Olympiad problems

2000 Romania Team Selection Test, 2

Tags: inequalities

Let $n\ge 1$ be a positive integer and $x_1,x_2\ldots ,x_n$ be real numbers such that $|x_{k+1}-x_k|\le 1$ for $k=1,2,\ldots ,n-1$. Prove that \[\sum_{k=1}^n|x_k|-\left|\sum_{k=1}^nx_k\right|\le\frac{n^2-1}{4}\] [i]Gh. Eckstein[/i]

2019 Purple Comet Problems, 26

Tags: 3d geometry , geometry

Let $D$ be a regular dodecahedron, which is a polyhedron with $20$ vertices, $30$ edges, and $12$ regular pentagon faces. A tetrahedron is a polyhedron with $4$ vertices, $6$ edges, and $4$ triangular faces. Find the number of tetrahedra with positive volume whose vertices are vertices of $D$. [img]https://cdn.artofproblemsolving.com/attachments/c/d/44d11fa3326780941d0b6756fb2e5989c2dc5a.png[/img]

1988 Greece National Olympiad, 4

Tags: algebra

Let $A\subseteq \mathbb{R}$ such that: i) If $a,b\in A$ then $\sqrt{ab} \in A$ ii) $1\in A$ and $2\in A$ Prove that $\sqrt[\displaystyle 2^{1453}]{2^{1821}}\in A$.

2008 Iran MO (2nd Round), 3

Tags: geometry , circumcircle , trigonometry

In triangle $ABC$, $H$ is the foot of perpendicular from $A$ to $BC$. $O$ is the circumcenter of $\Delta ABC$. $T,T'$ are the feet of perpendiculars from $H$ to $AB,AC$, respectively. We know that $AC=2OT$. Prove that $AB=2OT'$.

2021/2022 Tournament of Towns, P1

Tags: combinatorics , cards

The wizards $A, B, C, D$ know that the integers $1, 2, \ldots, 12$ are written on 12 cards, one integer on each card, and that each wizard will get three cards and will see only his own cards. Having received the cards, the wizards made several statements in the following order. [list=A] [*]“One of my cards contains the number 8”. [*]“All my numbers are prime”. [*]“All my numbers are composite and they all have a common prime divisor”. [*]“Now I know all the cards of each wizard”. [/list] What were the cards of $A{}$ if everyone was right? [i]Mikhail Evdokimov[/i]

2018 Tuymaada Olympiad, 6

Tags: invariant , parity , blackboard , combinatorics

The numbers $1, 2, 3, \dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after ten such operations. Determine all its possible values. [i]Proposed by A. Golovanov[/i]

1955 Miklós Schweitzer, 3

Tags: function

[b]3.[/b] Let the density function $f(x)$ of the random variable $\xi$ bean even function; let further $f(x)$ be monotonically non-increasing for $x>0$. Suppose that $D^{2}= \int_{-\infty }^{\infty }x^{2}f(x) dx$ exists. Prove that for every non negative $\lambda $ $P(\left |\xi \right |\geq \lambda D)\leq \frac{1}{1+\lambda ^{2}}$. [b](P. 7)[/b]

1990 AIME Problems, 11

Tags: factorial

Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.

2000 AMC 8, 2

Tags:

Which of these numbers is less than its reciprocal? $\textbf{(A)}\ -2\qquad \textbf{(B)}\ -1\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2$

2009 Iran Team Selection Test, 8

Tags: polynomial , algebra

Find all polynomials $ P(x,y)$ such that for all reals $ x$ and $y$, \[P(x^{2},y^{2}) =P\left(\frac {(x + y)^{2}}{2},\frac {(x - y)^{2}}{2}\right).\]

2005 France Team Selection Test, 5

Tags: trigonometry , geometry

Let $ABC$ be a triangle such that $BC=AC+\frac{1}{2}AB$. Let $P$ be a point of $AB$ such that $AP=3PB$. Show that $\widehat{PAC} = 2 \widehat{CPA}.$

2006 AMC 10, 16

Tags: modular arithmetic

Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur? $ \textbf{(A) } \text{Tuesday} \qquad \textbf{(B) } \text{Wednesday} \qquad \textbf{(C) } \text{Thursday} \qquad \textbf{(D) } \text{Friday} \qquad \textbf{(E) } \text{Saturday}$

1993 All-Russian Olympiad, 4

Tags: 3d geometry , prism , geometry

Prove that any two rectangular prisms with equal volumes can be placed in a space such that any horizontal plain that intersects one of the prisms will intersect the other forming a polygon with the same area.

