Olympiad problems

1991 AMC 12/AHSME, 13

Tags: probability

Horses X, Y and Z are entered in a three-horse race in which ties are not possible. If the odds against X winning are $3-to-1$ and the odds against Y winning are $2-to-3$, what are the odds against Z winning? (By "[i]odds against H winning are p-to-q[/i]" we mean that probability of H winning the race is $\frac{q}{p+q}$.) $ \textbf{(A)}\ 3-to-20\qquad\textbf{(B)}\ 5-to-6\qquad\textbf{(C)}\ 8-to-5\qquad\textbf{(D)}\ 17-to-3\qquad\textbf{(E)}\ 20-to-3 $

Kvant 2023, M2758

Tags: operation , combinatorics

The numbers $2,4,\ldots,2^{100}$ are written on a board. At a move, one may erase the numbers $a,b$ from the board and replace them with $ab/(a+b).$ Prove that the last numer on the board will be greater than 1. [i]From the folklore[/i]

2016 Auckland Mathematical Olympiad, 3

Tags: square , geometry , area

Triangle $XYZ$ is inside square $KLMN$ shown below so that its vertices each lie on three different sides of the square. It is known that: $\bullet$ The area of square $KLMN$ is $1$. $\bullet$ The vertices of the triangle divide three sides of the square up into these ratios: $KX : XL = 3 : 2$ $KY : YN = 4 : 1$ $NZ : ZM = 2 : 3$ What is the area of the triangle $XYZ$? (Note that the sketch is not drawn to scale). [img]https://cdn.artofproblemsolving.com/attachments/8/0/38e76709373ba02346515f9949ce4507ed4f8f.png[/img]

2011 Turkey MO (2nd round), 4

Tags: number theory , legendre s theorem , modular arithmetic

$a_{1}=5$ and $a_{n+1}=a_{n}^{3}-2a_{n}^{2}+2$ for all $n\geq1$. $p$ is a prime such that $p=3(mod 4)$ and $p|a_{2011}+1$. Show that $p=3$.

2007 District Olympiad, 1

Tags: limit , real analysis , induction

Let $a_1\in (0,1)$ and $(a_n)_{n\ge 1}$ a sequence of real numbers defined by $a_{n+1}=a_n(1-a_n^2),\ (\forall)n\ge 1$. Evaluate $\lim_{n\to \infty} a_n\sqrt{n}$.

2011 Hanoi Open Mathematics Competitions, 1

Tags: divisible , number theory

An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$. How many integers between $1$ and $100$ are octal? (A): $22$, (B): $24$, (C): $27$, (D): $30$, (E): $33$

2003 China Team Selection Test, 3

Tags: function , number theory

Let $x_0+\sqrt{2003}y_0$ be the minimum positive integer root of Pell function $x^2-2003y^2=1$. Find all the positive integer solutions $(x,y)$ of the equation, such that $x_0$ is divisible by any prime factor of $x$.

2019 IFYM, Sozopol, 2

Tags: game theory , coloring , board , combinatorics , game strategy

Let $n$ be a natural number. At first the cells of a table $2n$ x $2n$ are colored in white. Two players $A$ and $B$ play the following game. First is $A$ who has to color $m$ arbitrary cells in red and after that $B$ chooses $n$ rows and $n$ columns and color their cells in black. Player $A$ wins, if there is at least one red cell on the board. Find the least value of $m$ for which $A$ wins no matter how $B$ plays.

2017 Junior Balkan Team Selection Tests - Romania, 4

Tags: coloring , combinatorics , combinatorial geometry

Consider an $m\times n$ board where $m, n \ge 3$ are positive integers, divided into unit squares. Initially all the squares are white. What is the minimum number of squares that need to be painted red such that each $3\times 3$ square contains at least two red squares? Andrei Eckstein and Alexandru Mihalcu

1987 Tournament Of Towns, (146) 3

Tags: combinatorics

In a certain city only simple (pairwise) exchanges of apartments are allowed (if two families exchange fiats , they are not allowed to participate in another exchange on the same day). Prove that any compound exchange may be effected in two days. It is assumed that under any exchange (simple or comp ound) each family occupies one fiat before and after the exchange and the family cannot split up . (A . Shnirelman , N .N . Konstantinov)

2014 China Team Selection Test, 5

Tags: complex number , inequalities

Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.

