Olympiad problems

2016 LMT, 18

Tags:

Let $\triangle ABC$ be a triangle with $AB=5, BC=6, CA=7$. Suppose $P$ is a point inside $\triangle ABC$ such that $\triangle BPA\sim \triangle APC$. If $AP$ intersects $BC$ at $X$, find $\frac{BX}{CX}$. [i]Proposed by Nathan Ramesh

2019 IFYM, Sozopol, 6

Tags: combinatorics

There are $n$ kids. From each two at least one of them has sent an SMS to the other. For each kid $A$, among the kids on which $A$ has sent an SMS, exactly 10% of them have sent an SMS to $A$. Determine the number of possible three-digit values of $n$.

2014 ELMO Shortlist, 5

Tags: function , combinatorics

Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate \[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by Sammy Luo[/i]

2000 IberoAmerican, 2

Tags: induction , number theory

There are a buch of 2000 stones. Two players play alternatively, following the next rules: ($a$)On each turn, the player can take 1, 2, 3, 4 or 5 stones [b]of[/b] the bunch. ($b$) On each turn, the player has forbidden to take the exact same amount of stones that the other player took just before of him in the last play. The loser is the player who can't make a valid play. Determine which player has winning strategy and give such strategy.

2002 Moldova National Olympiad, 2

Tags:

From a set of consecutive natural numbers one number is excluded so that the aritmetic mean of the remaining numbers is $ 50.55$. Find the initial set of numbers and the excluded number.

1980 VTRMC, 2

Tags:

The sum of the first $n$ terms of the sequence $$1,1+2,1+2+2^2,\ldots,1+2+\cdots+2^{k-1},\ldots$$ is of the form $2^{n+R}+Sn^2+Tn+U$ for all $n>0.$ Find $R,S,T,$ and $U.$

1966 Dutch Mathematical Olympiad, 5

Tags: algebra , function , combinatorics

The image that maps $x$ to $1 - x$ is called [i]complement[/i], the image that maps $x$ to $\frac{1}{x}$ is called [i]invert[/i]. Two numbers $x$ and $y$ are called related if they can be transferred into each other by means of [i]complementation [/i]and/or [i]inversion[/i]. A [i]family [/i] is a collection of numbers where every two elements are related. Determine the maximum size $n$ of such a family. Show that the number line can be divided into $n$ parts, such that each of those $n$ parts contains exactly one number from each $n$-number family.

1974 AMC 12/AHSME, 21

Tags: geometric series

In a geometric series of positive terms the difference between the fifth and fourth terms is $576$, and the difference between the second and first terms is $9$. What is the sum of the first five terms of this series? $ \textbf{(A)}\ 1061 \qquad\textbf{(B)}\ 1023 \qquad\textbf{(C)}\ 1024 \qquad\textbf{(D)}\ 768 \qquad\textbf{(E)}\ \text{none of these} $

2022 MOAA, 7

Tags: geometry

A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$.

2005 China Western Mathematical Olympiad, 4

Tags: algebra

Given is the positive integer $n > 2$. Real numbers $\mid x_i \mid \leq 1$ ($i = 1, 2, ..., n$) satisfying $\mid \sum_{i=1}^{n}x_i \mid > 1$. Prove that there exists positive integer $k$ such that $\mid \sum_{i=1}^{k}x_i - \sum_{i=k+1}^{n}x_i \mid \leq 1$.

2019 Sharygin Geometry Olympiad, 21

Tags: geometry

An ellipse $\Gamma$ and its chord $AB$ are given. Find the locus of orthocenters of triangles $ABC$ inscribed into $\Gamma$.

2019 May Olympiad, 2

Tags: combinatorics , number theory

More than five competitors participated in a chess tournament. Each competitor played exactly once against each of the other competitors. Five of the competitors they each lost exactly two games. All other competitors each won exactly three games. There were no draws in the tournament. Determine how many competitors there were and show a tournament that verifies all conditions.

2013 ELMO Shortlist, 7

Tags: function , induction , 3n-3 theorem , continuity , algebra

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

2012 Online Math Open Problems, 42

Tags: trigonometry , geometry , circumcircle , vector , analytic geometry

In triangle $ABC,$ $\sin \angle A=\frac{4}{5}$ and $\angle A<90^\circ$ Let $D$ be a point outside triangle $ABC$ such that $\angle BAD=\angle DAC$ and $\angle BDC = 90^{\circ}.$ Suppose that $AD=1$ and that $\frac{BD} {CD} = \frac{3}{2}.$ If $AB+AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are pairwise relatively prime integers, find $a+b+c$. [i]Author: Ray Li[/i]

2014 IMO Shortlist, N7

Tags: number theory , sequence , prime divisor

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

2019 Pan-African Shortlist, N6

Tags: binomial coefficient , number theory

Find the $2019$th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.

2017 Harvard-MIT Mathematics Tournament, 3

Tags:

There are $2017$ jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>0$ such that $N$ of the jars have exactly $d$ coins). What is the maximum possible value of $N$?

2021 IMO Shortlist, G2

Tags: geometry

Let $\Gamma$ be a circle with centre $I$, and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $A I C$. The extension of $B A$ beyond $A$ meets $\Omega$ at $X$, and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$. The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$, respectively. Prove that \[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\] [i]Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland[/i]

2022 Brazil National Olympiad, 4

Tags: geometry

Let $ABC$ a triangle with $AB=BC$ and incircle $\omega$. Let $M$ the mindpoint of $BC$; $P, Q$ points in the sides $AB, AC$ such that $PQ\parallel BC$, $PQ$ is tangent to $\omega$ and $\angle CQM=\angle PQM$. Find the perimeter of triangle $ABC$ knowing that $AQ=1$.

2024 Chile TST Ibero., 2

Tags: algebra

A collection of regular polygons with sides of equal length is said to "fit" if, when arranged around a common vertex, they exactly complete the surrounding area of the point on the plane. For example, a square fits with two octagons. Determine all possible collections of regular polygons that fit.

Kvant 2020, M2618

Tags: algebra , floor function

For a given number $\alpha{}$ let $f_\alpha$ be a function defined as \[f_\alpha(x)=\left\lfloor\alpha x+\frac{1}{2}\right\rfloor.\]Let $\alpha>1$ and $\beta=1/\alpha$. Prove that for any natural $n{}$ the relation $f_\beta(f_\alpha(n))=n$ holds. [i]Proposed by I. Dorofeev[/i]

2008 Iran MO (3rd Round), 8

Tags: inequalities , number theory

In an old script found in ruins of Perspolis is written: [code] This script has been finished in a year whose 13th power is 258145266804692077858261512663 You should know that if you are skilled in Arithmetics you will know the year this script is finished easily.[/code] Find the year the script is finished. Give a reason for your answer.

2012 EGMO, 4

Tags: algorithm , induction , absolute value , combinatorics

A set $A$ of integers is called [i]sum-full[/i] if $A \subseteq A + A$, i.e. each element $a \in A$ is the sum of some pair of (not necessarily different) elements $b,c \in A$. A set $A$ of integers is said to be [i]zero-sum-free[/i] if $0$ is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of $A$. Does there exist a sum-full zero-sum-free set of integers? [i]Romania (Dan Schwarz)[/i]

2004 India National Olympiad, 3

Tags: algebra

If $a$ is a real root of $x^5 - x^3 + x - 2 = 0$, show that $[a^6] =3$

2006 CHKMO, 2

Tags: inequalities , triangle inequality , geometry

Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.