2016 Peru IMO TST, 9

Tags: function , number theory , greatest common divisor

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. [i]Proposed by James Rickards, Canada[/i]

2002 Tournament Of Towns, 2

Tags: number theory , greatest common divisor , combinatorics

All the species of plants existing in Russia are catalogued (numbered by integers from $2$ to $2000$ ; one after another, without omissions or repetitions). For any pair of species the gcd of their catalogue numbers was calculated and recorded but the catalogue numbers themselves were lost. Is it possible to restore the catalogue numbers from the data in hand?

2012 AMC 10, 2

Tags: geometry , rectangle

A square with side length $8$ is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles? $ \textbf{(A)}\ 2\text{ by }4 \qquad\textbf{(B)}\ 2\text{ by }6 \qquad\textbf{(C)}\ 2\text{ by }8 \qquad\textbf{(D)}\ 4\text{ by }4 \qquad\textbf{(E)}\ 4\text{ by }8 $

Kvant 2024, M2804

Tags: geometry

There are two equal circles of radius $1$ placed inside the triangle $ABC$ with side $BC = 6$. The circles are tangent to each other, one is inscribed in angle $B$, the other one is inscribed in angle $C$. (a) Prove that the centroid $M$ of the triangle $ABC$ does not lie inside any of the given circles. (b) Prove that if $M$ lies on one of the circles, then the triangle $ABC$ is isosceles.

2011 Gheorghe Vranceanu, 3

Tags: number theory , other bases , binary representation

Prova that any integer $ Z $ has a unique representation $$ a_0+a_12+a_22^2+a_32^3+\cdots +a_n2^n, $$ where $ n $ is natural, $ a_i\in\{ -1,0,+1\} $ for $ i=\overline{0,n} $ and $ a_ka_{k-1}=0 $ for $ k=\overline{1,n} . $

2009 Oral Moscow Geometry Olympiad, 5

Tags: convex , geometry , polyhedron

Prove that any convex polyhedron has three edges that can be used to form a triangle. (Barbu Bercanu, Romania)

2017 German National Olympiad, 1

Tags: quadratic , system of equations , equation , quadratic equation , algebra

Given two real numbers $p$ and $q$, we study the following system of equations with variables $x,y \in \mathbb{R}$: \begin{align*} x^2+py+q&=0,\\ y^2+px+q&=0. \end{align*} Determine the number of distinct solutions $(x,y)$ in terms of $p$ and $q$.

2001 All-Russian Olympiad Regional Round, 9.7

Tags: geometry , tangent

A circle inscribed in an angle with vertex $O$ touches its sides at points $A$ and $B$, $K$ is an arbitrary point on the smaller of the two arcs $AB$ of this circle. On the line $OB$ a point $L$ is taken such that the lines $OA$ and $KL$ are parallel. Let $M$ be the intersection point of the circle $\omega$ circumscribed around triangle $KLB$, with line $AK$, with $M$ different from $K$. Prove that line $OM$ touches circle $\omega$.

Gheorghe Țițeica 2024, P2

Tags: number theory

Prove that the number $$\bigg\lfloor\frac{2024}{1}\bigg\rfloor+\bigg\lfloor\frac{2023}{2}\bigg\rfloor+\bigg\lfloor\frac{2022}{3}\bigg\rfloor+\dots +\bigg\lfloor\frac{1013}{1012}\bigg\rfloor$$ is even.

2019 District Olympiad, 1

Tags: diophantine equation , algebra , number theory

Determine the integers $a, b, c$ for which $$\frac{a+1}{3}=\frac{b+2}{4}=\frac{5}{c+3}$$

2005 National Olympiad First Round, 2

Tags: modular arithmetic

Let $a_1, a_2, \dots, a_n$ be positive integers such that none of them is a multiple of $5$. What is the largest integer $n<2005$, such that $a_1^4 + a_2^4 + \cdots + a_n^4$ is divisible by $5$? $ \textbf{(A)}\ 2000 \qquad\textbf{(B)}\ 2001 \qquad\textbf{(C)}\ 2002 \qquad\textbf{(D)}\ 2003 \qquad\textbf{(E)}\ 2004 $

2023 Indonesia TST, C

Tags: combinatorics , trivial

Let $A$ and $B$ be nonempty subsets of $\mathbb{N}$. The sum of $2$ distinct elements in $A$ is always an element of $B$. Furthermore, the result of the division of $2$ distinct elements in $B$ (where the larger number is divided by the smaller number) is always a member of $A$. Determine the maximum number of elements in $A \cup B$.