Olympiad problems

2002 Kazakhstan National Olympiad, 4

Prove that there is a set $ A $ consisting of $2002$ different natural numbers satisfying the condition: for each $ a \in A $, the product of all numbers from $ A $, except $ a $, when divided by $ a $ gives the remainder $1$.

V Soros Olympiad 1998 - 99 (Russia), 11.6

Tags: rectangle , geometry

Cut the $10$ cm $x 20$ cm rectangle into two pieces with one straight cut so that they can be placed inside the $19.4$ cm diameter circle without intersecting.

2014 NIMO Problems, 2

Tags: symmetry , number theory , perimeter , relatively prime , probability , analytic geometry , geometry

Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Rajiv Movva[/i]

2008 iTest Tournament of Champions, 2

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Find the value of $|xy|$ given that $x$ and $y$ are integers and \[6x^2y^2+5x^2-18y^2=17253.\]

2017 IFYM, Sozopol, 4

Tags: number theory

Find all pairs of natural numbers $(a,n)$, $a\geq n \geq 2,$ for which $a^n+a-2$ is a power of $2$.

1972 Bundeswettbewerb Mathematik, 4

Tags: combinatorics

$p>2$ persons participate at a chess tournament, two players play at most one game against each other. After $n$ games were made, no more game is running and in every subset of three players, we can find at least two that havem't played against each other. Show that $n \leq \frac{p^{2}}4$.

2018 Harvard-MIT Mathematics Tournament, 2

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Compute the positive real number $x$ satisfying $$x^{(2x^6)}=3.$$

2018 Sharygin Geometry Olympiad, 8

Tags: incenter , perpendicular bisector , geometry

Let $I$ be the incenter of fixed triangle $ABC$, and $D$ be an arbitrary point on $BC$. The perpendicular bisector of $AD$ meets $BI,CI$ at $F$ and $E$ respectively. Find the locus of orthocenters of $\triangle IEF$ as $D$ varies.

1990 National High School Mathematics League, 3

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There are $n$ schools in a city. $i$th school dispatches $C_i(1\leq C_i\leq39,1\leq i\leq n)$ students to watch a football match. The number of all students $\sum_{i=1}^{n}C_{i}=1990$. In each line, there are $199$ seats, but students from the same school must sit in the same line. So, how many lines of seats we need to have to make sure all students have a seat.

1999 Belarusian National Olympiad, 7

Tags: angle bisector , exterior angle , geometry

Let [i]O[/i] be the center of circle[i] W[/i]. Two equal chords [i]AB[/i] and [i]CD [/i]of[i] W [/i]intersect at [i]L [/i]such that [i]AL>LB [/i]and [i]DL>LC[/i]. Let [i]M [/i]and[i] N [/i]be points on [i]AL[/i] and [i]DL[/i] respectively such that ([i]ALC[/i])=2*([i]MON[/i]). Prove that the chord of [i]W[/i] passing through [i]M [/i]and [i]N[/i] is equal to [i]AB[/i] and [i]CD[/i].

1978 Swedish Mathematical Competition, 2

Tags: algebra , number theory , sum , digit

Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find \[ s_1 + s_2 + \cdots + s_n \]

2012 Princeton University Math Competition, A6

Tags: algebra

Let an be a sequence such that $a_0 = 0$ and: $a_{3n+1} = a_{3n} + 1 = a_n + 1$ $a_{3n+2} = a_{3n} + 2 = a_n + 2$ for all natural numbers $n$. How many $n$ less than $2012$ have the property that $a_n = 7$?

2022 Germany Team Selection Test, 3

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Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

2006 Baltic Way, 7

Tags: combinatorics

A photographer took some pictures at a party with $10$ people. Each of the $45$ possible pairs of people appears together on exactly one photo, and each photo depicts two or three people. What is the smallest possible number of photos taken?

