Olympiad problems

2013 Turkey Junior National Olympiad, 3

Let $ABC$ be a triangle such that $AC>AB.$ A circle tangent to the sides $AB$ and $AC$ at $D$ and $E$ respectively, intersects the circumcircle of $ABC$ at $K$ and $L$. Let $X$ and $Y$ be points on the sides $AB$ and $AC$ respectively, satisfying \[ \frac{AX}{AB}=\frac{CE}{BD+CE} \quad \text{and} \quad \frac{AY}{AC}=\frac{BD}{BD+CE} \] Show that the lines $XY, BC$ and $KL$ are concurrent.

2017 ASDAN Math Tournament, 23

Tags: 2017 , Guts Round

Ben creates an $8\times8$ grid of coins, where each coin faces heads with probability $\tfrac{1}{2}$, and tails with probability $\tfrac{1}{2}$. Ben then makes a series of moves; each move consists of selecting a coin in the grid and flipping over all coins in the same row and column as the selected coin. Suppose that in Ben’s current grid of coins, it is possible to make a series of moves so that all coins in the grid are heads, and that Ben will make the fewest number of moves to do so. What is the expected number of moves that Ben makes?

1949 Moscow Mathematical Olympiad, 168

Tags: Sum , divisible , number theory

Prove that some (or one) of any $100$ integers can always be chosen so that the sum of the chosen integers is divisible by $100$.

2003 District Olympiad, 1

Tags: function , geometry , 3D geometry , algebra proposed , algebra

Find all functions $\displaystyle f : \mathbb N^\ast \to \mathbb N^\ast$ ($\displaystyle N^\ast = \{ 1,2,3,\ldots \}$) with the property that, for all $\displaystyle n \geq 1$, \[ f(1) + f(2) + \ldots + f(n) \] is a perfect cube $\leq n^3$. [i]Dinu Teodorescu[/i]

2004 Estonia National Olympiad, 4

Tags: functional equation , functional , algebra

Find all functions $f$ which are defined on all non-negative real numbers, take nonnegative real values only, and satisfy the condition $x \cdot f(y) + y\cdot f(x) = f(x) \cdot f(y) \cdot (f(x) + f(y))$ for all non-negative real numbers $x, y$.

LMT Team Rounds 2010-20, B1

Tags: algebra

Four $L$s are equivalent to three $M$s. Nine $M$s are equivalent to fourteen $T$ s. Seven $T$ s are equivalent to two $W$ s. If Kevin has thirty-six $L$s, how many $W$ s would that be equivalent to?

2020 Dutch BxMO TST, 2

Tags: geometry , Concyclic , cyclic quadrilateral

In an acute-angled triangle $ABC, D$ is the foot of the altitude from $A$. Let $D_1$ and $D_2$ be the symmetric points of $D$ wrt $AB$ and $AC$, respectively. Let $E_1$ be the intersection of $BC$ and the line through $D_1$ parallel to $AB$ . Let $E_2$ be the intersection of$ BC$ and the line through $D_2$ parallel to $AC$. Prove that $D_1, D_2, E_1$ and $E_2$ on one circle whose center lies on the circumscribed circle of $\vartriangle ABC$.

1989 India National Olympiad, 5

Tags: number theory unsolved , number theory

For positive integers $ n$, define $ A(n)$ to be $ \frac {(2n)!}{(n!)^{2}}$. Determine the sets of positive integers $ n$ for which (a) $ A(n)$ is an even number, (b) $ A(n)$ is a multiple of $ 4$.

2011 Today's Calculation Of Integral, 687

Tags: calculus , integration , quadratics , function , algebra , domain , analytic geometry

(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

2019 Kyiv Mathematical Festival, 2

Tags: Kyiv mathematical festival , inequalities , BPSQ

Let $a,b,c>0$ and $abc\ge1.$ Prove that $a^4+b^3+c^2\ge a^3+b^2+c.$

1995 Austrian-Polish Competition, 1

Tags: algebra , system of equations

Determine all real solutions $(a_1,...,a_n)$ of the following system of equations: $$\begin{cases}a_3 = a_2 + a_1\\ a_4 = a_3 + a_2\\ ...\\ a_n = a_{n-1} + a_{n-2}\\ a_1= a_n +a_{n-1} \\ a_2 = a_1 + a_n \end{cases}$$

2014 Canadian Mathematical Olympiad Qualification, 5

Tags: algebra , polynomial , irreducibility

Let $f(x) = x^4 + 2x^3 - x - 1$. (a) Prove that $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients. (b) Find the exact values of the 4 roots of $f(x)$.

2024 Putnam, A2

Tags: Putnam

For which real polynomials $p$ is there a real polynomial $q$ such that \[ p(p(x))-x=(p(x)-x)^2q(x) \] for all real $x$?

