Olympiad problems

2016 CMIMC, 2

Tags: algebra

Determine the value of the sum \[\left|\sum_{1\leq i<j\leq 50}ij(-1)^{i+j}\right|.\]

2014 China Team Selection Test, 2

Tags: combinatorics

Let $A_1A_2...A_{101}$ be a regular $101$-gon, and colour every vertex red or blue. Let $N$ be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the $101$-gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour. $(1)$ Find the largest possible value of $N$. $(2)$ Find the number of ways to colour the vertices such that maximum $N$ is acheived. (Two colourings a different if for some $A_i$ the colours are different on the two colouring schemes).

2025 India STEMS Category B, 1

Tags: algebra , polynomial

Let $\mathcal{P}$ be the set of all polynomials with coefficients in $\{0, 1\}$. Suppose $a, b$ are non-zero integers such that for every $f \in \mathcal{P}$ with $f(a)\neq 0$, we have $f(a) \mid f(b)$. Prove that $a=b$. [i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]

2010 Contests, 4

Tags: inequalities

Given $n$ positive real numbers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n \ge 0$ and $x_1^2+x_2^2+\cdots+x_n^2=1$, prove that \[\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.\]

2006 Moldova Team Selection Test, 1

Tags: number theory

Let $(a_n)$ be the Lucas sequence: $a_0=2,a_1=1, a_{n+1}=a_n+a_{n-1}$ for $n\geq 1$. Show that $a_{59}$ divides $(a_{30})^{59}-1$.

2015 India IMO Training Camp, 1

Tags: reflection , incenter , geometry

In a triangle $ABC$, a point $D$ is on the segment $BC$, Let $X$ and $Y$ be the incentres of triangles $ACD$ and $ABD$ respectively. The lines $BY$ and $CX$ intersect the circumcircle of triangle $AXY$ at $P\ne Y$ and $Q\ne X$, respectively. Let $K$ be the point of intersection of lines $PX$ and $QY$. Suppose $K$ is also the reflection of $I$ in $BC$ where $I$ is the incentre of triangle $ABC$. Prove that $\angle BAC=\angle ADC=90^{\circ}$.

1994 Argentina National Olympiad, 2

Tags: number theory , perfect square , algebra

For what positive integer values of $x$ is $x^4 + 6x^3 + 11x^2 + 3x + 31$ a perfect square?

2019 SEEMOUS, 1

Tags: real analysis

A sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is called "Devin" if for any $f\in C[0,1]$ $$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(x_i)=\int_0^1 f(x)\,dx $$ Prove that a sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is "Devin" if and only if for any non-negative integer $k$ it holds $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i^k=\frac{1}{k+1}.$$ [b]Remark[/b]. I left intact the text as it was proposed. Devin is a Bulgarian city and SPA resort, where this competition took place.

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]