Olympiad problems

2021 All-Russian Olympiad, 6

Tags: algebra , polynomial

Given is a polynomial $P(x)$ of degree $n>1$ with real coefficients. The equation $P(P(P(x)))=P(x)$ has $n^3$ distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.

2003 Iran MO (3rd Round), 26

Tags: vector , trigonometry , trapezoid , absolute value , trig identities , law of cosines , geometry

Circles $ C_1,C_2$ intersect at $ P$. A line $ \Delta$ is drawn arbitrarily from $ P$ and intersects with $ C_1,C_2$ at $ B,C$. What is locus of $ A$ such that the median of $ AM$ of triangle $ ABC$ has fixed length $ k$.

2016 CCA Math Bonanza, L1.4

Tags:

A triangle has a perimeter of $4$ [i]yards[/i] and an area of $6$ square [i]feet[/i]. If one of the angles of the triangle is right, what is the length of the largest side of the triangle, in feet? [i]2016 CCA Math Bonanza Lightning #1.4[/i]

2019 Novosibirsk Oral Olympiad in Geometry, 4

Tags: square , geometry , isosceles , collinear

Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]

2006 Hanoi Open Mathematics Competitions, 9

Tags: minimum , inequalities , algebra

What is the smallest possible value of $x^2 + y^2 - x -y - xy$?

2016 China Northern MO, 5

Tags: algebra

Let $\theta_{i}\in(0,\frac{\pi}{2})(i=1,2,\cdots,n)$. Prove: $$(\sum_{i=1}^n\tan\theta_{i})(\sum_{i=1}^n\cot\theta_{i})\geq(\sum_{i=1}^n\sin\theta_{i})^2+(\sum_{i=1}^n\cos\theta_{i})^2.$$

1978 Miklós Schweitzer, 4

Tags: function , limit , real analysis

Let $ \mathbb{Q}$ and $ \mathbb{R}$ be the set of rational numbers and the set of real numbers, respectively, and let $ f : \mathbb{Q} \rightarrow \mathbb{R}$ be a function with the following property. For every $ h \in \mathbb{Q} , \;x_0 \in \mathbb{R}$, \[ f(x\plus{}h)\minus{}f(x) \rightarrow 0\] as $ x \in \mathbb{Q}$ tends to $ x_0$. Does it follow that $ f$ is bounded on some interval? [i]M. Laczkovich[/i]

2020 USMCA, 3

Tags:

For a word $w$ consisting of $n$ lowercase letters, an [i]edit[/i] is specified by a pair $(i,c)$ where $i\in \{1,\ldots, n\}$ and $c$ is a lowercase letter, and transforms $w$ by replacing its $i^{\text{th}}$ letter with $c$. It is possible that $c$ is the same as the letter it replaced. How many sequences of six edits transform $\verb|frog|$ into $\verb|goat|$? Note that on the word $\verb|abcd|$, the edits $(1,\verb|a|)$ and $(2,\verb|b|)$ are considered distinct, even though both result in the word $\verb|abcd|$.

MathLinks Contest 5th, 6.2

Tags: number theory

We say that a positive integer $n$ is nice if $\frac{4}{n}$ cannot be written as $\frac{1}{x}+\frac{1}{xy}+\frac{1}{z}$ for any positive integers $x, y, z$. Let us denote by $ a_n$ the number of nice numbers smaller than $n$. Prove that the sequence $\frac{n}{a_n}$ is not bounded.

1955 Moscow Mathematical Olympiad, 309

Tags: combinatorial geometry , combinatorics , number theory , convex polygon , enumeration

A point $O$ inside a convex $n$-gon $A_1A_2 . . .A_n$ is connected with segments to its vertices. The sides of this $n$-gon are numbered $1$ to $n$ (distinct sides have distinct numbers). The segments $OA_1,OA_2, . . . ,OA_n$ are similarly numbered. a) For $n = 9$ find a numeration such that the sum of the sides’ numbers is the same for all triangles $A_1OA_2, A_2OA_3, . . . , A_nOA_1$. b) Prove that for $n = 10$ there is no such numeration.

