Olympiad problems

1974 Miklós Schweitzer, 3

Prove that a necessary and sufficient for the existence of a set $ S \subset \{1,2,...,n \}$ with the property that the integers $ 0,1,...,n\minus{}1$ all have an odd number of representations in the form $ x\minus{}y, x,y \in S$, is that $ (2n\minus{}1)$ has a multiple of the form $ 2.4^k\minus{}1$ [i]L. Lovasz, J. Pelikan[/i]

2013 AMC 10, 25

Tags: search , modular arithmetic

Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N=749$, Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum $S=13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$? ${ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D}}\ 20\qquad\textbf{(E)}\ 25 $

1992 Baltic Way, 15

Tags: search , combinatorics

Noah has 8 species of animals to fit into 4 cages of the ark. He plans to put species in each cage. It turns out that, for each species, there are at most 3 other species with which it cannot share the accomodation. Prove that there is a way to assign the animals to their cages so that each species shares with compatible species.

1990 India National Olympiad, 5

Tags: triangle inequality , limit , algebra , search , inequalities

Let $ a$, $ b$, $ c$ denote the sides of a triangle. Show that the quantity \[ \frac{a}{b\plus{}c}\plus{}\frac{b}{c\plus{}a}\plus{}\frac{c}{a\plus{}b}\] must lie between the limits $ 3/2$ and 2. Can equality hold at either limits?

2009 Indonesia TST, 1

Tags: search , number theory , pigeonhole principle

a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

1998 Taiwan National Olympiad, 2

Tags: search , function , number theory

Does there exist a solution $(x,y,z,u,v)$ in integers greater than $1998$ to the equation $x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65$?

1988 AMC 8, 23

Tags: search

Maria buys computer disks at a price of 4 for 5 dollars and sells them at a price of 3 for 5 dollars. How many computer disks must she sell in order to make a profit of 100 dolars? $ \text{(A)}\ 100\qquad\text{(B)}\ 120\qquad\text{(C)}\ 200\qquad\text{(D)}\ 240\qquad\text{(E)}\ 1200 $

2008 IMC, 2

Tags: geometry , trigonometry , search , ellipse , conic , analytic geometry , ratio

Two different ellipses are given. One focus of the first ellipse coincides with one focus of the second ellipse. Prove that the ellipses have at most two points in common.

2007 Baltic Way, 20

Tags: search , function , number theory , geometry

Let $a$ and $b$ be positive integers, $b<a$, such that $a^3+b^3+ab$ is divisible by $ab(a-b)$. Prove that $ab$ is a perfect cube.

2002 AMC 12/AHSME, 13

Tags: prime factorization , search , modular arithmetic , number theory

What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1,k_2,\ldots,k_n$ for which \[k_1^2+k_2^2+\ldots+k_n^2=2002?\] $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

2012 Indonesia TST, 4

Tags: quadratic , search , pigeonhole principle , number theory , induction , modular arithmetic , floor function

Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$. Remark: "Natural numbers" is the set of positive integers.

1972 Miklós Schweitzer, 4

Tags: abstract algebra , superior algebra , search , group theory

Let $ G$ be a solvable torsion group in which every Abelian subgroup is finitely generated. Prove that $ G$ is finite. [i]J. Pelikan[/i]

1983 Miklós Schweitzer, 2

Tags: superior algebra , search

Let $ I$ be an ideal of the ring $ R$ and $ f$ a nonidentity permutation of the set $ \{ 1,2,\ldots, k \}$ for some $ k$. Suppose that for every $ 0 \not\equal{} a \in R, \;aI \not\equal{} 0$ and $ Ia \not\equal{}0$ hold; furthermore, for any elements $ x_1,x_2,\ldots ,x_k \in I$, \[ x_1x_2\ldots x_k\equal{}x_{1f}x_{2f}\ldots x_{kf}\] holds. Prove that $ R$ is commutative. [i]R. Wiegandt[/i]

PEN D Problems, 6

Tags: induction , euler , search , congruence , modular arithmetic

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[2, \; 2^{2}, \; 2^{2^{2}}, \; 2^{2^{2^{2}}}, \cdots \pmod{n}\] is eventually constant.

