Olympiad problems

1984 IMO Longlists, 66

Let $1=d_1<d_2<....<d_k=n$ be all different divisors of positive integer n written in ascending order. Determine all n such that: \[d_6^{2} +d_7^{2} - 1=n\]

Let $G$ be a simple $k$ edge-connected graph on $n$ vertices and let $u$ and $v$ be different vertices of $G$. Prove that there are $k$ edge-disjoint paths from $u$ to $v$ each having at most $\frac{20n}{k}$ edges.

2011 India IMO Training Camp, 3

Tags: rectangle , induction , combinatorics

Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.

1989 Bulgaria National Olympiad, Problem 2

Tags: limit , series , floor function , algebra , sequence

Prove that the sequence $(a_n)$, where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.

2001 Italy TST, 2

Tags: algebra , inequalities

Let $0\le a\le b\le c$ be real numbers. Prove that \[(a+3b)(b+4c)(c+2a)\ge 60abc \]

2009 District Olympiad, 1

Tags: geometry , vector geometry

On the sides $ AB $ and $ AC $ of the triangle $ ABC $ consider the points $ D, $ respectively, $ E, $ such that $$ \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} . $$ If $ T $ is the intersection of $ DC $ and $ BE, $ determine the real number $ \alpha $ so that: $$ \overrightarrow{TB} +\overrightarrow{TC} =\alpha\cdot\overrightarrow{TA} . $$

2015 SG Originals, N6

Tags: function , number theory , periodic function , arrows

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties: (i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$; (ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite. Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic. [i]Proposed by Ang Jie Jun, Singapore[/i]

1993 Romania Team Selection Test, 2

Tags: geometry

Suppose that $ D,E,F$ are points on sides $ BC,CA,AB$ of a triangle $ ABC$ respectively such that $ BD\equal{}CE\equal{}AF$ and $ \angle BAD\equal{}\angle CBE\equal{}\angle ACF$.Prove that the triangle $ ABC$ is equilateral.

1985 Bundeswettbewerb Mathematik, 3

Tags: 3d geometry , sphere , geometry

From a point in space, $n$ rays are issuing, whereas the angle among any two of these rays is at least $30^{\circ}$. Prove that $n < 59$.

2006 IberoAmerican Olympiad For University Students, 4

Tags: calculus , integration , algebra , polynomial , function , real analysis

Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval \[a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b\] such that \[\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots\] for all polynomials $f$ with real coefficients and degree less than $n$.

2004 Baltic Way, 8

Tags: algebra , polynomial , inequalities , function , number theory , prime number , logarithm

Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.

2003 National Olympiad First Round, 18

Tags: modular arithmetic

What is the least integer $n>2003$ such that $5^n + n^5$ is a multiple of $11$? $ \textbf{(A)}\ 2010 \qquad\textbf{(B)}\ 2011 \qquad\textbf{(C)}\ 2012 \qquad\textbf{(D)}\ 2014 \qquad\textbf{(E)}\ \text{None of the preceding} $

2018 Regional Olympiad of Mexico Northeast, 1

Tags: algebra

$N$ different positive integers are arranged around a circle , in such a way that the sum of every $5$ consecutive numbers in the circle is a multiple of $13$. Let $A $ be the smallest possible sum of the $n$ numbers. Calculate the value of $A$ for $\bullet$ $n = 99$, $\bullet$ $n = 100$.

2003 National Olympiad First Round, 19

Tags:

At least how many elements does the set which contains all of the midpoints of segments connecting $2003$ different points in a plane have? $ \textbf{(A)}\ 2006 \qquad\textbf{(B)}\ 4001 \qquad\textbf{(C)}\ 4003 \qquad\textbf{(D)}\ 4006 \qquad\textbf{(E)}\ \text{None of the preceeding} $

2001 CentroAmerican, 2

Tags:

Let $ AB$ be the diameter of a circle with a center $ O$ and radius $ 1$. Let $ C$ and $ D$ be two points on the circle such that $ AC$ and $ BD$ intersect at a point $ Q$ situated inside of the circle, and $ \angle AQB\equal{} 2 \angle COD$. Let $ P$ be a point that intersects the tangents to the circle that pass through the points $ C$ and $ D$. Determine the length of segment $ OP$.

2006 Pre-Preparation Course Examination, 3

Tags: combinatorics

The bell number $b_n$ is the number of ways to partition the set $\{1,2,\ldots,n\}$. For example $b_3=5$. Find a recurrence for $b_n$ and show that $b_n=e^{-1}\sum_{k\geq 0} \frac{k^n}{k!}$. Using a combinatorial proof show that the number of ways to partition $\{1,2,\ldots,n\}$, such that now two consecutive numbers are in the same block, is $b_{n-1}$.

2011 ISI B.Math Entrance Exam, 2

Tags: 3d geometry , quadratic , number theory

Given two cubes $R$ and $S$ with integer sides of lengths $r$ and $s$ units respectively . If the difference between volumes of the two cubes is equal to the difference in their surface areas , then prove that $r=s$.

2004 France Team Selection Test, 1

Tags: number theory , quadratic , modular arithmetic

If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$. Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?

Gheorghe Țițeica 2025, P4

Tags: ring theory , abstract algebra

Let $R$ be a ring. Let $x,y\in R$ such that $x^2=y^2=0$. Prove that if $x+y-xy$ is nilpotent, so is $xy$. [i]Janez Šter[/i]

1939 Moscow Mathematical Olympiad, 048

Tags: factoring , algebra , factor

Factor $a^{10} + a^5 + 1$ into nonconstant polynomials with integer coefficients

2013 AMC 10, 4

Tags: algebra

A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? $ \textbf {(A) } 35 \qquad \textbf {(B) } 40 \qquad \textbf {(C) } 45 \qquad \textbf {(D) } 50 \qquad \textbf {(E) } 55 $

1998 Harvard-MIT Mathematics Tournament, 4

Tags: ratio , number theory , relatively prime

Given that $r$ and $s$ are relatively prime positive integers such that $\dfrac{r}{s}=\dfrac{2(\sqrt{2}+\sqrt{10})}{5\left(\sqrt{3+\sqrt{5}}\right)}$, find $r$ and $s$.

2003 Italy TST, 2

Tags: circumcircle , geometry

Let $B\not= A$ be a point on the tangent to circle $S_1$ through the point $A$ on the circle. A point $C$ outside the circle is chosen so that segment $AC$ intersects the circle in two distinct points. Let $S_2$ be the circle tangent to $AC$ at $C$ and to $S_1$ at some point $D$, where $D$ and $B$ are on the opposite sides of the line $AC$. Let $O$ be the circumcentre of triangle $BCD$. Show that $O$ lies on the circumcircle of triangle $ABC$.

2022 ISI Entrance Examination, 8

Tags: trigonometry , algebra , inequalities

Find the minimum value of $$\big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big|$$ for real numbers $x$ not multiple of $\frac{\pi}{2}$.

2012 China Team Selection Test, 1

Tags: inequalities , induction , number theory

Given an integer $n\ge 2$. Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions: [list] [*] $a_1>a_2>\ldots>a_n$; [*] $\gcd (a_1,a_2,\ldots,a_n)=1$; [*] $a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$,where $a_{n+1}=a_1$.[/list]