Olympiad problems

2014 Contests, 1

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

1987 Vietnam National Olympiad, 3

Tags: pyramid , geometry , 3d geometry , sphere , vector , analytic geometry

Prove that among any five distinct rays $ Ox$, $ Oy$, $ Oz$, $ Ot$, $ Or$ in space there exist two which form an angle less than or equal to $ 90^{\circ}$.

Kvant 2020, M2612

Tags: combinatorics , combinatorial game

Peter and Basil play the following game on a horizontal table $1\times{2019}$. Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by $s$ cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by $s$ cells neither to the left, nor to the right, the coin stays in the current cell. Find the least $n$ such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.

1992 Tournament Of Towns, (333) 1

Tags: perfect square , number theory

Prove that the product of all integers from $2^{1917} +1$ up to $2^{1991} -1$ is not the square of an integer. (V. Senderov, Moscow)

1993 Poland - Second Round, 5

Tags: geometry , inradius , sum , incircle

Let $D,E,F$ be points on the sides $BC,CA,AB$ of a triangle $ABC$, respectively. Suppose that the inradii of the triangles $AEF,BFD,CDE$ are all equal to $r_1$. If $r_2$ and $r$ are the inradii of triangles $DEF$ and $ABC$ respectively, prove that $r_1 +r_2 =r$.

1957 Moscow Mathematical Olympiad, 359

Tags: geometry , locus , perpendicular

Straight lines $OA$ and $OB$ are perpendicular. Find the locus of endpoints $M$ of all broken lines $OM$ of length $\ell$ which intersect each line parallel to $OA$ or $OB$ at not more than one point.

1994 Tournament Of Towns, (425) 2

Tags: combinatorics

An $8$ by $8$ square is divided into $64$ $1$ by $1$ squares, and must be covered by $64$ black and $64$ white, isosceles, right-angled triangles (each square must be covered by two triangles). A covering is said to be “fine” if any two neighbouring triangles (i.e. having a common side) are of different colours. How many different fine coverings are there? (NB Vassiliev)

2021 IMO, 3

Tags: geometry , concurrency , circumcenter , moving points

Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD.$ The point $E$ on the segment $AC$ satisfies $\angle ADE =\angle BCD,$ the point $F$ on the segment $AB$ satisfies $\angle FDA =\angle DBC,$ and the point $X$ on the line $AC$ satisfies $CX = BX.$ Let $O_1$ and $O_2$ be the circumcenters of the triangles $ADC$ and $EXD,$ respectively. Prove that the lines $BC, EF,$ and $O_1O_2$ are concurrent.

2015 Paraguay Juniors, 1

Tags: geometry

The rectangle in the figure has dimensions $16$ x $20$ and is divided into $10$ smaller equal rectangles. What is the perimeter of each of the $10$ smaller rectangles?

1999 Portugal MO, 3

Tags: geometry , chord

If two parallel chords of a circumference, $10$ mm and $14$ mm long, with distance $6$ mm from each other, how long is the chord equidistant from these two?

2012 Canada National Olympiad, 2

Tags: number theory , least common multiple

For any positive integers $n$ and $k$, let $L(n,k)$ be the least common multiple of the $k$ consecutive integers $n,n+1,\ldots ,n+k-1$. Show that for any integer $b$, there exist integers $n$ and $k$ such that $L(n,k)>bL(n+1,k)$.

1969 Miklós Schweitzer, 4

Tags: inequalities , logarithm , real analysis

Show that the following inequality hold for all $ k \geq 1$, real numbers $ a_1,a_2,...,a_k$, and positive numbers $ x_1,x_2,...,x_k.$ \[ \ln \frac {\sum\limits_{i \equal{} 1}^kx_i}{\sum\limits_{i \equal{} 1}^kx_i^{1 \minus{} a_i}} \leq \frac {\sum\limits_{i \equal{} 1}^ka_ix_i \ln x_i}{\sum\limits_{i \equal{} 1}^kx_i} . \] [i]L. Losonczi[/i]

2019 Paraguay Mathematical Olympiad, 3

Tags: number theory , digit

Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?

