Olympiad problems

2022 Miklós Schweitzer, 9

Tags:

Plane vectors form a group for addition. Show that this group has a generator system of every set $S$ that contains a Borel subset of positive linear measure of a circular arc.

2009 Canada National Olympiad, 2

Tags: rotation , geometry

Two circles of different radii are cut out of cardboard. Each circle is subdivided into $200$ equal sectors. On each circle $100$ sectors are painted white and the other $100$ are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Show that one can rotate the small circle so that the sectors on the two circles line up and at least $100$ sectors on the small circle lie over sectors of the same color on the big circle.

2019 Tuymaada Olympiad, 3

Tags: combinatorics , minimum value , minimum , chessboard

The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to rooms adjacent by side. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess rook (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 1$)?

2019 Centers of Excellency of Suceava, 2

Tags: limit , convergence , real analyssi , real analysis , sequence

Let be two real numbers $ b>a>0, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ with $ x_2>x_1>0 $ and such that $$ ax_{n+2}+bx_n\ge (a+b)x_{n+1} , $$ for any natural numbers $ n. $ Prove that $ \lim_{n\to\infty } x_n=\infty . $ [i]Dan Popescu[/i]

2016 Bulgaria National Olympiad, Problem 4

Tags: combinatorics

Determine whether there exist a positive integer $n<10^9$, such that $n$ can be expressed as a sum of three squares of positive integers by more than $1000$ distinct ways?

2022 Swedish Mathematical Competition, 1

Tags: combinatorics , tiling , tiles , combinatorial geometry

What sizes of squares with integer sides can be completely covered without overlap by identical tiles consisting of three squares with side $1$ joined together in one $L$ shape? [center][img]https://cdn.artofproblemsolving.com/attachments/3/f/9fe95b05527857f7e44dfd033e6fb01e5d25a2.png[/img][/center]

2002 Tournament Of Towns, 1

Tags: trigonometry , calculus , integration , geometry

In a triangle $ABC$ it is given $\tan A,\tan B,\tan C$ are integers. Find their values.

2003 China Team Selection Test, 3

Tags: analytic geometry , inequalities , graph theory , number theory

Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.

1999 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , integration , trigonometry , function

Find \[\int_{-4\pi\sqrt{2}}^{4\pi\sqrt{2}}\left(\dfrac{\sin x}{1+x^4}+1\right)dx.\]

2008 Indonesia TST, 2

Tags: number theory , factorial , positive integer

Find all positive integers $1 \le n \le 2008$ so that there exist a prime number $p \ge n$ such that $$\frac{2008^p + (n -1)!}{n}$$ is a positive integer.

2019 AMC 12/AHSME, 21

Tags: complex number

Let $$z=\frac{1+i}{\sqrt{2}}.$$ What is $$(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) \cdot (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})?$$ $\textbf{(A) } 18 \qquad \textbf{(B) } 72-36\sqrt2 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 72+36\sqrt2$

1990 AMC 12/AHSME, 1

Tags:

If $\dfrac{x/4}{2}=\dfrac{4}{x/2}$ then $x=$ $\textbf{(A) }\pm 1/2\qquad \textbf{(B) }\pm 1\qquad \textbf{(C) }\pm 2\qquad \textbf{(D) }\pm 4\qquad \textbf{(E) }\pm 8$

1991 Hungary-Israel Binational, 4

Tags: quadratic , system of equations , algebra

Find all the real values of $ \lambda$ for which the system of equations $ x\plus{}y\plus{}z\plus{}v\equal{}0$ and $ \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0$, has a unique real solution.

Estonia Open Senior - geometry, 2018.1.1

Tags: geometry , lattice points , equilateral

Is there an equilateral triangle in the coordinate plane, both coordinates of each vertex of which are integers?

2001 Mexico National Olympiad, 3

Tags: geometry , cyclic quadrilateral , equal segments

$ABCD$ is a cyclic quadrilateral. $M$ is the midpoint of $CD$. The diagonals meet at $P$. The circle through $P$ which touches $CD$ at $M$ meets $AC$ again at $R$ and $BD$ again at $Q$. The point $S$ on $BD$ is such that $BS = DQ$. The line through $S$ parallel to $AB$ meets $AC$ at $T$. Show that $AT = RC$.

2019 Durer Math Competition Finals, 1

Tags: algebra , sequence , sum , inequalities

Let $a_o,a_1,a_2,..,a_ n$ be a non-decreasing sequence of $n+1$ real numbers where $a_0 = 0$ and for every $j > i $ we have $a_j - a_i \le j - i$. Show that $$\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3$$

1967 IMO Shortlist, 6

Tags: inequality , polynomial , algebra , n-variable inequality

Prove the following inequality: \[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1} x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in \mathbb{N}.$

2020 Turkey Junior National Olympiad, 1

Tags: quadratic equation , number theory

Determine all real number $(x,y)$ pairs that satisfy the equation. $$2x^2+y^2+7=2(x+1)(y+1)$$

1985 IMO Longlists, 46

Tags: analytic geometry , geometry

Let $C$ be the curve determined by the equation $y = x^3$ in the rectangular coordinate system. Let $t$ be the tangent to $C$ at a point $P$ of $C$; t intersects $C$ at another point $Q$. Find the equation of the set $L$ of the midpoints $M$ of $PQ$ as $P$ describes $C$. Is the correspondence associating $P$ and $M$ a bijection of $C$ on $L$ ? Find a similarity that transforms $C$ into $L.$

2007 Kyiv Mathematical Festival, 4

Tags: geometric transformation , reflection , trapezoid , geometry

The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$

2020 China Northern MO, BP5

Tags: number theory , relatively prime , combinatorics , set

It is known that subsets $A_1,A_2, \cdots , A_n$ of set $I=\{1,2,\cdots ,101\}$ satisfy the following condition $$\text{For any } i,j \text{ } (1 \leq i < j \leq n) \text{, there exists } a,b \in A_i \cap A_j \text{ so that } (a,b)=1$$ Determine the maximum positive integer $n$. *$(a,b)$ means $\gcd (a,b)$

2022 JHMT HS, 4

Tags: combinatorics

For a nonempty set $A$ of integers, let $\mathrm{range} \, A=\max A-\min A$. Find the number of subsets $S$ of \[ \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \] such that $\mathrm{range} \, S$ is an element of $S$.

2022 Bangladesh Mathematical Olympiad, 8

Tags: number theory

Solve the following problems - A) Find any $158$ consecutive integers such that the sum of digits for any of the numbers is not divisible by $17.$ B) Prove that, among any $159$ consecutive integers there will always be at least one integer whose sum of digits is divisible by $17.$

2006 Pre-Preparation Course Examination, 4

Tags: abstract algebra , algebra

If $d\in \mathbb{Q}$, is there always an $\omega \in \mathbb{C}$ such that $\omega ^n=1$ for some $n\in \mathbb{N}$ and $\mathbb{Q}(\sqrt{d})\subseteq \mathbb{Q}(\omega)$?

2012 Portugal MO, 1

Tags: floor function , number theory

Find the number of positive integers $n$ such that $1\leq n\leq 1000$ and $n$ is divisible by $\lfloor \sqrt[3]{n} \rfloor$.