Olympiad problems

1972 IMO Longlists, 17

Tags: geometry

A solid right circular cylinder with height $h$ and base-radius $r$ has a solid hemisphere of radius $r$ resting upon it. The center of the hemisphere $O$ is on the axis of the cylinder. Let $P$ be any point on the surface of the hemisphere and $Q$ the point on the base circle of the cylinder that is furthest from $P$ (measuring along the surface of the combined solid). A string is stretched over the surface from $P$ to $Q$ so as to be as short as possible. Show that if the string is not in a plane, the straight line $PO$ when produced cuts the curved surface of the cylinder.

Kvant 2022, M2718

Tags: rectangle , combinatorics , parity

$m\times n$ grid is tiled by mosaics $2\times2$ and $1\times3$ (horizontal and vertical). Prove that the number of ways to choose a $1\times2$ rectangle (horizontal and vertical) such that one of its cells is tiled by $2\times2$ mosaic and the other cell is tiled by $1\times3$ mosaic [horizontal and vertical] is an even number.

2014 IMO Shortlist, N8

Tags: number theory

For every real number $x$, let $||x||$ denote the distance between $x$ and the nearest integer. Prove that for every pair $(a, b)$ of positive integers there exist an odd prime $p$ and a positive integer $k$ satisfying \[\displaystyle\left|\left|\frac{a}{p^k}\right|\right|+\left|\left|\frac{b}{p^k}\right|\right|+\left|\left|\frac{a+b}{p^k}\right|\right|=1.\] [i]Proposed by Geza Kos, Hungary[/i]

1994 AMC 12/AHSME, 22

Tags:

Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 630 $

1998 IberoAmerican, 2

Tags: combinatorics

Find the maximal possible value of $n$ such that there exist points $P_1,P_2,P_3,\ldots,P_n$ in the plane and real numbers $r_1,r_2,\ldots,r_n$ such that the distance between any two different points $P_i$ and $P_j$ is $r_i+r_j$.

2011 Romanian Masters In Mathematics, 1

Tags: algebra , function

Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing. [i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]

2020 South Africa National Olympiad, 2

Tags: rhombus , geometry , square , area

Let $S$ be a square with sides of length $2$ and $R$ be a rhombus with sides of length $2$ and angles measuring $60^\circ$ and $120^\circ$. These quadrilaterals are arranged to have the same centre and the diagonals of the rhombus are parallel to the sides of the square. Calculate the area of the region on which the figures overlap.

2019 Iran Team Selection Test, 4

Tags: geometry , circumcircle , geometric transformation , reflection

Given an acute-angled triangle $ABC$ with orthocenter $H$. Reflection of nine-point circle about $AH$ intersects circumcircle at points $X$ and $Y$. Prove that $AH$ is the external bisector of $\angle XHY$. [i]Proposed by Mohammad Javad Shabani[/i]

2020 Jozsef Wildt International Math Competition, W14

Tags: number theory , fibonacci

Let $\{F_n\}_{n\ge1}$ be the Fibonacci sequence defined by $F_1=F_2=1$ and for all $n\ge3$, $F_n=F_{n-1}+F_{n-2}$. Prove that among the first $10000000000000002$ terms of the sequence there is one term that ends up with $8$ zeroes. [i]Proposed by José Luis Díaz-Barrero[/i]

2008 Harvard-MIT Mathematics Tournament, 3

Tags: counting , distinguishability

There are $ 5$ dogs, $ 4$ cats, and $ 7$ bowls of milk at an animal gathering. Dogs and cats are distinguishable, but all bowls of milk are the same. In how many ways can every dog and cat be paired with either a member of the other species or a bowl of milk such that all the bowls of milk are taken?

2010 AMC 10, 12

Tags:

At the beginning of the school year, $ 50\%$ of all students in Mr. Well's math class answered "Yes" to the question "Do you love math", and $ 50\%$ answered "No." At the end of the school year, $ 70\%$ answered "Yes" and $ 30\%$ answered "No." Altogether, $ x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $ x$? $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 80$

2015 Saudi Arabia JBMO TST, 3

Tags: geometry , circumcircle

Let $ABC$ be an acute-angled triangle inscribed in the circle $(O)$. Let $AD$ be the diameter of $(O)$. The points $M,N$ are chosen on $BC$ such that $OM\parallel AB, ON\parallel AC$. The lines $DM,DN$ cut $(O)$ again at $P,Q$. Prove that $BC=DP=DQ$. Tran Quang Hung, Vietnam

2015 Princeton University Math Competition, 10

Tags:

Let $S$ be the set of integer triplets $(a, b, c)$ with $1 \le a \le b \le c$ that satisfy $a + b + c = 77$ and: \[\frac{1}{a} +\frac{1}{b}+\frac{1}{c}= \frac{1}{5}.\]What is the value of the sum $\sum_{a,b,c \in S} a\cdot b \cdot c$?

