Olympiad problems

1993 Tournament Of Towns, (376) 4

Positive integers are written on the blackboard one after another. The next integer $a_{n+1}$ (to be written after $a_1$,$a_2$,$...$,$a_n$) is an arbitrary integer not representable as a sum of several previous integers taken one or more times (i.e. $a_{n+1}$ is not of the form $k_1 *a_i + k_2 *a_2 + ... + k_n *a_n$ where$ k_1$, $k_2$,$..$, $k_n$ are non-negative integers). Prove that the process of writing cannot be infinite. (A Belov)

2006 Romania National Olympiad, 1

Tags: abstract algebra , algebra , polynomial , group theory , superior algebra

Let $\displaystyle \mathcal K$ be a finite field. Prove that the following statements are equivalent: (a) $\displaystyle 1+1=0$; (b) for all $\displaystyle f \in \mathcal K \left[ X \right]$ with $\displaystyle \textrm{deg} \, f \geq 1$, $\displaystyle f \left( X^2 \right)$ is reducible.

2012 Iran MO (3rd Round), 2

Tags: inequalities , continued fraction , algebra

Suppose $N\in \mathbb N$ is not a perfect square, hence we know that the continued fraction of $\sqrt{N}$ is of the form $\sqrt{N}=[a_0,\overline{a_1,a_2,...,a_n}]$. If $a_1\neq 1$ prove that $a_i\le 2a_0$.

2010 VJIMC, Problem 3

Tags: matrix , linear algebra

Let $A$ and $B$ be two $n\times n$ matrices with integer entries such that all of the matrices $$A,\enspace A+B,\enspace A+2B,\enspace A+3B,\enspace\ldots,\enspace A+(2n)B$$are invertible and their inverses have integer entries, too. Show that $A+(2n+1)B$ is also invertible and that its inverse has integer entries.

2014 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: combinatorics , acquaintance

At the beginning of school year in one of the first grade classes: $i)$ every student had exatly $20$ acquaintances $ii)$ every two students knowing each other had exactly $13$ mutual acquaintances $iii)$ every two students not knowing each other had exactly $12$ mutual acquaintances Find number of students in this class

1990 IMO Longlists, 42

Tags: inequalities , geometry

Find $n$ points $p_1, p_2, \ldots, p_n$ on the circumference of a unit circle, such that $\sum_{1\leq i< j \leq n} p_i p_j$ is maximal.

2019 German National Olympiad, 6

Tags: algebra , system of equations

Suppose that real numbers $x,y$ and $z$ satisfy the following equations: \begin{align*} x+\frac{y}{z} &=2,\\ y+\frac{z}{x} &=2,\\ z+\frac{x}{y} &=2. \end{align*} Show that $s=x+y+z$ must be equal to $3$ or $7$. [i]Note:[/i] It is not required to show the existence of such numbers $x,y,z$.

2006 Sharygin Geometry Olympiad, 8.2

Tags: polygon , cut , minimum , combinatorial geometry , geometry

What $n$ is the smallest such that “there is a $n$-gon that can be cut into a triangle, a quadrilateral, ..., a $2006$-gon''?

2008 Ukraine Team Selection Test, 12

Tags: polynomial , induction , algebra

Prove that for all natural $ m$, $ n$ polynomial $ \sum_{i \equal{} 0}^{m}\binom{n\plus{}i}{n}\cdot x^i$ has at most one real root.

2009 National Olympiad First Round, 4

Tags:

Let $ a,b$ be integers greater than $ 1$. What is the largest $ n$ which cannot be written in the form $ n \equal{} 7a \plus{} 5b$ ? $\textbf{(A)}\ 82 \qquad\textbf{(B)}\ 47 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 42 \qquad\textbf{(E)}\ \text{None}$

2007 Switzerland - Final Round, 4

Tags: geometry , perpendicular , orthocenter

Let $ABC$ be an acute-angled triangle with $AB> AC$ and orthocenter $H$. Let $D$ the projection of $A$ on $BC$. Let $E$ be the reflection of $C$ wrt $D$. The lines $AE$ and $BH$ intersect at point $S$. Let $N$ be the midpoint of $AE$ and let $M$ be the midpoint of $BH$. Prove that $MN$ is perpendicular to $DS$.

2020 Abels Math Contest (Norwegian MO) Final, 4b

Tags: geometry , circumcircle , angle bisector

The triangle $ABC$ has a right angle at $A$. The centre of the circumcircle is called $O$, and the base point of the normal from $O$ to $AC$ is called $D$. The point $E$ lies on $AO$ with $AE = AD$. The angle bisector of $\angle CAO$ meets $CE$ in $Q$. The lines $BE$ and $OQ$ intersect in $F$. Show that the lines $CF$ and $OE$ are parallel.

2020 Israel Olympic Revenge, G

Tags: angle bisector , apollonius circle , nine point center , geometry

Let $ABC$ be an acute triangle with $AB\neq AC$. The angle bisector of $\angle BAC$ intersects with $BC$ at a point $D$. $BE,CF$ are the altitudes of the triangle and $Ap_1,Ap_2$ are the isodynamic points of triangle $ABC$.Let the $A$-median of $ABC$ intersect $EF$ at $T$. Show that the line connecting $T$ with the nine-point center of $ABC$ is perpendicular to $BC$ if and only if $\angle Ap_1DAp_2=90^\circ$.

