Olympiad problems

2007 Paraguay Mathematical Olympiad, 1

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A list with $2007$ positive integers is written on a board, such that the arithmetic mean of all the numbers is $12$. Then, seven consecutive numbers are erased from the board. The arithmetic mean of the remaining numbers is $11.915$. The seven erased numbers have this property: the sixth number is half of the seventh, the fifth number is half of the sixth, and so on. Determine the $7$ erased numbers.

2007 Balkan MO Shortlist, A2

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Find all values of $a \in \mathbb{R}$ for which the polynomial \begin{align*} f(x)=x^4-2x^3 + \left(5-6a^2 \right)x^2 + \left(2a^2-4 \right)x + \left(a^2 -2 \right)^2 \end{align*} has exactly three real roots.

2021 Kyiv Mathematical Festival, 3

Tags: geometry

Let $\omega$ be the circumcircle of a triangle $ABC$ ($AB>AC$), $E$ be the midpoint of the arc $AC$ which does not contain point $B,$ аnd $F$ the midpoint of the arc $AB$ which does not contain point $C.$ Lines $AF$ and $BE$ meet at point $P,$ line $CF$ and $AE$ meet at point $R,$ and the tangent to $\omega$ at point $A$ meets line $BC$ at point $Q.$ Prove that points $P,Q,R$ are collinear. (M. Kurskiy)

2012 AMC 8, 21

Tags: geometry , 3d geometry

Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet? $\textbf{(A)}\hspace{.05in}5\sqrt2 \qquad \textbf{(B)}\hspace{.05in}10 \qquad \textbf{(C)}\hspace{.05in}10\sqrt2 \qquad \textbf{(D)}\hspace{.05in}50 \qquad \textbf{(E)}\hspace{.05in}50\sqrt2 $

2024 Princeton University Math Competition, A6 / B8

Tags: combinatorics

Ezzie is walking around the perimeter of a regular hexagon. Each vertex of the hexagon has an instruction telling him to move clockwise or counterclockwise around the hexagon. However, when he leaves a vertex the instruction switches from clockwise to counterclockwise on that vertex, or vice versa. We say that a configuration $C$ of Ezzie’s position and the instructions on the vertices is [I]irrepeatable[/I] if, when starting from configuration $C,$ configuration $C$ only appears finitely many more times. Find the number of irrepeatable configurations.

MBMT Team Rounds, 2015 E13

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A blind ant is walking on the coordinate plane. It is trying to reach an anthill, placed at all points where both the $x$-coordinate and $y$-coordinate are odd. The ant starts at the origin, and each minute it moves one unit either up, down, to the right, or to the left, each with probability $\frac{1}{4}$. The ant moves $3$ times and doesn't reach an anthill during this time. On average, how many additional moves will the ant need to reach an anthill? (Compute the expected number of additional moves needed.)

2012 South East Mathematical Olympiad, 4

Tags: trigonometry , inequalities

Let $a, b, c, d$ be real numbers satisfying inequality $a\cos x+b\cos 2x+c\cos 3x+d\cos 4x\le 1$ holds for arbitrary real number $x$. Find the maximal value of $a+b-c+d$ and determine the values of $a,b,c,d$ when that maximum is attained.

2005 Georgia Team Selection Test, 8

Tags: geometry , circumcircle

In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.

2023 Sharygin Geometry Olympiad, 15

Tags: geometry , concurrency

Let $ABCD$ be a convex quadrilateral. Points $X$ and $Y$ lie on the extensions beyond $D$ of the sides $CD$ and $AD$ respectively in such a way that $DX = AB$ and $DY = BC$. Similarly points $Z$ and $T$ lie on the extensions beyond $B$ of the sides $CB$ and $AB$ respectively in such a way that $BZ = AD$ and $BT = DC$. Let $M_1$ be the midpoint of $XY$, and $M_2$ be the midpoint of $ZT$. Prove that the lines $DM_1, BM_2$ and $AC$ concur.

2001 Greece National Olympiad, 4

Tags: combinatorics

The numbers $1$ to $500$ are written on a board. Two pupils $A$ and $B$ play the following game: A player in turn deletes one of the numbers from the board. The game is over when only two numbers remain. Player $B$ wins if the sum of the two remaining numbers is divisible by $3,$ otherwise $A$ wins. If $A$ plays ﬁrst, show that $B$ has a winning strategy.

2020 Taiwan TST Round 1, 2

Tags: number theory

Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.

2007 Today's Calculation Of Integral, 172

Tags: calculus , integration , trigonometry , euler , calculus computations

Evaluate $\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.$

1974 Poland - Second Round, 1

Tags: combinatorics , subset , set

Let $ Z $ be a set of $ n $ elements. Find the number of such pairs of sets $ (A, B) $ such that $ A $ is contained in $ B $ and $ B $ is contained in $ Z $. We assume that every set also contains itself and the empty set.

