Olympiad problems

1980 Miklós Schweitzer, 8

Let $ f(x)$ be a nonnegative, integrable function on $ (0,2\pi)$ whose Fourier series is $ f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x)$, where none of the positive integers $ n_k$ divides another. Prove that $ |a_k| \leq a_0$. [i]G. Halasz[/i]

2007 Harvard-MIT Mathematics Tournament, 2

Tags: probability

A candy company makes $5$ colors of jellybeans, which come in equal proportions. If I grab a random sample of $5$ jellybeans, what is the probability that I get exactly $2$ distinct colors?

1982 AMC 12/AHSME, 15

Tags: function , floor function

Let $[z]$ denote the greatest integer not exceeding $z$. Let $x$ and $y$ satisfy the simultaneous equations \[ \begin{array}{c} y=2[x]+3, \\ y=3[x-2]+5. \end{array} \]If $x$ is not an integer, then $x+y$ is $\textbf {(A) } \text{an integer} \qquad \textbf {(B) } \text{between 4 and 5} \qquad \textbf {(C) } \text{between -4 and 4} \qquad \textbf {(D) } \text{between 15 and 16} \qquad \textbf {(E) } 16.5$

2002 Tournament Of Towns, 2

Tags: 3d geometry , geometry , inequalities

A cube is cut by a plane such that the cross section is a pentagon. Show there is a side of the pentagon of length $\ell$ such that the inequality holds: \[ |\ell-1|>\frac{1}{5} \]

2024 Iranian Geometry Olympiad, 4

Tags: geometry

Eric has assembled a convex polygon $P$ from finitely many centrally symmetric (not necessarily congruent or convex) polygonal tiles. Prove that $P$ is centrally symmetric. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

2010 Contests, 2

Tags: combinatorics

Exactly $4n$ numbers in set $A= \{ 1,2,3,...,6n \} $ of natural numbers painted in red, all other in blue. Proved that exist $3n$ consecutive natural numbers from $A$, exactly $2n$ of which numbers is red.

1966 IMO Shortlist, 22

Tags: geometry , parallelogram , vector

Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$ Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $% P^{\prime }.$

2023 Assam Mathematics Olympiad, 9

Tags:

What is the smallest positive integer having $24$ positive divisors?

2012 NIMO Problems, 8

Tags: geometry , geometric transformation , reflection , integration , trigonometry , probability , expected value

Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$. [i]Proposed by Lewis Chen[/i]

2015 Miklos Schweitzer, 10

Tags: functional analysis , spectral analysis

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable,strictly convex function.Let $H$ be a Hilbert space and $A,B$ be bounded,self adjoint linear operators on $H$.Prove that,if $f(A)-f(B)=f'(B)(A-B)$ then $A=B$.

2003 National High School Mathematics League, 5

Tags:

If $x,y\in(-2,2),xy=-1$, then the minumum value of $u=\frac{4}{4-x^2}+\frac{9}{9-y^2}$ is $\text{(A)}\frac{8}{5}\qquad\text{(B)}\frac{24}{11}\qquad\text{(C)}\frac{12}{7}\qquad\text{(D)}\frac{12}{5}\qquad$

2019 Regional Olympiad of Mexico Center Zone, 5

Tags: number theory

A serie of positive integers $a_{1}$,$a_{2}$,. . . ,$a_{n}$ is $auto-delimited$ if for every index $i$ that holds $1\leq i\leq n$, there exist at least $a_{i}$ terms of the serie such that they are all less or equal to $i$. Find the maximum value of the sum $a_{1}+a_{2}+\cdot \cdot \cdot+a_{n}$, where $a_{1}$,$a_{2}$,. . . ,$a_{n}$ is an $auto-delimited$ serie.