2006 India Regional Mathematical Olympiad, 6

Tags: search , number theory

Prove that there are infinitely many positive integers $ n$ such that $ n(n\plus{}1)$ can be represented as a sum of two positive squares in at least two different ways. (Here $ a^{2}\plus{}b^{2}$ and $ b^{2}\plus{}a^{2}$ are considered as the same representation.)

2009 Today's Calculation Of Integral, 398

Tags: geometry , search , calculus computations , calculus , integration , inequalities

In $ xyz$ space, find the volume of the solid expressed by the sytem of inequality: $ 0\leqq x\leqq 1,\ 0\leqq y\leqq 1,\ 0\leqq z\leqq 1$ $ x^2 \plus{} y^2 \plus{} z^2 \minus{} 2xy \minus{} 1\geqq 0$

1994 All-Russian Olympiad, 4

Tags: search , combinatorics , geometry

In a regular $ 6n\plus{}1$-gon, $ k$ vertices are painted in red and the others in blue. Prove that the number of isosceles triangles whose vertices are of the same color does not depend on the arrangement of the red vertices.

2005 Irish Math Olympiad, 5

Tags: search , inequalities

Let $ a,b,c$ be nonnegative real numbers. Prove that: $ \frac{1}{3}((a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2) \le a^2\plus{}b^2\plus{}c^2\minus{}3 \sqrt[3]{a^2 b^2 c^2 } \le (a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2.$

2010 Contests, 2

Tags: search , inequalities

$a,b,c$ are positive real numbers. prove the following inequality: $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{(a+b+c)^2}\ge \frac{7}{25}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c})^2$ (20 points)

1971 Miklós Schweitzer, 1

Tags: superior algebra , topology , search

Let $ G$ be an infinite compact topological group with a Hausdorff topology. Prove that $ G$ contains an element $ g \not\equal{} 1$ such that the set of all powers of $ g$ is either everywhere dense in $ G$ or nowhere dense in $ G$. [i]J. Erdos[/i]

2005 MOP Homework, 2

Tags: incenter , geometry , search , circumcircle

Let $I$ be the incenter of triangle $ABC$, and let $A_1$, $B_1$, and $C_1$ be arbitrary points lying on segments $AI$,$BI$, and $CI$, respectively. The perpendicular bisectors of segments $AA_1$, $BB_1$, and $CC_1$ form triangles $A_2B_2C_2$. Prove that the circumcenter of triangle $A_2B_2C_2$ coincides with the circumcenter of triangle $ABC$ if and only if $I$ is the orthocenter of triangle $A_1B_1C_1$.

2002 AMC 10, 24

Tags: number theory , prime factorization , search , modular arithmetic

What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1,k_2,\ldots,k_n$ for which \[k_1^2+k_2^2+\ldots+k_n^2=2002?\] $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

2005 South East Mathematical Olympiad, 2

Tags: analytic geometry , search , geometry

Circle $C$ (with center $O$) does not have common point with line $l$. Draw $OP$ perpendicular to $l$, $P \in l$. Let $Q$ be a point on $l$ ($Q$ is different from $P$), $QA$ and $QB$ are tangent to circle $C$, and intersect the circle at $A$ and $B$ respectively. $AB$ intersects $OP$ at $K$. $PM$, $PN$ are perpendicular to $QB$, $QA$, respectively, $M \in QB$, $N \in QA$. Prove that segment $KP$ is bisected by line $MN$.

2009 Indonesia TST, 1

Tags: combinatorics , algebra , search , polynomial

Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.

2010 District Olympiad, 4

Tags: algebra , function , search

Consider the sequence $ a_n\equal{}\left|z^n\plus{}\frac{1}{z^n}\right|\ ,\ n\ge 1$, where $ z\in \mathbb{C}^*$ is given. i) Prove that if $ a_1>2$, then: \[ a_{n\plus{}1}<\frac{a_n\plus{}a_{n\plus{}2}}{2}\ ,\ (\forall)n\in \mathbb{N}^*\] ii) Prove that if there is a $ k\in \mathbb{N}^*$ such that $ a_k\le 2$, then $ a_1\le 2$.