2003 Swedish Mathematical Competition, 3

Tags: floor function , algebra , equation

Find all real solutions $x$ of the equation $$\lfloor x^2-2 \rfloor +2 \lfloor x \rfloor = \lfloor x \rfloor ^2. $$ .

1997 Polish MO Finals, 1

Tags: number theory

The positive integers $x_1, x_2, ... , x_7$ satisfy $x_6 = 144$, $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n = 1, 2, 3, 4$. Find $x_7$.

2005 South africa National Olympiad, 4

Tags: geometry , analytic geometry , power of a point , barycentric coordinates

The inscribed circle of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. Let $Q$ denote the other point of intersection of $AD$ and the inscribed circle. Prove that $EQ$ extended passes through the midpoint of $AF$ if and only if $AC = BC$.

2020 Hong Kong TST, 2

Tags: combinatorial geometry , combinatorics

Suppose there are $2019$ distinct points in a plane and the distances between pairs of them attain $k$ different values. Prove that $k$ is at least $44$.

Champions Tournament Seniors - geometry, 2013.3

Tags: geometry , geometric inequality , 3d geometry , pyramid

On the base of the $ABC$ of the triangular pyramid $SABC$ mark the point $M$ and through it were drawn lines parallel to the edges $SA, SB$ and $SC$, which intersect the side faces at the points $A1_, B_1$ and $C_1$, respectively. Prove that $\sqrt{MA_1}+ \sqrt{MB_1}+ \sqrt{MC_1}\le \sqrt{SA+SB+SC}$

2017 Harvard-MIT Mathematics Tournament, 14

Tags: combinatorics

Mrs. Toad has a class of $2017$ students, with unhappiness levels $1, 2, \dots, 2017$ respectively. Today in class, there is a group project and Mrs. Toad wants to split the class in exactly $15$ groups. The unhappiness level of a group is the average unhappiness of its members, and the unhappiness of the class is the sum of the unhappiness of all $15$ groups. What's the minimum unhappiness of the class Mrs. Toad can achieve by splitting the class into $15$ groups?

2009 Jozsef Wildt International Math Competition, W. 17

Tags: function , inequalities

If $a$, $b$, $c>0$ and $abc=1$, $\alpha = max\{a,b,c\}$; $f,g : (0, +\infty )\to \mathbb{R}$, where $f(x)=\frac{2(x+1)^2}{x}$ and $g(x)= (x+1)\left (\frac{1}{\sqrt{x}}+1\right )^2$, then $$(a+1)(b+1)(c+1)\geq min\{ \{f(x),g(x) \}\ |\ x\in\{a,b,c\} \backslash \{ \alpha \}\} $$

2002 Putnam, 4

Tags: linear algebra , matrix , symmetry , putnam games

In Determinant Tic-Tac-Toe, Player $1$ enters a $1$ in an empty $3 \times 3$ matrix. Player $0$ counters with a $0$ in a vacant position and play continues in turn intil the $ 3 \times 3 $ matrix is completed with five $1$’s and four $0$’s. Player $0$ wins if the determinant is $0$ and player $1$ wins otherwise. Assuming both players pursue optimal strategies, who will win and how?

2016 NIMO Problems, 4

Tags: algebra , dice , probability , quadratic , number theory

A fair 100-sided die is rolled twice, giving the numbers $a$ and $b$ in that order. If the probability that $a^2-4b$ is a perfect square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $100m+n$. [i] Proposed by Justin Stevens [/i]

2014 Contests, 1

Tags: combinatorics

In Sikinia we only pay with coins that have a value of either $11$ or $12$ Kulotnik. In a burglary in one of Sikinia's banks, $11$ bandits cracked the safe and could get away with $5940$ Kulotnik. They tried to split up the money equally - so that everyone gets the same amount - but it just doesn't worked. After a while their leader claimed that it actually isn't possible. Prove that they didn't get any coin with the value $12$ Kulotnik.

2023 Harvard-MIT Mathematics Tournament, 3

Tags:

Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle BCD = \theta$ for some acute angle $\theta$. Point $X$ lies inside the quadrilateral such that $\angle XAD = \angle XDA = 90^{\circ}-\theta$. Prove that $BX = XC$.

2005 USAMTS Problems, 1

Tags: ratio , geometry

$\overline{AB}$ is a diameter of circle $C_1$. Point $P$ is on $C_1$ such that $AP>BP$. Circle $C_2$ is centered at $P$ with radius $PB$. The extension of $\overline{AP}$ past $P$ meets $C_2$ at $Q$. Circle $C_3$ is centered at $A$ and is externally tangent to $C_2$. Circle $C_4$ passes through $A$, $Q$, and $R$. Find, with proof, the ratio between the area of $C_4$ and the area of $C_1$, and show that this ratio is the same for all points $P$ on $C_1$ such that $AP>BP$.