Olympiad problems

1987 Austrian-Polish Competition, 3

A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.

2019 CHMMC (Fall), 2

Tags: combinatorics

Alex, Bob, Charlie, Daniel, and Ethan are five classmates. Some pairs of them are friends. How many possible ways are there for them to be friends such that everyone has at least one friend, and such that there is exactly one loop of friends among the five classmates? Note: friendship is two-way, so if person x is friends with person y then person y is friends with person $x$.

2001 Singapore Team Selection Test, 3

Tags: combinatorics , game

A game of Jai Alai has eight players and starts with players $P_1$ and $P_2$ on court and the other players $P_3, P_4, P_5, P_6, P_7, P_8$ waiting in a queue. After each point is played, the loser goes to the end of the queue; the winner adds $1$ point to his score and stays on the court; and the player at the head of the queue comes on to contest the next point. Play continues until someone has scored $7$ points. At that moment, we observe that a total of $37$ points have been scored by all eight players. Determine who has won and justify your answer.

2013 Stanford Mathematics Tournament, 3

Tags: algebra , polynomial , quadratic

Karl likes the number $17$ his favorite polynomials are monic quadratics with integer coefficients such that $17$ is a root of the quadratic and the roots differ by no more than $17$. Compute the sum of the coefficients of all of Karl's favorite polynomials. (A monic quadratic is a quadratic polynomial whose $x^2$ term has a coefficient of $1$.)

2004 Greece National Olympiad, 1

Tags: rearrangement inequality , inequalities

Find the greatest value of $M$ $\in \mathbb{R}$ such that the following inequality is true $\forall$ $x, y, z$ $\in \mathbb{R}$ $x^4+y^4+z^4+xyz(x+y+z)\geq M(xy+yz+zx)^2$.

1998 Mediterranean Mathematics Olympiad, 1

Tags: inequalities , geometry

A square $ABCD$ is inscribed in a circle. If $M$ is a point on the shorter arc $AB$, prove that \[MC \cdot MD > 3\sqrt{3} \cdot MA \cdot MB.\]

2014 Bulgaria National Olympiad, 1

Tags: trigonometry , geometry , orthocenter , locus

Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies. [i]Proposed by T. Vitanov, E. Kolev[/i]

2011 F = Ma, 15

Tags:

A vertical mass-spring oscillator is displaced $\text{2.0 cm}$ from equilibrium. The $\text{100 g}$ mass passes through the equilibrium point with a speed of $\text{0.75 m/s}$. What is the spring constant of the spring? (A) $\text{90 N/m}$ (B) $\text{100 N/m}$ (C) $\text{110 N/m}$ (D) $\text{140 N/m}$ (E) $\text{160 N/m}$

1991 Arnold's Trivium, 10

Tags: trigonometry

Investigate the asymptotic behaviour of the solutions $y$ of the equation $x^5 + x^2y^2 = y^6$ that tend to zero as $x\to0$.

1947 Moscow Mathematical Olympiad, 123

Tags: remainder , polynomial , algebra

Find the remainder after division of the polynomial $x+x^3 +x^9 +x^{27} +x^{81} +x^{243}$ by $x-1$.

2016 Hanoi Open Mathematics Competitions, 8

Tags: number theory , diophantine equation , diophantine

Find all positive integers $x,y,z$ such that $x^3 - (x + y + z)^2 = (y + z)^3 + 34$

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 $A[ADX]+A[BCY] = A[PXOY]$.