2024 UMD Math Competition Part I, #24

Tags: geometry , probability , number theory

Let $n\ge3$ be an integer. A regular $n$-gon $P$ is given. We randomly select three distinct vertices of $P$. The probability that these three vertices form an isosceles triangle is $1/m$, where $m$ is an integer. How many such integers $n\le 2024$ are there? \[\rm a. ~674\qquad \mathrm b. ~675\qquad \mathrm c. ~682 \qquad\mathrm d. ~684\qquad\mathrm e. ~685\]

2017 CCA Math Bonanza, I4

Tags:

Cole is trying to solve the [i]Collatz conjecture[/i]. She decides to make a model with a piece of wood with a hole for every natural number. For every even number there is a rope from $n$ to $\frac{n}{2}$ and for every odd number there is a rope from $n$ to $3n+1$. She wants to bring her model to a convention but in order to do that she needs to cut off the part containing the first $240$ holes. How many ropes did she break? [i]2017 CCA Math Bonanza Individual Round #4[/i]

2024 Francophone Mathematical Olympiad, 2

Tags: combinatorics , winning strategy , game

Given $n \ge 2$ points on a circle, Alice and Bob play the following game. Initially, a tile is placed on one of the points and no segment is drawn. The players alternate in turns, with Alice to start. In a turn, a player moves the tile from its current position $P$ to one of the $n-1$ other points $Q$ and draws the segment $PQ$. This move is not allowed if the segment $PQ$ is already drawn. If a player cannot make a move, the game is over and the opponent wins. Determine, for each $n$, which of the two players has a winning strategy.

2020 Latvia Baltic Way TST, 12

Tags: rhombus , geometry

There are rhombus $ABCD$ and circle $\Gamma_B$, which is centred at $B$ and has radius $BC$, and circle $\Gamma_C$, which is centred at $C$ and has radius $BC$. Circles $\Gamma_B$ and $\Gamma_C$ intersect at point $E$. The line $ED$ intersects $\Gamma_B$ at point $F$. Find all possible values of $\angle AFB$.

2024 Azerbaijan Senior NMO, 3

Tags: geometry

In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.

2011 IMO Shortlist, 1

Tags: weight , combinatorics

Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways in which this can be done. [i]Proposed by Morteza Saghafian, Iran[/i]

2007 Poland - Second Round, 2

Tags: power of a point , geometry , radical axis , cyclic quadrilateral

We are given a cyclic quadrilateral $ABCD \quad AB\not=CD$. Quadrilaterals $AKDL$ and $CMBN$ are rhombuses with equal sides. Prove, that $KLMN$ is cyclic

2003 Polish MO Finals, 1

Tags: homothety , symmetry , rotation , trapezoid , geometric transformation , geometry , circumcircle

In an acute-angled triangle $ABC, CD$ is the altitude. A line through the midpoint $M$ of side $AB$ meets the rays $CA$ and $CB$ at $K$ and $L$ respectively such that $CK = CL.$ Point $S$ is the circumcenter of the triangle $CKL.$ Prove that $SD = SM.$

2009 National Olympiad First Round, 24

Tags: rectangle , geometry

In $ xy \minus{}$plane, there are $ b$ blue and $ r$ red rectangles whose sides are parallel to the axis. Any parallel line to the axis can intersect at most one rectangle with same color. For any two rectangle with different colors, there is a line which is parallel to the axis and which intersects only these two rectangles. $ (b,r)$ cannot be ? $\textbf{(A)}\ (1,7) \qquad\textbf{(B)}\ (2,6) \qquad\textbf{(C)}\ (3,4) \qquad\textbf{(D)}\ (3,3) \qquad\textbf{(E)}\ \text{None}$

1973 Polish MO Finals, 6

Tags: ellipse , geometric inequalities , conic , geometry

Prove that for every centrally symmetric polygon there is at most one ellipse containing the polygon and having the minimal area.

1999 French Mathematical Olympiad, Problem 3

Tags: inradius , geometric inequalities , triangle , ratio , geometry

For which acute-angled triangles is the ratio of the smallest side to the inradius the maximum?

2016 Latvia Baltic Way TST, 1

Tags: combinatorics

$2016$ numbers written on the board: $\frac{1}{2016}, \frac{2}{2016}, \frac{3}{2016}, ..., \frac{2016}{2016}$. In one move, it is allowed to choose any two numbers $a$ and $b$ written on the board, delete them, and write the number $3ab - 2a - 2b + 2$ instead. Determine what number will remain written on the board after $2015$ moves.

2001 Irish Math Olympiad, 5

Tags: inequalities

Prove that for all real numbers $ a,b$ with $ ab>0$ we have: $ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}10ab\plus{}b^2}{12}$ and find the cases of equality. Hence, or otherwise, prove that for all real numbers $ a,b$ $ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}ab\plus{}b^2}{3}$ and find the cases of equality.

1991 China Team Selection Test, 1

Tags: function , inequalities , algebra , polynomial

Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have \[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]