Olympiad problems

2012 Online Math Open Problems, 30

The Lattice Point Jumping Frog jumps between lattice points in a coordinate plane that are exactly $1$ unit apart. The Lattice Point Jumping Frog starts at the origin and makes $8$ jumps, ending at the origin. Additionally, it never lands on a point other than the origin more than once. How many possible paths could the frog have taken? [i]Author: Ray Li[/i] [hide="Clarifications"] [list=1][*]The Lattice Jumping Frog is allowed to visit the origin more than twice. [*]The path of the Lattice Jumping Frog is an ordered path, that is, the order in which the Lattice Jumping Frog performs its jumps matters.[/list][/hide]

2007 France Team Selection Test, 1

Tags: number theory

For a positive integer $a$, $a'$ is the integer obtained by the following method: the decimal writing of $a'$ is the inverse of the decimal writing of $a$ (the decimal writing of $a'$ can begin by zeros, but not the one of $a$); for instance if $a=2370$, $a'=0732$, that is $732$. Let $a_{1}$ be a positive integer, and $(a_{n})_{n \geq 1}$ the sequence defined by $a_{1}$ and the following formula for $n \geq 1$: \[a_{n+1}=a_{n}+a'_{n}. \] Can $a_{7}$ be prime?

2020 Turkey EGMO TST, 5

Tags: geometry

$A, B, C, D, E$ points are on $\Gamma$ cycle clockwise. $[AE \cap [CD = \{M\}$ and $[AB \cap [DC = \{N\}$. The line parallels to $EC$ and passes through $M$ intersects with the line parallels to $BC$ and passes through $N$ on $K$. Similarly, the line parallels to $ED$ and passes through $M$ intersects with the line parallels to $BD$ and passes through $N$ on $L$. Show that the lines $LD$ and $KC$ intersect on $\Gamma$.

2014 Contests, 3

Tags: greatest common divisor , least common multiple , inequalities , number theory

Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$

2017 Iran Team Selection Test, 3

Tags: algebra , functional equation , function

Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$ $$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$ $$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$ [i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]

2008 Princeton University Math Competition, A6

Tags: geometry , analytic geometry

Find the coordinates of the point in the plane at which the sum of the distances from it to the three points $(0, 0)$, $(2, 0)$, $(0, \sqrt{3})$ is minimal.

2009 CHKMO, 3

Tags: geometry , trigonometry , ratio , incenter , inradius , circumcircle , parallelogram

$ \Delta ABC$ is a triangle such that $ AB \neq AC$. The incircle of $ \Delta ABC$ touches $ BC, CA, AB$ at $ D, E, F$ respectively. $ H$ is a point on the segment $ EF$ such that $ DH \bot EF$. Suppose $ AH \bot BC$, prove that $ H$ is the orthocentre of $ \Delta ABC$. Remark: the original question has missed the condition $ AB \neq AC$

2015 Switzerland - Final Round, 6

Tags: combinatorics , combinatorial geometry

We have an $8\times 8$ board. An [i]interior [/i] edge is an edge between two $1 \times 1$ cells. we cut the board into $1 \times 2$ dominoes. For an inner edge $k$, $N(k)$ denotes the number of ways to cut the board so that it cuts along edge $k$. Calculate the last digit of the sum we get if we add all $N(k)$, where $k$ is an inner edge.

2020 Sharygin Geometry Olympiad, 7

Tags: geometry

Prove that the medial lines of triangle $ABC$ meets the sides of triangle formed by its excenters at six concyclic points.

2021 Purple Comet Problems, 15

Tags: number theory

Let $m$ and $n$ be positive integers such that $$(m^3 -27)(n^3- 27) = 27(m^2n^2 + 27):$$ Find the maximum possible value of $m^3 + n^3$.

1990 Iran MO (2nd round), 2

Tags: number theory

Find all integer solutions to the equation \[(x^2-x)(x^2-2x+2)=y^2-1\]

2010 Balkan MO Shortlist, A2

Tags: algebra

Let the sequence $(a_n)_{n \in \mathbb{N}}$, where $\mathbb{N}$ denote the set of natural numbers, is given with $a_1=2$ and $a_{n+1}$ $=$ $a_n^2$ $-$ $a_n+1$. Find the minimum real number $L$, such that for every $k$ $\in$ $\mathbb{N}$ \begin{align*} \sum_{i=1}^k \frac{1}{a_i} < L \end{align*}

2023 Math Prize for Girls Problems, 5

Tags:

Acute triangle $ABC$ has area 870. The triangle whose vertices are the feet of the altitudes of $\triangle ABC$ has area 48. Determine \[ \sin^2 A + \sin^2 B + \sin^2 C . \]

2018 Irish Math Olympiad, 10

Tags: game strategy , combinatorics

The game of Greed starts with an initial configuration of one or more piles of stones. Player $1$ and Player $2$ take turns to remove stones, beginning with Player $1$. At each turn, a player has two choices: • take one stone from any one of the piles (a simple move); • take one stone from each of the remaining piles (a greedy move). The player who takes the last stone wins. Consider the following two initial configurations: (a) There are $2018$ piles, with either $20$ or $18$ stones in each pile. (b) There are four piles, with $17, 18, 19$, and $20$ stones, respectively. In each case, find an appropriate strategy that guarantees victory to one of the players.

