Olympiad problems

2018 CCA Math Bonanza, L1.3

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$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\angle{ACE}$ is $90^{\circ}$. What is the average of the measures, in degrees, of $\angle{ABC}$ and $\angle{CDE}$? [i]2018 CCA Math Bonanza Lightning Round #1.3[/i]

1972 AMC 12/AHSME, 25

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Inscribed in a circle is a quadrilateral having sides of lengths $25,~39,~52,$ and $60$ taken consecutively. The diameter of this circle has length $\textbf{(A) }62\qquad\textbf{(B) }63\qquad\textbf{(C) }65\qquad\textbf{(D) }66\qquad \textbf{(E) }69$

2010 Contests, 3

Let $a_0, a_1, \ldots, a_9$ and $b_1 , b_2, \ldots,b_9$ be positive integers such that $a_9<b_9$ and $a_k \neq b_k, 1 \leq k \leq 8.$ In a cash dispenser/automated teller machine/ATM there are $n\geq a_9$ levs (Bulgarian national currency) and for each $1 \leq i \leq 9$ we can take $a_i$ levs from the ATM (if in the bank there are at least $a_i$ levs). Immediately after that action the bank puts $b_i$ levs in the ATM or we take $a_0$ levs. If we take $a_0$ levs from the ATM the bank doesn’t put any money in the ATM. Find all possible positive integer values of $n$ such that after finite number of takings money from the ATM there will be no money in it.

2014 Peru IMO TST, 10

Tags: geometry

Let $ABCDEF$ be a convex hexagon that does not have two parallel sides, such that $\angle AF B = \angle F DE, \angle DF E = \angle BDC$ and $\angle BFC = \angle ADF.$ Prove that the lines $ AB, FC$ and $DE$ are concurrent if and only if the lines $ AF, BE$ and $CD$ are concurrent.

2005 Tournament of Towns, 1

Tags: analytic geometry

On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis. [i](4 points)[/i]

1987 IMO Longlists, 8

Tags: number theory

Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros. [hide="Note"]Note. Here (and generally in MathLinks) natural numbers supposed to be positive.[/hide]

2020 Dürer Math Competition (First Round), P2

Tags: table , combinatorics

Initially we have a $2 \times 2$ table with at least one grain of wheat on each cell. In each step we may perform one of the following two kinds of moves: $i.$ If there is at least one grain on every cell of a row, we can take away one grain from each cell in that row. $ii.$ We can double the number of grains on each cell of an arbitrary column. a) Show that it is possible to reach the empty table using the above moves, starting from the position down below. b) Show that it is possible to reach the empty table from any starting position. c) Prove that the same is true for the $8 \times 8$ tables as well.

2013 AMC 12/AHSME, 23

Tags: geometric transformation , rotation , geometry , number theory , greatest common divisor

$ ABCD$ is a square of side length $ \sqrt{3} + 1 $. Point $ P $ is on $ \overline{AC} $ such that $ AP = \sqrt{2} $. The square region bounded by $ ABCD $ is rotated $ 90^{\circ} $ counterclockwise with center $ P $, sweeping out a region whose area is $ \frac{1}{c} (a \pi + b) $, where $a $, $b$, and $ c $ are positive integers and $ \text{gcd}(a,b,c) = 1 $. What is $ a + b + c $? $\textbf{(A)} \ 15 \qquad \textbf{(B)} \ 17 \qquad \textbf{(C)} \ 19 \qquad \textbf{(D)} \ 21 \qquad \textbf{(E)} \ 23 $

1979 Chisinau City MO, 176

Tags: trigonometry , algebra

Indicate all the roots of the equation $x^2+1 = \cos x$.

2023 Bangladesh Mathematical Olympiad, P2

Tags: algebra

Let {$a_1, a_2,\cdots,a_n$} be a set of $n$ real numbers whos sym equals S. It is known that each number in the set is less than $\frac{S}{n-1}$. Prove that for any three numbers $a_i$, $a_j$ and $a_k$ in the set, $a_i+a_j>a_k$.

2022 JHMT HS, 2

Tags: trigonometry , polynomial , algebra

The polynomial $P(x)=3x^3-2x^2+ax-b$ has roots $\sin^2\theta$, $\cos^2\theta$, and $\sin\theta\cos\theta$ for some angle $\theta$. Find $P(1)$.

2016 Brazil Team Selection Test, 3

Tags: game , combinatorics

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

PEN G Problems, 6

Tags: irrational number

Prove that for any irrational number $\xi$, there are infinitely many rational numbers $\frac{m}{n}$ $\left( (m,n) \in \mathbb{Z}\times \mathbb{N}\right)$ such that \[\left\vert \xi-\frac{n}{m}\right\vert < \frac{1}{\sqrt{5}m^{2}}.\]

1980 Yugoslav Team Selection Test, Problem 3

Tags: recurrence relation , number theory , sequence

A sequence $(x_n)$ satisfies $x_{n+1}=\frac{x_n^2+a}{x_{n-1}}$ for all $n\in\mathbb N$. Prove that if $x_0,x_1$, and $\frac{x_0^2+x_1^2+a}{x_0x_1}$ are integers, then all the terms of sequence $(x_n)$ are integers.