2022 CMIMC, 9

Tags: team

For natural numbers $n$, let $r(n)$ be the number formed by reversing the digits of $n$, and take $f(n)$ to be the maximum value of $\frac{r(k)}k$ across all $n$-digit positive integers $k$. If we define $g(n)=\left\lfloor\frac1{10-f(n)}\right\rfloor$, what is the value of $g(20)$? [i]Proposed by Adam Bertelli[/i]

2023 Canadian Mathematical Olympiad Qualification, 4

Tags: number theory , periodic sequence , periodic

Let $a_1$, $a_2$, $ ...$ be a sequence of numbers, each either $1$ or $-1$. Show that if $$\frac{a_1}{3}+\frac{a_2}{3^2} + ... =\frac{p}{q}$$ for integers $p$ and $q$ such that $3$ does not divide $q$, then the sequence $a_1$, $a_2$, $ ...$ is periodic; that is, there is some positive integer $n$ such that $a_i = a_{n+i}$ for $i = 1$, $2$,$...$.

2019 Saint Petersburg Mathematical Olympiad, 2

Tags: combinatorics , number theory , greatest common divisor

On the blackboard there are written $100$ different positive integers . To each of these numbers is added the $\gcd$ of the $99$ other numbers . In the new $100$ numbers , is it possible for $3$ of them to be equal. [i] (С. Берлов)[/i]

1989 AIME Problems, 2

Tags:

Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices?

2006 AMC 12/AHSME, 7

Tags:

Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible? $ \textbf{(A) } 4 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 24 \qquad \textbf{(E) } 48$

2019 Bosnia and Herzegovina Junior BMO TST, 3

Tags: combinatorics , winning strategy

$3.$ Let $S$ be the set of all positive integers from $1$ to $100$ included. Two players play a game. The first player removes any $k$ numbers he wants, from $S$. The second player's goal is to pick $k$ different numbers, such that their sum is $100$. Which player has the winning strategy if : $a)$ $k=9$? $b)$ $k=8$?

1998 Baltic Way, 8

Tags: polynomial , algebra

Let $P_k(x)=1+x+x^2+\ldots +x^{k-1}$. Show that \[ \sum_{k=1}^n \binom{n}{k} P_k(x)=2^{n-1} P_n \left( \frac{x+1}{2} \right) \] for every real number $x$ and every positive integer $n$.

2023 New Zealand MO, 4

Tags: combinatorics

For any positive integer $n$, let $f(n)$ be the number of subsets of $\{1, 2, . . . , n\}$ whose sum is equal to $n$. Does there exist infinitely many positive integers $m$ such that $f(m) = f(m + 1)$? (Note that each element in a subset must be distinct.)

2013 USAMTS Problems, 1

Tags: graph theory

Alex is trying to open a lock whose code is a sequence that is three letters long, with each of the letters being one of $\text A$, $\text B$ or $\text C$, possibly repeated. The lock has three buttons, labeled $\text A$, $\text B$ and $\text C$. When the most recent $3$ button-presses form the code, the lock opens. What is the minimum number of total button presses Alex needs to guarantee opening the lock?

1997 Slovenia Team Selection Test, 3

Tags: combinatorics , combinatorial geometry , point , coloring

Let $A_1,A_2,...,A_n$ be $n \ge 2$ distinct points on a circle. Find the number of colorings of these points with $p \ge 2$ colors such that every two adjacent points receive different colors

2018 CIIM, Problem 5

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Consider the transformation $$T(x,y,z) = (\sin y + \sin z - \sin x,\sin z + \sin x - \sin y,\sin x +\sin y -\sin z).$$ Determine all the points $(x,y,z) \in [0,1]^3$ such that $T^n(x,y,z) \in [0,1]^3,$ for every $n \geq 1$.

2024 Bulgarian Winter Tournament, 12.2

Tags: geometry

Let $ABC$ be scalene and acute triangle with $CA>CB$ and let $P$ be an internal point, satisfying $\angle APB=180^{\circ}-\angle ACB$; the lines $AP, BP$ meet $BC, CA$ at $A_1, B_1$. If $M$ is the midpoint of $A_1B_1$ and $(A_1B_1C)$ meets $(ABC)$ at $Q$, show that $\angle PQM=\angle BQA_1$.