2006 Tournament of Towns, 2

Tags: altitude , geometry

Suppose $ABC$ is an acute triangle. Points $A_1, B_1$ and $C_1$ are chosen on sides $BC, AC$ and $AB$ respectively so that the rays $A_1A, B_1B$ and $C_1C$ are bisectors of triangle $A_1B_1C_1$. Prove that $AA_1, BB_1$ and $CC_1$ are altitudes of triangle $ABC$. (6)

2007 Mathematics for Its Sake, 3

Tags: modular arithmetic , number theory

Prove that there exists only one pair $ (p,q) $ of odd primes satisfying the properties that $ p^2\equiv 4\pmod q $ and $ q^2\equiv 1\pmod p. $ [i]Ana Maria Acu[/i]

1988 AMC 8, 10

Tags:

Chris' birthday is on a Thursday this year. What day of the week will it be $60$ days after her birthday? $ \text{(A)}\ \text{Monday}\qquad\text{(B)}\ \text{Wednesday}\qquad\text{(C)}\ \text{Thursday}\qquad\text{(D)}\ \text{Friday}\qquad\text{(E)}\ \text{Saturday} $

1979 IMO Longlists, 76

Tags: geometry , geometry proposed

Suppose that a triangle whose sides are of integer lengths is inscribed in a circle of diameter $6.25$. Find the sides of the triangle.

2013 F = Ma, 23

Tags:

A man with mass $m$ jumps off of a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance $H$ at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant $k$, and stretches from an original length of $L_0$ to a final length $L = L_0 + h$. The maximum tension in the Bungee cord is four times the weight of the man. Determine the spring constant $k$. $\textbf{(A) } \frac{mg}{h}\\ \\ \textbf{(B) } \frac{2mg}{h}\\ \\ \textbf{(C) } \frac{mg}{H}\\ \\ \textbf{(D) } \frac{4mg}{H}\\ \\ \textbf{(E) } \frac{8mg}{H}$

2003 Switzerland Team Selection Test, 9

Tags: number theory , combinatorics , divides

Given integers $0 < a_1 < a_2 <... < a_{101} < 5050$, prove that one can always choose for different numbers $a_k,a_l,a_m,a_n$ such that $5050 | a_k +a_l -a_m -a_n$

2023 CMWMC, R7

Tags: CMWMC , geometry , combinatorics , number theory , algebra

[b]p19.[/b] Sequences $a_n$ and $b_n$ of positive integers satisfy the following properties: (1) $a_1 = b_1 = 1$ (2) $a_5 = 6, b_5 \ge 7$ (3) Both sequences are strictly increasing (4) In each sequence, the difference between consecutive terms is either $1$ or $2$ (5) $\sum^5_{n=1}na_n =\sum^5_{n=1}nb_n = S$ Compute $S$. [b]p20.[/b] Let $A$, $B$, and $C$ be points lying on a line in that order such that $AB = 4$ and $BC = 2$. Let $I$ be the circle centered at B passing through $C$, and let $D$ and $E$ be distinct points on $I$ such that $AD$ and $AE$ are tangent to $I$. Let $J$ be the circle centered at $C$ passing through $D$, and let $F$ and $G$ be distinct points on $J$ such that $AF$ and $AG$ are tangent to $J$ and $DG < DF$. Compute the area of quadrilateral $DEFG$. [b]p21.[/b] Twain is walking randomly on a number line. They start at $0$, and flip a fair coin $10$ times. Every time the coin lands heads, they increase their position by 1, and every time the coin lands tails, they decrease their position by $1$. What is the probability that at some point the absolute value of their position is at least $3$? PS. You should use hide for answers.

2012 Putnam, 4

Tags: Putnam , logarithms , limit , inequalities , induction , calculus , integration

Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$)

XMO (China) 2-15 - geometry, 13.3

Tags: geometry , Orthocenter and Circumcenter , Concyclic

Let O be the circumcenter of triangle ABC. Let H be the orthocenter of triangle ABC. the perpendicular bisector of AB meet AC,BC at D,E. the circumcircle of triangle DEH meet AC,BC,OH again at F,G,L. CH meet FG at T,and ABCT is concyclic. Prove that LHBC is concyclic. graph: https://cdn.luogu.com.cn/upload/image_hosting/w6z6mvm4.png

1982 All Soviet Union Mathematical Olympiad, 329

Tags: number theory , sum of digits , divisible , exponential

a) Let $m$ and $n$ be natural numbers. For some nonnegative integers $k_1, k_2, ... , k_n$ the number $$2^{k_1}+2^{k_2}+...+2^{k_n}$$ is divisible by $(2^m-1)$. Prove that $n \ge m$. b) Can you find a number, divisible by $111...1$ ($m$ times "$1$"), that has the sum of its digits less than $m$?

2021 Brazil EGMO TST, 2

Tags: number theory

Let $a,b,k$ be positive integers such that $gcd(a,b)^2+lcm(a,b)^2+a^2b^2=2020^k$ Prove that $k$ is an even number.

2023 Princeton University Math Competition, A2

Tags: combinatorics

On an infinite triangular lattice, there is a single atom at a lattice point. We allow for four operations as illustrated in Figure 1. In words, one could take an existing atom, split it into three atoms, and place them at adjacent lattice points in one of the two displayed fashions (a “split”). One could also reverse the process, i.e. taking three existing atoms in the displayed configurations, and merge them into a single atom at the center (a “merge”). [center][img]https://cdn.artofproblemsolving.com/attachments/2/5/41abc4dc8fb8235e5eb0c98638f9e4a0896c05.png[/img][/center] Figure 1: The four possible operations on an atom. Assume that, after finitely many operations, there is again only a single atom remaining on the lattice. Show that this is possible if and only if the final atom is contained in the sublattice implied by Figure 2. [center][img]https://cdn.artofproblemsolving.com/attachments/b/4/7a7bd10a1862947c250fa07571c061367a5a71.png[/img][/center] Figure 2: The possible positions for the final atom is the green sublattice. The position of the original atom is marked in purple.