MathLinks Contest 6th, 3.3

Tags: combinatorics , geometry

We say that a set of points $M$ in the plane is convex if for any two points $A, B \in M$, all the points from the segment $(AB)$ also belong to $M$. Let $n \ge 2$ be an integer and let $F$ be a family of $n$ disjoint convex sets in the plane. Prove that there exists a line $\ell$ in the plane, a set $M \in F$ and a subset $S \subset F$ with at least $\lceil \frac{n}{12} \rceil $ elements such that $M$ is contained in one closed half-plane determined by $\ell$, and all the sets $N \in S$ are contained in the complementary closed half-plane determined by $\ell$.

2012 Stars of Mathematics, 2

Tags: number theory

Prove the value of the expression $$\displaystyle \dfrac {\sqrt{n + \sqrt{0}} + \sqrt{n + \sqrt{1}} + \sqrt{n + \sqrt{2}} + \cdots + \sqrt{n + \sqrt{n^2-1}} + \sqrt{n + \sqrt{n^2}}} {\sqrt{n - \sqrt{0}} + \sqrt{n - \sqrt{1}} + \sqrt{n - \sqrt{2}} + \cdots + \sqrt{n - \sqrt{n^2-1}} + \sqrt{n - \sqrt{n^2}}}$$ is constant over all positive integers $n$. ([i]Folklore (also Philippines Olympiad)[/i])

2013 Stanford Mathematics Tournament, 1

Tags: incenter , circumcircle , geometry

A circle of radius $2$ is inscribed in equilateral triangle $ABC$. The altitude from $A$ to $BC$ intersects the circle at a point $D$ not on $BC$. $BD$ intersects the circle at a point $E$ distinct from $D$. Find the length of $BE$.

2016 Taiwan TST Round 2, 2

Tags: divisibility , number theory

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

2005 Thailand Mathematical Olympiad, 16

Tags: sum , algebra

Compute the sum of roots of $(2 - x)^{2005} + x^{2005} = 0$.

1972 Poland - Second Round, 5

Tags: concurrency , cyclic quadrilateral , geometry

Prove that in a convex quadrilateral inscribed in a circle, straight lines passing through the midpoints of the sides and perpendicular to the opposite sides intersect at one point.

2021 Moldova Team Selection Test, 5

Tags: algebra , function , geometry

Let $ABC$ be an equilateral triangle. Find all positive integers $n$, for which the function $f$, defined on all points $M$ from the circle $S$ circumscribed to triangle $ABC$, defined by the formula $f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n$, is a constant function.

2013 BMT Spring, 2

Tags: algebra

Find the sum of all positive integers $N$ such that $s =\sqrt[3]{2 + \sqrt{N}} + \sqrt[3]{2 - \sqrt{N}}$ is also a positive integer

1990 AMC 8, 5

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Which of the following is closest to the product $ (.48017)(.48017)(.48017) $? $ \text{(A)}\ 0.011\qquad\text{(B)}\ 0.110\qquad\text{(C)}\ 1.10\qquad\text{(D)}\ 11.0\qquad\text{(E)}\ 110 $

2000 Bundeswettbewerb Mathematik, 4

Tags: algebra , 3d geometry , geometry

Consider the sums of the form $\sum_{k=1}^{n} \epsilon_k k^3,$ where $\epsilon_k \in \{-1, 1\}.$ Is any of these sums equal to $0$ if [b](a)[/b] $n=2000;$ [b](b)[/b] $n=2001 \ ?$

2012 May Olympiad, 4

Tags: game , combinatorics

Pedro has $111$ blue chips and $88$ white chips. There is a machine that for every $14$ blue chips , it gives $11$ white pieces and for every $7$ white chips, it gives $13$ blue pieces. Decide if Pedro can achieve, through successive operations with the machine, increase the total number of chips by $33$, so that the number of blue chips equals $\frac53$ of the amount of white chips. If possible, indicate how to do it. If not, indicate why.