1997 India Regional Mathematical Olympiad, 6

Tags: search

Find the number of unordered pairs $\{ A,B \}$ of subsets of an n-element set $X$ that satisfies the following: (a) $A \not= B$ (b) $A \cup B = X$

1992 Dutch Mathematical Olympiad, 2

Tags:

In the fraction below and its decimal notation (with period of length $ 4$) every letter represents a digit, and different letters denote different digits. The numerator and denominator are coprime. Determine the value of the fraction: $ \frac{ADA}{KOK}\equal{}0.SNELSNELSNELSNEL...$ $ Note.$ Ada Kok is a famous dutch swimmer, and "snel" is Dutch for "fast".

2017 Czech-Polish-Slovak Junior Match, 4

Tags: trapezoid , construction , geometry

Bolek draw a trapezoid $ABCD$ trapezoid ($AB // CD$) on the board, with its midsegment line $EF$ in it. Point intersection of his diagonal $AC, BD$ denote by $P,$ and his rectangular projection on line $AB$ denote by $Q$. Lolek, wanting to tease Bolek, blotted from the board everything except segments $EF$ and $PQ$. When Bolek saw it, wanted to complete the drawing and draw the original trapezoid, but did not know how to do it. Can you help Bolek?

2018-IMOC, N6

Tags: algebra , polynomial , number theory

If $f$ is a polynomial sends $\mathbb Z$ to $\mathbb Z$ and for $n\in\mathbb N_{\ge2}$, there exists $x\in\mathbb Z$ so that $n\nmid f(x)$, show that for every $k\in\mathbb Z$, there is a non-negative integer $t$ and $a_1,\ldots,a_t\in\{-1,1\}$ such that $$a_1f(1)+\ldots+a_tf(t)=k.$$

2007 IMS, 3

Tags: search , real analysis

Prove that $\mathbb R^{2}$ has a dense subset such that has no three collinear points.

1997 Czech and Slovak Match, 3

Tags: functional equation , algebra

Find all functions $f : R\rightarrow R$ such that $f ( f (x)+y) = f (x^2 -y)+4 f (x)y$ for all $x,y \in R$ .

2022 Stanford Mathematics Tournament, 8

Tags:

Given that $20^{22}+1$ has exactly $4$ prime divisors $p_1<p_2<p_3<p_4$, determine $p_1+p_2$.

2021/2022 Tournament of Towns, P4

Tags: geometry

Consider a white 100×100 square. Several cells (not necessarily neighbouring) were painted black. In each row or column that contains some black cells their number is odd. Hence we may consider the middle black cell for this row or column (with equal numbers of black cells in both opposite directions). It so happened that all the middle black cells of such rows lie in different columns and all the middle black cells of the columns lie in different rows. a) Prove that there exists a cell that is both the middle black cell of its row and the middle black cell of its column. b) Is it true that any middle black cell of a row is also a middle black cell of its column?

2018 Moscow Mathematical Olympiad, 11

Tags: geometry

Ivan want to paint ball. Ivan can put ball in the glass with some paint, and then one half of ball will be painted. Ivan use $5$ glasses to paint glass competely. Prove, that one glass was not needed, and Ivan can paint ball with $4$ glasses, putting ball in it by same way.

2014 Contests, 1

Tags: function , floor function , algebra

Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\] in which $a,~ b,~ c$, and $d$ vary over the set of positive integers. (Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)

2013 IMAC Arhimede, 2

Tags: number theory , perfect cube , divisible , exponential

For all positive integer $n$, we consider the number $$a_n =4^{6^n}+1943$$ Prove that $a_n$ is dividible by $2013$ for all $n\ge 1$, and find all values of $n$ for which $a_n - 207$ is the cube of a positive integer.

2007 Croatia Team Selection Test, 5

Tags: symmetry , ratio , geometry

Let there be two circles. Find all points $M$ such that there exist two points, one on each circle such that $M$ is their midpoint.

2018 Belarusian National Olympiad, 10.8

Tags: combinatorial game , combinatorics

The vertices of the regular $n$-gon and its center are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center. The winner I a player if after his maveit is possible to get any marked point from any other moving along the segments. For each $n>2$ determine who has a winning strategy.