Olympiad problems

2009 ITAMO, 1

Tags: combinatorics

A flea is initially at the point $(0, 0)$ in the Cartesian plane. Then it makes $n$ jumps. The direction of the jump is taken in a choice of the four cardinal directions. The first step is of length $1$, the second of length $2$, the third of length $4$, and so on. The $n^{th}$-jump is of length $2^{n-1}$. Prove that, if you know the final position flea, then it is possible to uniquely determine its position after each of the $n$ jumps.

2009 South africa National Olympiad, 5

Tags: invariant , absolute value , combinatorics

A game is played on a board with an infinite row of holes labelled $0, 1, 2, \dots$. Initially, $2009$ pebbles are put into hole $1$; the other holes are left empty. Now steps are performed according to the following scheme: (i) At each step, two pebbles are removed from one of the holes (if possible), and one pebble is put into each of the neighbouring holes. (ii) No pebbles are ever removed from hole $0$. (iii) The game ends if there is no hole with a positive label that contains at least two pebbles. Show that the game always terminates, and that the number of pebbles in hole $0$ at the end of the game is independent of the specific sequence of steps. Determine this number.

2005 Germany Team Selection Test, 3

Tags: inradius , trigonometry , inequalities , function , triangle inequality , geometry

Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

2024 Harvard-MIT Mathematics Tournament, 25

Tags: guts

Point $P$ is inside a square $ABCD$ such that $\angle APB = 135^\circ, PC=12,$ and $PD=15.$ Compute the area of this square.

2021 Princeton University Math Competition, A2 / B4

Tags: algebra

For a bijective function $g : R \to R$, we say that a function $f : R \to R$ is its superinverse if it satisfies the following identity $(f \circ g)(x) = g^{-1}(x)$, where $g^{-1}$ is the inverse of $g$. Given $g(x) = x^3 + 9x^2 + 27x + 81$ and $f$ is its superinverse, find $|f(-289)|$.

VII Soros Olympiad 2000 - 01, 8.4

Tags: geometry , 3d geometry , combinatorics , equilateral , coloring

Paint the maximum number of vertices of the cube red so that you cannot select three of the red vertices that form an equilateral triangle.

2018 AMC 12/AHSME, 19

Tags:

Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\tfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$? $\textbf{(A) } 324 \qquad \textbf{(B) } 330 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 361 \qquad \textbf{(E) } 646$

2014 Contests, 1

Tags: basic probability , probability

1. What is the probability that a randomly chosen word of this sentence has exactly four letters?

2013 Thailand Mathematical Olympiad, 8

Tags: algebra , polynomial

Let $p(x) = x^{2013} + a_{2012}x^{2012} + a_{2011}x^{2011} +...+ a_1x + a_0$ be a polynomial with real coefficients with roots $- b_{1006}, - b_{1005}, ... , -b_1, 0, b_1, ... , b_{1005}, b_{1006}$, where $b_1, b_2, ... , b_{1006}$ are positive reals with product $1$. Show that $a_3a_{2011} \le 1012036$

1989 National High School Mathematics League, 1

Tags:

On complex plane, if $A,B$ are two angles of acute triangle $ABC$, then the point $z=(\cos B-\sin A)+\text{i}(\sin B-\cos A)$ corresponding to is in $\text{(A)}$Quadrant I $\text{(B)}$Quadrant II $\text{(C)}$Quadrant III $\text{(D)}$Quadrant IV

2008 Teodor Topan, 3

Tags: limit , geometry , geometric transformation , calculus , calculus computations , logarithm

Consider the sequence $ a_n\equal{}\sqrt[3]{n^3\plus{}3n^2\plus{}2n\plus{}1}\plus{}a\sqrt[5]{n^5\plus{}5n^4\plus{}1}\plus{}\frac{ln(e^{n^2}\plus{}n\plus{}2)}{n\plus{}2}\plus{}b$. Find $ a,b \in \mathbb{R}$ such that $ \displaystyle\lim_{n\to\infty}a_n\equal{}5$.