2009 Today's Calculation Of Integral, 463

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{e^{\frac{1}{\cos ^ 2 x}}\sin x}{\cos ^ 3 x}\ dx$.

2016 ASDAN Math Tournament, 9

Tags: geometry test

A cyclic quadrilateral $ABCD$ has side lengths $AB=14$, $BC=19$, $CD=26$, and $DA=29$. Compute the sine of the smaller angle between diagonals $AC$ and $BD$.

2014 China Northern MO, 8

Tags: combinatorics , game , winning strategy

Two people, $A$ and $B$, play the game of blowing up a balloon. The balloon will explode only when the volume of the balloon $V>2014$ mL. $A$ blows in $1$ mL first, and then they takes turns blowing. It is agreed that the gas blown by each person must not be less than the gas blown by the other party last time and should not be more than twice the amount of gas the other party blew last time. The agreement is that the person who blows up the balloon loses. Who has a winning strategy ? Briefly explain it. (Do not consider the change in volume caused by the change in tension when the balloon is inflated).

2016 Taiwan TST Round 1, 1

Tags: geometry

Let $AB$ be a chord on a circle $O$, $M$ be the midpoint of the smaller arc $AB$. From a point $C$ outside the circle $O$ draws two tangents to the circle $O$ at the points $S$ and $T$. Suppose $MS$ intersects with $AB$ at the point $E$, $MT$ intersects with $AB$ at the point $F$. From $E,F$ draw a line perpendicular to $AB$ that intersects with $OS,OT$ at the points $X,Y$, respectively. Draw another line from $C$ which intersects with the circle $O$ at the points $P$ and $Q$. Let $R$ be the intersection point of $MP$ and $AB$. Finally, let $Z$ be the circumcenter of triangle $PQR$. Prove that $X$,$Y$ and $Z$ are collinear.

Brazil L2 Finals (OBM) - geometry, 2007.1

Tags: geometry , circumcircle , angle chasing , circumcenter , angle

Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be the intersection of straight lines $BO$ and $AC$ and $\omega$ be the circumcircle of triangle $AOP$. Suppose that $BO = AP$ and that the measure of the arc $OP$ in $\omega$, that does not contain $A$, is $40^o$. Determine the measure of the angle $\angle OBC$. [img]https://3.bp.blogspot.com/-h3UVt-yrJ6A/XqBItXzT70I/AAAAAAAAL2Q/7LVv0gmQWbo1_3rn906fTn6wosY1-nIfwCK4BGAYYCw/s1600/2007%2Bomb%2Bl2.png[/img]

2010 Sharygin Geometry Olympiad, 8

Tags: point , geometry , combinatorics , polygon , regular polygon , perimeter , minimum

Given is a regular polygon. Volodya wants to mark $k$ points on its perimeter so that any another regular polygon (maybe having a different number of sides) doesn’t contain all marked points on its perimeter. Find the minimal $k$ sufficient for any given polygon.

2022 Harvard-MIT Mathematics Tournament, 6

Tags: number theory

Let f be a function from $\{1, 2, . . . , 22\}$ to the positive integers such that $mn | f(m) + f(n)$ for all $m, n \in \{1, 2, . . . , 22\}$. If $d$ is the number of positive divisors of $f(20)$, compute the minimum possible value of $d$.

1983 IMO Longlists, 5

Tags: analytic geometry , graphing lines , slope , combinatorial geometry , algebra

Consider the set $\mathbb Q^2$ of points in $\mathbb R^2$, both of whose coordinates are rational. [b](a)[/b] Prove that the union of segments with vertices from $\mathbb Q^2$ is the entire set $\mathbb R^2$. [b](b)[/b] Is the convex hull of $\mathbb Q^2$ (i.e., the smallest convex set in $\mathbb R^2$ that contains $\mathbb Q^2$) equal to $\mathbb R^2$ ?

2018 Istmo Centroamericano MO, 4

Tags: trinomial , quadratic , perfect square , number theory

Let $t$ be an integer. Suppose the equation $$x^2 + (4t - 1) x + 4t^2 = 0$$ has at least one positive integer solution $n$. Show that $n$ is a perfect square.

2013 Greece Team Selection Test, 4

Tags: trapezoid , combinatorics , tiling

Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids". Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".

2012 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra

$\begin{cases} x^3-ax^2+b^3=0 \\x^3-bx^2+c^3=0 \\ x^3-cx^2+a^3=0 \end{cases}$ Prove that system hasn`t solutions if $a,b,c$ are different.

1958 Miklós Schweitzer, 1

Tags: group theory

[b]1.[/b] Find the groups every generating system of which contains a basis. (A basis is a set of elements of the group such that the direct product of the cyclic groups generated by them is the group itself.) [b](A. 14)[/b]