Olympiad problems

2001 AMC 10, 15

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$.

MOAA Team Rounds, 2019.7

Tags: number theory , team

Suppose $ABC$ is a triangle inscribed in circle $\omega$ . Let $A'$ be the point on $\omega$ so that $AA'$ is a diameter, and let $G$ be the centroid of $ABC$. Given that $AB = 13$, $BC = 14$, and $CA = 15$, let $x$ be the area of triangle $AGA'$ . If $x$ can be expressed in the form $m/n$ , where m and n are relatively prime positive integers, compute $100n + m$.

2006 AMC 12/AHSME, 5

Tags:

John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John? $ \textbf{(A) } 30 \qquad \textbf{(B) } 50 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } 120$

1987 AMC 8, 21

Tags:

Suppose $n^{*}$ means $\frac{1}{n}$, the reciprocal of $n$. For example, $5^{*}=\frac{1}{5}$. How many of the following statements are true? i) $3^*+6^*=9^*$ ii) $6^*-4^*=2^*$ iii) $2^*\cdot 6^*=12^*$ iv) $10^*\div 2^* =5^*$ $\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4$

Mathematical Minds 2024, P4

Tags: inequalities

Let $a$, $b$, $c$ be positive real numbers such that $a+b+c=3$. Prove that $$\sqrt[3]{\frac{a^3+b^3}{2}}+\sqrt[3]{\frac{b^3+c^3}{2}}+\sqrt[3]{\frac{c^3+a^3}{2}}\leqslant a^2+b^2+c^2.$$ [i]Proposed by Andrei Vila[/i]

2025 India National Olympiad, P5

Tags: combinatorics , game

Greedy goblin Griphook has a regular $2000$-gon, whose every vertex has a single coin. In a move, he chooses a vertex, removes one coin each from the two adjacent vertices, and adds one coin to the chosen vertex, keeping the remaining coin for himself. He can only make such a move if both adjacent vertices have at least one coin. Griphook stops only when he cannot make any more moves. What is the maximum and minimum number of coins he could have collected? [i]Proposed by Pranjal Srivastava and Rohan Goyal[/i]

2008 China Team Selection Test, 3

Tags: polynomial , algebra

Let $ n>m>1$ be odd integers, let $ f(x)\equal{}x^n\plus{}x^m\plus{}x\plus{}1$. Prove that $ f(x)$ can't be expressed as the product of two polynomials having integer coefficients and positive degrees.

2017 NZMOC Camp Selection Problems, 8

Tags: system of equations , system , algebra

Find all possible real values for $a, b$ and $c$ such that (a) $a + b + c = 51$, (b) $abc = 4000$, (c) $0 < a \le 10$ and $c \ge 25$.

2021 Czech-Polish-Slovak Junior Match, 1

Tags: equal area , geometry , trapezoid

Consider a trapezoid $ABCD$ with bases $AB$ and $CD$ satisfying $| AB | > | CD |$. Let $M$ be the midpoint of $AB$. Let the point $P$ lie inside $ABCD$ such that $| AD | = | PC |$ and $| BC | = | PD |$. Prove that if $| \angle CMD | = 90^o$, then the quadrilaterals $AMPD$ and $BMPC$ have the same area.

2006 China Western Mathematical Olympiad, 1

Tags: number theory

Let $S=\{n|n-1,n,n+1$ can be expressed as the sum of the square of two positive integers.$\}$. Prove that if $n$ in $S$, $n^{2}$ is also in $S$.

2009 Postal Coaching, 2

Tags: number theory , remainder

Let $a > 2$ be a natural number. Show that there are infinitely many natural numbers n such that $a^n \equiv -1$ (mod $n^2$).