2022 IMC, 3

Tags: function , counting techniques , generating functions

Let $p$ be a prime number. A flea is staying at point $0$ of the real line. At each minute, the flea has three possibilities: to stay at its position, or to move by $1$ to the left or to the right. After $p-1$ minutes, it wants to be at $0$ again. Denote by $f(p)$ the number of its strategies to do this (for example, $f(3) = 3$: it may either stay at $0$ for the entire time, or go to the left and then to the right, or go to the right and then to the left). Find $f(p)$ modulo $p$.

2009 Purple Comet Problems, 16

Tags: trigonometry , complex number

Let the complex number $z = \cos\tfrac{1}{1000} + i \sin\tfrac{1}{1000}.$ Find the smallest positive integer $n$ so that $z^n$ has an imaginary part which exceeds $\tfrac{1}{2}.$

1992 India Regional Mathematical Olympiad, 5

Tags: geometry

$ABCD$ is a quadrilateral and $P,Q$ are the midpoints of $CD, AB, AP, DQ$ meet at $X$ and $BP, CQ$ meet at $Y$. Prove that $A[ADX]+A[BCY] = A[PXOY]$.

2018 Iran Team Selection Test, 5

Tags: combinatorics

$2n-1$ distinct positive real numbers with sum $S $ are given. Prove that there are at least $\binom {2n-2}{n-1}$ different ways to choose $n $ numbers among them such that their sum is at least $\frac {S}{2}$. [i]Proposed by Amirhossein Gorzi[/i]

2009 Harvard-MIT Mathematics Tournament, 3

Tags: parallelogram , geometry

Let $T$ be a right triangle with sides having lengths $3$, $4$, and $5$. A point $P$ is called [i]awesome[/i] if P is the center of a parallelogram whose vertices all lie on the boundary of $T$. What is the area of the set of awesome points?

2016 Purple Comet Problems, 9

Tags:

Find the sum of all perfect squares that divide 2016.

2023 Bangladesh Mathematical Olympiad, P7

Tags: number theory , strong induction

Prove that every positive integer can be represented in the form $$3^{m_1}\cdot 2^{n_1}+3^{m_2}\cdot 2^{n_2} + \dots + 3^{m_k}\cdot 2^{n_k}$$ where $m_1 > m_2 > \dots > m_k \geq 0$ and $0 \leq n_1 < n_2 < \dots < n_k$ are integers.

1993 IMO Shortlist, 2

Tags: modular arithmetic , number theory , divisibility , prime number , composite number

A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

2023 LMT Spring, 10

Tags: algebra

The sequence $a_0,a_1,a_2,...$ is defined such that $a_0 = 2+ \sqrt3$, $a_1 =\sqrt{5-2\sqrt5}$, and $$a_n a_{n-1}a_{n-2} - a_n + a_{n-1} + a_{n-2} = 0.$$ Find the least positive integer $n$ such that $a_n = 1$.

1949-56 Chisinau City MO, 26

Tags: congruent triangles , congruent , geometry

Formulate a criterion for the conguence of triangles by two medians and an altitude.

2014 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , positive integer , equation

Solve the equation $$x^2+y^2+z^2=686$$ where $x$, $y$ and $z$ are positive integers

2006 Princeton University Math Competition, 7

Tags:

Find one complex value of $x$ that satisfies the equation $\sqrt{3}x^7+x^4+2=0$.

2019 HMNT, 2

Tags: number theory

Meghana writes two (not necessarily distinct) primes $q$ and $r$ in base $10$ next to each other on a blackboard, resulting in the concatenation of $q$ and $r$ (for example, if $q = 13$ and $r = 5$, the number on the blackboard is now $135$). She notices that three more than the resulting number is the square of a prime $p$. Find all possible values of $p$.

2010 Princeton University Math Competition, 3

Tags: geometry , ratio , angle bisector

Triangle $ABC$ has $AB = 4$, $AC = 5$, and $BC = 6$. An angle bisector is drawn from angle $A$, and meets $BC$ at $M$. What is the nearest integer to $100 \frac{AM}{CM}$?