2005 AMC 12/AHSME, 22

Tags: geometry , 3d geometry , sphere

A rectangular box $ P$ is inscribed in a sphere of radius $ r$. The surface area of $ P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $ r$? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

2024/2025 TOURNAMENT OF TOWNS, P4

Tags: number theory , geometry

Given $2N$ real numbers. It is known that if they are arbitrarily divided into two groups of $N$ numbers each then the products of the numbers of each group differ by $2$ at most. Is it necessarily true that if we arbitrarily place these numbers along a circle then there are two neighboring numbers that differ by $2$ at most, for a) $N=50$; (3 marks) b) $N=25$? (5 marks)

2009 China Team Selection Test, 2

Tags: induction , ratio , inequalities

Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$

2011 Cuba MO, 7

Tags: number theory , lcm , least common multiple

Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than $2011$.

2014 Dutch BxMO/EGMO TST, 3

Tags: geometry

In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

2010 Turkey MO (2nd round), 3

Tags: inequalities , analytic geometry , combinatorics

Let $K$ be the set of all sides and diagonals of a convex $2010-gon$ in the plane. For a subset $A$ of $K,$ if every pair of line segments belonging to $A$ intersect, then we call $A$ as an [i]intersecting set.[/i] Find the maximum possible number of elements of union of two [i]intersecting sets.[/i]

2011 Purple Comet Problems, 2

Tags: geometry , perimeter

The diagram below shows a $12$-sided figure made up of three congruent squares. The figure has total perimeter $60$. Find its area. [asy] size(150); defaultpen(linewidth(0.8)); path square=unitsquare; draw(rotate(360-135)*square^^rotate(345)*square^^rotate(105)*square); [/asy]

Cono Sur Shortlist - geometry, 2009.G3

Tags: geometry

We have a convex polygon $P$ in the plane and two points $S,T$ in the boundary of $P$, dividing the perimeter in a proportion $1:2$. Three distinct points in the boundary, denoted by $A,B,C$ start to move simultaneously along the boundary, in the same direction and with the same speed. Prove that there will be a moment in which one of the segments $AB, BC, CA$ will have a length smaller or equal than $ST$.

2016 BMT Spring, 9

Tags: combinatorics

On $5 \times 5$ grid of lattice points, every point is uniformly randomly colored blue, red, or green. Find the expected number of monochromatic triangles T with vertices chosen from the lattice grid, such that some two sides of $T$ are parallel to the axis.

2011 ELMO Shortlist, 5

Tags: geometry , inequalities

Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that \[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$. [i]Victor Wang.[/i]

1999 National High School Mathematics League, 1

Tags: geometry

In convex quadrilateral $ABCD$, $\angle BAC=\angle CAD$. $E$ lies on segment $CD$, $BE$ and $AC$ intersect at $F,$ $DF$ and $BC$ intersect at $G.$ Prove that $\angle GAC=\angle EAC$.

PEN M Problems, 34

Tags: modular arithmetic , recursive sequences

The sequence of integers $\{ x_{n}\}_{n\ge1}$ is defined as follows: \[x_{1}=1, \;\; x_{n+1}=1+{x_{1}}^{2}+\cdots+{x_{n}}^{2}\;(n=1,2,3 \cdots).\] Prove that there are no squares of natural numbers in this sequence except $x_{1}$.

2018 Costa Rica - Final Round, LRP3

Tags: combinatorics , game , game strategy

Jordan is in the center of a circle whose radius is $100$ meters and can move one meter at a time, however, there is a giant who at every step can force you to move in the opposite direction to the one he chose (it does not mean returning to the place of departure, but advance but in the opposite direction to the chosen one). Determine the minimum number of steps that Jordan must give to get out of the circle.

2004 Miklós Schweitzer, 10

Tags: probability , normal distribution

Let $\mathcal{N}_p$ stand for a $p$ dimensional random variable of standard normal distribution. For $a\in\mathbb{R}^p$, let $H_p(a)$ stand for the expectation $E|\mathcal{N}_p+a|$. For $p>1$, prove that $$H_p(a)=(p-1)\int_0^{\infty} H_1\left( \frac{|a|}{\sqrt{r^2+1}}\right) \frac{r^{p-2}}{\sqrt{(r^2+1)^p}} \mathrm{d}r$$

2015 Switzerland - Final Round, 3

Tags: function , algebra , functional equation

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, such that for arbitrary $x,y \in \mathbb{R}$: \[ (y+1)f(x)+f(xf(y)+f(x+y))=y.\]