Olympiad problems

2012 Peru IMO TST, 5

Tags: geometry

Let $ABCD$ be a parallelogram such that $\angle{ABC} > 90^{\circ}$, and $\mathcal{L}$ the line perpendicular to $BC$ that passes through $B$. Suppose that the segment $CD$ does not intersect $\mathcal{L}$. Of all the circumferences that pass through $C$ and $D$, there is one that is tangent to $\mathcal{L}$ at $P$, and there is another one that is tangent to $\mathcal{L}$ at $Q$ (where $P \neq Q$). If $M$ is the midpoint of $AB$, prove that $\angle{PMD} = \angle{QMD}$.

2016 Fall CHMMC, 6

Tags: counting

How many binary strings of length $10$ do not contain the substrings $101$ or $010$?

2018 Adygea Teachers' Geometry Olympiad, 4

Tags: geometry , 3d geometry , cube , angle , plane

Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$. a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$. b) Find the angle between the plane $a$ and the plane $BB_1C_1$.

2020 Princeton University Math Competition, A1/B3

Tags: number theory

Compute the last two digits of $$9^{2020} + 9^{2020^2}+ ... + 9^{2020^{2020}}$$

2008 Philippine MO, 2

Tags: number theory

Find the largest integer $n$ for which $\frac{n^{2007}+n^{2006}+\cdots+n^2+n+1}{n+2007}$ is an integer.

Dumbest FE I ever created, 5.

Tags: function , algebra , algebruh , rutthee ahh function

Find all non decreasing function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ and $m,n \in \mathbb{N}_0$ such that $m+n \neq 0$ there exist $m',n' \in \mathbb{N}_0$ such that $m'+n'=m+n+1$ and $$f(f^m(x)+f^n(y))=f^{m'}(x)+f^{n'}(y)$$ . Note : $f^0(x)=x$ and $f^{n}(x)=f(f^{n-1}(x))$ for all $n \in \mathbb{N}$ . [hide=original]Find all non decreasing functions $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ $$ f(x+f(y))=f(x)+f(y) \text{ or } f(f(x))+y$$ .[/hide]

2003 Federal Math Competition of S&M, Problem 4

Tags: induction , ellipse , vector , combinatorics , conic

Let $ n$ be an even number, and $ S$ be the set of all arrays of length $ n$ whose elements are from the set $ \left\{0,1\right\}$. Prove that $ S$ can be partitioned into disjoint three-element subsets such that for each three arrays $ \left(a_i\right)_{i \equal{} 1}^n$, $ \left(b_i\right)_{i \equal{} 1}^n$, $ \left(c_i\right)_{i \equal{} 1}^n$ which belong to the same subset and for each $ i\in\left\{1,2,...,n\right\}$, the number $ a_i \plus{} b_i \plus{} c_i$ is divisible by $ 2$.

2012 QEDMO 11th, 10

Tags: combinatorics

Let there be three cups $A, B$ and $C$, which start with $a, b$ and $c$ (all of them are natural numbers) units of gallium filled. It is also believed that all cups are large enough to contain the total amount of gallium available. It is now allowed to move gallium from one cup to another cup, provided that the contents of the latter cup are exactly double. (a) For which starting positions is it possible to empty one of the cups? (b) For which starting positions is it possible to put all of the gallium in one cup?

Durer Math Competition CD Finals - geometry, 2022.C3

Tags: geometry , tangent , equilateral , square

To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.

2024 Iran MO (3rd Round), 2

Tags: cell , combinatorics , table , tabular triangle

Consider the main diagonal and the cells above it in an $ n \times n $ grid. These cells form what we call a tabular triangle of length $ n $. We want to place a real number in each cell of a tabular triangle of length $ n $ such that for each cell, the sum of the numbers in the cells in the same row and the same column (including itself) is zero. For example, the sum of the cells marked with a circle is zero. It is known that the number in the topmost and leftmost cell is $1.$ Find all possible ways to fill the remaining cells.

1962 AMC 12/AHSME, 25

Tags:

Given square $ ABCD$ with side $ 8$ feet. A circle is drawn through vertices $ A$ and $ D$ and tangent to side $ BC.$ The radius of the circle, in feet, is: $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 4 \sqrt{2} \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 5 \sqrt{2} \qquad \textbf{(E)}\ 6$

1984 All Soviet Union Mathematical Olympiad, 383

Tags: integer root , trinomial , polynomial , game , combinatorics

The teacher wrote on a blackboard: $$x^2 + 10x + 20$$ Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: $$x^2 + 20x + 10$$ Is it true that some time during the process there was written the square polynomial with the integer roots?

2023 Stanford Mathematics Tournament, 7

Tags: geometry

Triangle $ABC$ has $AC = 5$. $D$ and $E$ are on side $BC$ such that $AD$ and $AE$ trisect $\angle BAC$, with $D$ closer to $B$ and $DE =\frac32$, $EC =\frac52$ . From $B$ and $E$, altitudes $BF$ and $EG$ are drawn onto side $AC$. Compute $\frac{CF}{CG}-\frac{AF}{AG}$ .

2010 Contests, 3

Tags: inequalities

Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$

2024 Junior Balkan Team Selection Tests - Romania, P3

Tags: algebra

Determine all positive integers $a,b,c,d,e,f$ satisfying the following condition: for any two of them, $x{}$ and $y{},$ two of the remaining numbers, $z{}$ and $t{},$ satisfy $x/y=z/t.$ [i]Cristi Săvescu[/i]

2007 AIME Problems, 6

Tags:

An integer is called [i]parity-monotonic[/i] if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ is $a_{i}$ is even. How many four-digit parity-monotonic integers are there?

2019 Greece National Olympiad, 4

Tags: combinatorics

Given a $n\times m$ grid we play the following game . Initially we place $M$ tokens in each of $M$ empty cells and at the end of the game we need to fill the whole grid with tokens.For that purpose we are allowed to make the following move:If an empty cell shares a common side with at least two other cells that contain a token then we can place a token in this cell.Find the minimum value of $M$ in terms of $m,n$ that enables us to win the game.

2003 France Team Selection Test, 2

Tags: analytic geometry , modular arithmetic , number theory

A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

2010 Harvard-MIT Mathematics Tournament, 7

Tags: calculus , inequalities , complex number

Let $a_1$, $a_2$, and $a_3$ be nonzero complex numbers with non-negative real and imaginary parts. Find the minimum possible value of \[\dfrac{|a_1+a_2+a_3|}{\sqrt[3]{|a_1a_2a_3|}}.\]

1996 Bundeswettbewerb Mathematik, 4

Tags: number theory , perfect square

Find all natural numbers $n$ for which $n2^{n-1} +1$ is a perfect square.

2008 Hong Kong TST, 4

Tags: geometry , geometric transformation , homothety , ratio , circumcircle , trapezoid , projective geometry

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.