1998 Gauss, 23

Tags: gauss

A cube measures $10 \text{cm} \times 10 \text{cm} \times10 \text{cm}$ . Three cuts are made parallel to the faces of the cube as shown creating eight separate solids which are then separated. What is the increase in the total surface area? $\textbf{(A)}\ 300 \text{cm}^2 \qquad \textbf{(B)}\ 800 \text{cm}^2 \qquad \textbf{(C)}\ 1200 \text{cm}^2 \qquad \textbf{(D)}\ 600 \text{cm}^2 \qquad \textbf{(E)}\ 0 \text{cm}^2$

STEMS 2023 Math Cat A, 5

Tags: algebra , polynomial , computational

Consider a polynomial $P(x) \in \mathbb{R}[x]$, with degree $2023$, such that $P(\sin^2(x))+P(\cos^2(x)) =1$ for all $x \in \mathbb{R}$. If the sum of all roots of $P$ is equal to $\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$?

1995 Abels Math Contest (Norwegian MO), 3

Tags: sum , algebra , natural

Show that there exists a sequence $x_1,x_2,...$ of natural numbers in which every natural number occurs exactly once, such that the sums $\sum_{i=1}^n \frac{1}{x_i}$, $n = 1,2,3,...$, include all natural numbers.

2018 MIG, 12

Tags:

A unit cube is sliced by a plane passing through two of its vertices and the midpoints of the edges it passes through. What is the area of the rhombus formed by this intersection? [center][img]https://cdn.artofproblemsolving.com/attachments/3/5/3ed19fa0b4d454a3afc16c6bcf9d69403f6b2c.png[/img][/center] $\textbf{(A) } \dfrac{\sqrt6}{2}\qquad\textbf{(B) }\sqrt2\qquad\textbf{(C) }\sqrt3\qquad\textbf{(D) }\sqrt6\qquad\textbf{(E) }2\sqrt6$

2015 Azerbaijan Team Selection Test, 1

Tags: geometry , circumcircle

Let $\omega$ be the circumcircle of an acute-angled triangle $ABC$. The lines tangent to $\omega$ at the points $A$ and $B$ meet at $K$. The line passing through $K$ and parallel to $BC$ intersects the side $AC$ at $S$. Prove that $BS=CS$

2022 Turkey EGMO TST, 2

Tags: combinatorics , extremal principle , set

We are given some three element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ [i]nice [/i] if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element [i]nice [/i] subset while keeping it nice, find the minimum value of $n$.

VI Soros Olympiad 1999 - 2000 (Russia), 11.2

Tags: algebra , derivative , calculus

Let $$f(x) = (...((x - 2)^2 - 2)^2 - 2)^2... - 2)^2$$ (here there are $n$ brackets $( )$). Find $f''(0)$

2011 ELMO Shortlist, 6

Tags: algebra , polynomial , counting , derangement , induction , geometric transformation , geometry

Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$, \[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$. [i]Calvin Deng.[/i]

2003 AMC 10, 2

Tags:

Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $ \$1$ more than a pink pill, and Al’s pills cost a total of $ \$546$ for the two weeks. How much does one green pill cost? $ \textbf{(A)}\ \$7 \qquad \textbf{(B)}\ \$14 \qquad \textbf{(C)}\ \$19 \qquad \textbf{(D)}\ \$20 \qquad \textbf{(E)}\ \$39$

2015 Singapore MO Open, 2

Tags: graph theory , sequence

A boy lives in a small island in which there are three roads at every junction. He starts from his home and walks along the roads. At each junction he would choose to turn to the road on his right or left alternatively, i.e., his choices would be . . ., left, right, left,... Prove that he will eventually return to his home.

1983 Vietnam National Olympiad, 2

Tags: inequalities

Decide whether $S_n$ or $T_n$ is larger, where \[S_n =\displaystyle\sum_{k=1}^n \frac{k}{(2n - 2k + 1)(2n - k + 1)}, T_n =\displaystyle\sum_{k=1}^n\frac{1}{k}\]