angle ABC = \angle KAD =\angle AKD$. Prove that $BK = 2DC$.

2012 AMC 12/AHSME, 17

Tags: analytic geometry , graphing lines , slope , trigonometry , geometric transformation , dilation , geometry

Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.2\qquad\textbf{(C)}\ 6.4\qquad\textbf{(D)}\ 6.6\qquad\textbf{(E)}\ 6.8 $

2009 Kazakhstan National Olympiad, 4

Tags: inequalities

Let $0<a_1 \leq a_2 \leq \cdots\leq a_n $ ($n \geq 3; n \in \mathbb{N}$) be $n$ real numbers. Prove the inequality \[\frac{a_1^2}{a_2}+\frac{a_2^3}{a_3^2}+\cdots+\frac{a_n^{n+1}}{a_1^n} \geq a_1+a_2+\cdots+a_n\]

2010 Kyrgyzstan National Olympiad, 7

Tags: number theory

Find all natural triples $(a,b,c)$, such that: $a - )\,a \le b \le c$ $b - )\,(a,b,c) = 1$ $c - )\,\left. {{a^2}b} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{b^2}c} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{c^2}a} \right|{a^3} + {b^3} + {c^3}$.

1994 Hong Kong TST, 1

Tags: trigonometry , geometry

In a $\triangle ABC$, $\angle C=2 \angle B$. $P$ is a point in the interior of $\triangle ABC$ satisfying that $AP=AC$ and $PB=PC$. Show that $AP$ trisects the angle $\angle A$.

2014 Contests, 1

Tags: algebra , system of equations

Determine the value of the expression $x^2 + y^2 + z^2$, if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.

Istek Lyceum Math Olympiad 2016, 2

Tags: inversion , circles , geometry

Let $\omega$ be the semicircle with diameter $PQ$. A circle $k$ is tangent internally to $\omega$ and to segment $PQ$ at $C$. Let $AB$ be the tangent to $K$ perpendicular to $PQ$, with $A$ on $\omega$ and $B$ on the segment $CQ$. Show that $AC$ bisects angle $\angle PAB$

2001 APMO, 5

Tags: geometry

Find the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1,\dots,~X_n$ in the plane with $AB\ne CD$ that satisfy the following condition: for each $i=1,2,\dots,n$ triangles $ABX_i$ and $CDX_i$ are equal.

2001 AMC 10, 15

Tags: geometry , parallelogram , similar triangles

A street has parallel curbs $ 40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $ 15$ feet and each stripe is $ 50$ feet long. Find the distance, in feet, between the stripes. $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 25$

2018 Junior Balkan Team Selection Tests - Moldova, 2

Tags: geometry

Let $ABC$ be an acute triangle.Let $OF \| BC$ where $O$ is the circumcenter and $F$ is between $A$ and $B$.Let $H$ be the orthocenter.Let $M$ be the midpoint of $AH$.Prove that $\angle FMC=90$.

2009 Puerto Rico Team Selection Test, 3

Tags: geometry , altitude

On an arbitrary triangle $ ABC$ let $ E$ be a point on the height from $ A$. Prove that $ (AC)^2 - (CE)^2 = (AB)^2 - (EB)^2$.

2013 National Olympiad First Round, 19

Tags: analytic geometry

What is the minimum value of \[\sqrt {x^2 - 4x + 7 - 2\sqrt 2} + \sqrt {x^2 - 8x + 27 - 6\sqrt 2}\] where $x$ is a real number? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3\sqrt 2 \qquad\textbf{(C)}\ 1 + \sqrt 2 \qquad\textbf{(D)}\ 2\sqrt 2 \qquad\textbf{(E)}\ \text{None of above} $

2016 India Regional Mathematical Olympiad, 2

Tags: number theory , greatest common divisor

Consider a sequence $(a_k)_{k \ge 1}$ of natural numbers defined as follows: $a_1=a$ and $a_2=b$ with $a,b>1$ and $\gcd(a,b)=1$ and for all $k>0$, $a_{k+2}=a_{k+1}+a_k$. Prove that for all natural numbers $n$ and $k$, $\gcd(a_n,a_{n+k}) <\frac{a_k}{2}$.

2012 Turkmenistan National Math Olympiad, 1

Tags: trigonometry , inequalities

Find the max and min value of $a\cos^2 x+b\sin x\cos x+c\sin^2 x$.

1949 Putnam, A6

Tags: trigonometry

Prove that for every real or complex $x$ $$\prod_{k=1}^{\infty} \frac{1+2\cos \frac{2x}{3^{k}}}{3} =\frac{\sin x}{x}.$$

2023 Bangladesh Mathematical Olympiad, P5

Tags: algebra , manipulation , identity

Let $m$, $n$ and $p$ are real numbers such that $\left(m+n+p\right)\left(\frac 1m + \frac 1n + \frac1p\right) =1$. Find all possible values of $$\frac 1{(m+n+p)^{2023}} -\frac 1{m^{2023}} -\frac 1{n^{2023}} -\frac 1{p^{2023}}.$$

1998 Poland - Second Round, 1

Tags: function , composite function , functional equation , algebra

Let $A_n = \{1,2,...,n\}$. Prove or disprove: For all integers $n \ge 2$ there exist functions $f,g : A_n \to A_n$ which satisfy $f(f(k)) = g(g(k)) = k$ for $1 \le k \le n$, and $g(f(k)) = k +1$ for $1 \le k \le n -1$.

2018 Putnam, A2

Tags: determinant

Let $S_1, S_2, \dots, S_{2^n - 1}$ be the nonempty subsets of $\{1, 2, \dots, n\}$ in some order, and let $M$ be the $(2^n - 1) \times (2^n - 1)$ matrix whose $(i, j)$ entry is \[m_{ij} = \left\{ \begin{array}{cl} 0 & \text{if $S_i \cap S_j = \emptyset$}, \\ 1 & \text{otherwise}. \end{array} \right.\] Calculate the determinant of $M$.

1995 South africa National Olympiad, 3

Tags: algebra

Suppose that $a_1,a_2,\dots,a_n$ are the numbers $1,2,3,\dots,n$ but written in any order. Prove that \[(a_1-1)^2+(a_2-2)^2+\cdots+(a_n-n)^2\] is always even.

2016 AMC 8, 5

Tags:

The number $N$ is a two-digit number. [list] [*]When $N$ is divided by $9$, the remainder is $1$. [*]When $N$ is divided by $10$, the remainder is $3$. [/list] What is the remainder when $N$ is divided by $11$? $\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }7$

2007 Oral Moscow Geometry Olympiad, 2

Tags: geometry , equal segments , right triangle , isosceles

An isosceles right-angled triangle $ABC$ is given. On the extensions of sides $AB$ and $AC$, behind vertices $B$ and $C$ equal segments $BK$ and $CL$ were laid. $E$ and F are the points of intersection of the segment $KL$ and the lines perpendicular to the $KC$ , passing through the points $B$ and $A$, respectively. Prove that $EF = FL$.

2022 JBMO Shortlist, N1

Tags: factorial , sum of integers , shortlist , number theory

Determine all pairs $(k, n)$ of positive integers that satisfy $$1! + 2! + ... + k! = 1 + 2 + ... + n.$$

2009 IMS, 1

Tags: superior algebra , group theory , abstract algebra

$ G$ is a group. Prove that the following are equivalent: 1. All subgroups of $ G$ are normal. 2. For all $ a,b\in G$ there is an integer $ m$ such that $ (ab)^m\equal{}ba$.