2004 Romania Team Selection Test, 8

Tags: geometric transformation , homothety , geometry

Let $\Gamma$ be a circle, and let $ABCD$ be a square lying inside the circle $\Gamma$. Let $\mathcal{C}_a$ be a circle tangent interiorly to $\Gamma$, and also tangent to the sides $AB$ and $AD$ of the square, and also lying inside the opposite angle of $\angle BAD$. Let $A'$ be the tangency point of the two circles. Define similarly the circles $\mathcal{C}_b$, $\mathcal{C}_c$, $\mathcal{C}_d$ and the points $B',C',D'$ respectively. Prove that the lines $AA'$, $BB'$, $CC'$ and $DD'$ are concurrent.

2017 BMT Spring, 7

Tags: combinatorics

A light has been placed on every lattice point (point with integer coordinates) on the (infinite) 2$D$ plane. Dene the Chebyshev distance between points $(x_1,y_1)$ and $(x_2, y_2)$ to be $\ max (|x_1 - x_2|, |y_1 -y_2|)$. Each light is turned on with probability $\frac{1}{2^{d/2}}$ , where $d$ is the Chebyshev distance from that point to the origin. What is expected number of lights that have all their directly adjacent lights turned on? (Adjacent points being points such that $|x_1-x_2|+|y_1- y_2| =1$.)

2007 Tournament Of Towns, 5

Tags: pyramid , 3d geometry , octahedron , geometry

From a regular octahedron with edge $1$, cut off a pyramid about each vertex. The base of each pyramid is a square with edge $\frac 13$. Can copies of the polyhedron so obtained, whose faces are either regular hexagons or squares, be used to tile space?

1994 Italy TST, 4

Tags: set , subset , algebra

Let $X$ be a set of $n$ elements and $k$ be a positive integer. Consider the family $S_k$ of all $k$-tuples $(E_1,...,E_k)$ with $E_i \subseteq X$ for each $i$. Evaluate the sums $\sum_{(E_1,...,E_k) \in S_k }|E_1 \cap ... \cap E_k|$ and $\sum_{(E_1,...,E_k) \in S_k }|E_1 \cup ... \cup E_k|$

2008 AMC 8, 19

Tags: probability

Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart? [asy] size((50)); dot((5,0)); dot((5,5)); dot((0,5)); dot((-5,5)); dot((-5,0)); dot((-5,-5)); dot((0,-5)); dot((5,-5)); [/asy] $ \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{2}{7}\qquad\textbf{(C)}\ \frac{4}{11}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{4}{7} $

2009 Finnish National High School Mathematics Competition, 3

Tags: geometry

The circles $\mathcal{Y}_0$ and $\mathcal{Y}_1$ lies outside each other. Let $O_0$ be the center of $\mathcal{Y}_0$ and $O_1$ be the center of $\mathcal{Y}_1$. From $O_0$, draw the rays which are tangents to $\mathcal{Y}_1$ and similarty from $O_1$, draw the rays which are tangents to $\mathcal{Y}_0$. Let the intersection points of rays and circle $\mathcal{Y}_i$ be $A_i$ and $B_i$. Show that the line segments $A_0B_0$ and $A_1B_1$ have equal lengths.

2020 CMIMC Combinatorics & Computer Science, 4

Tags: computer science , combinatorics

The continent of Trianglandia is an equilateral triangle of side length $9$, divided into $81$ triangular countries of side length $1$. Each country has the resources to choose at most $1$ of its $3$ sides and build a “wall” covering that entire side. However, since all the countries are at war, no two countries are willing to have their walls touch, even at a corner. What is the maximum number of walls that can be built in Trianglandia?

2010 Iran MO (3rd Round), 4

Tags: ceiling function , floor function , combinatorics

suppose that $\mathcal F\subseteq X^{(K)}$ and $|X|=n$. we know that for every three distinct elements of $\mathcal F$ like $A,B$ and $C$ we have $A\cap B \not\subset C$. a)(10 points) Prove that : \[|\mathcal F|\le \dbinom{k}{\lfloor\frac{k}{2}\rfloor}+1\] b)(15 points) if elements of $\mathcal F$ do not necessarily have $k$ elements, with the above conditions show that: \[|\mathcal F|\le \dbinom{n}{\lceil\frac{n-2}{3}\rceil}+2\]

1975 Chisinau City MO, 93

Tags: algebra

Prove that $(a^2 + b^2 + c^2)^ 2 = 2 (a^4 + b^4 + c^4)$ if $a + b + c = 0$.

2010 Indonesia TST, 4

Tags: geometric transformation , trigonometry , circumcircle , angle bisector , reflection , geometry , similar triangles

Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP\equal{}\angle PBQ\equal{}\angle QBC,\] (a) prove that $ ABC$ is a right-angled triangle, and (b) calculate $ \dfrac{BP}{CH}$. [i]Soewono, Bandung[/i]

2000 Flanders Math Olympiad, 2

Tags: ratio , geometry , vector

Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.