1989 AMC 8, 2

Tags: least common multiple , number theory

$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000} =$ $\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$

2016 Brazil Team Selection Test, 4

Tags: combinatorics , additive combinatorics

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

2014 ASDAN Math Tournament, 13

Tags: team test

Let $\alpha,\beta,\gamma$ be the three real roots of the polynomial $x^3-x^2-2x+1=0$. Find all possible values of $\tfrac{\alpha}{\beta}+\tfrac{\beta}{\gamma}+\tfrac{\gamma}{\alpha}$.

2013 Junior Balkan Team Selection Tests - Moldova, 4

Tags: angle , algebra

A train from stop $A$ to stop $B$ is traveled in $X$ minutes ($0 <X <60$). It is known that when starting from $A$, as well as when arriving at $B$, the angle formed by the hour and the minute had measure equal to $X$ degrees. Find $X $.

1990 IMO Longlists, 88

Tags: cauchy schwarz , holder , inequality , 4-variable inequality , algebra

Let $ w, x, y, z$ are non-negative reals such that $ wx \plus{} xy \plus{} yz \plus{} zw \equal{} 1$. Show that $ \frac {w^3}{x \plus{} y \plus{} z} \plus{} \frac {x^3}{w \plus{} y \plus{} z} \plus{} \frac {y^3}{w \plus{} x \plus{} z} \plus{} \frac {z^3}{w \plus{} x \plus{} y}\geq \frac {1}{3}$.

2012 USAJMO, 5

Tags:

For distinct positive integers $a, b<2012$, define $f(a, b)$ to be the number of integers $k$ with $1\le k<2012$ such that the remainder when $ak$ divided by $2012$ is greater than that of $bk$ divided by $2012$. Let $S$ be the minimum value of $f(a, b)$, where $a$ and $b$ range over all pairs of distinct positive integers less than $2012$. Determine $S$.

2023 Singapore Junior Math Olympiad, 2

Tags: number theory

What is the maximum number of integers that can be chosen from $1,2,\dots,99$ so that the chosen integers can be arranged in a circle with the property that the product of every pair of neighbouring integers is 3-digit number?

2004 Olympic Revenge, 4

Tags: functional equation , functional , algebra

Find all functions $f:R \rightarrow R$ such that for any reals $x,y$, $f(x^2+y)=f(x)f(x+1)+f(y)+2x^2y$.

2013 Princeton University Math Competition, 1

Tags: princeton , college

Including the original, how many ways are there to rearrange the letters in PRINCETON so that no two vowels (I, E, O) are consecutive and no three consonants (P, R, N, C, T, N) are consecutive?

1999 Turkey MO (2nd round), 6

Tags: combinatorics , induction

We wish to ﬁnd the sum of $40$ given numbers utilizing $40$ processors. Initially, we have the number $0$ on the screen of each processor. Each processor adds the number on its screen with a number entered directly (only the given numbers could be entered directly to the processors) or transferred from another processor in a unit time. Whenever a number is transferred from a processor to another, the former processor resets. Find the least time needed to ﬁnd the desired sum.

2011 IMC, 4

Tags: algebra , polynomial , induction

Let $f$ be a polynomial with real coefficients of degree $n$. Suppose that $\displaystyle \frac{f(x)-f(y)}{x-y}$ is an integer for all $0 \leq x<y \leq n$. Prove that $a-b | f(a)-f(b)$ for all distinct integers $a,b$.

1992 Tournament Of Towns, (356) 5

Tags: geometry

The bisector of the angle $A$ of triangle $ABC$ intersects its circumscribed circle at the point $D$. Suppose $P$ is the point symmetric to the incentre of the triangle with respect to the midpoint of the side $BC$, and $M$ is the second intersection point of the line $PD$ with the circumscribed circle. Prove that one of the distances $AM$, $BM$, $CM$ is equal to the sum of two other distances. (VO Gordon)