Olympiad problems

2008 Iran MO (3rd Round), 4

Tags: function , combinatorics

Let $ S$ be a sequence that: \[ \left\{ \begin{array}{cc} S_0\equal{}0\hfill\\ S_1\equal{}1\hfill\\ S_n\equal{}S_{n\minus{}1}\plus{}S_{n\minus{}2}\plus{}F_n& (n>1) \end{array} \right.\] such that $ F_n$ is Fibonacci sequence such that $ F_1\equal{}F_2\equal{}1$. Find $ S_n$ in terms of Fibonacci numbers.

2000 Czech And Slovak Olympiad IIIA, 6

Tags: number theory , digit

Find all four-digit numbers $\overline{abcd}$ (in decimal system) such that $\overline{abcd}= (\overline{ac}+1).(\overline{bd} +1)$

1999 Greece Junior Math Olympiad, 1

Tags: inequalities , algebra

Show that if $a,b$ are positive real numbers such that $a^{2000}+b^{2000}=a^{1998}+b^{1998}$ then $a^2+b^2 \le 2$.

1994 Poland - Second Round, 6

Tags: number theory , divide , prime

Let $p$ be a prime number. Prove that there exists $n \in Z$ such that $p | n^2 -n+3$ if and only if there exists $m \in Z$ such that $p | m^2 -m+25$.

2019 Yasinsky Geometry Olympiad, p2

Tags: geometry , rectangle , inradius , circumcircle

Given the equilateral triangle $ABC$. It is known that the radius of the inscribed circle is in this triangle is equal to $1$. The rectangle $ABDE$ is such that point $C$ belongs to its side $DE$. Find the radius of the circle circumscribed around the rectangle $ABDE$.

2010 India IMO Training Camp, 10

Tags: geometry , parallelogram , inequalities , analytic geometry , trigonometry , complex number

Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$

2010 Contests, 4

Tags: number theory

Let $r$ be a positive integer and let $N$ be the smallest positive integer such that the numbers $\frac{N}{n+r}\binom{2n}{n}$, $n=0,1,2,\ldots $, are all integer. Show that $N=\frac{r}{2}\binom{2r}{r}$.

2011 Thailand Mathematical Olympiad, 5

Tags: induction , search , number theory

Find all $n$ such that \[n = d (n) ^ 4\] Where $d (n)$ is the number of divisors of $n$, for example $n = 2 \cdot 3\cdot 5\implies d (n) = 2 \cdot 2\cdot 2$.

1975 Chisinau City MO, 89

Tags: cyclic , geometry , circles

A closed line on a plane is such that any quadrangle inscribed in it has the sum of opposite angles equal to $180^o$. Prove that this line is a circle.

2005 Oral Moscow Geometry Olympiad, 3

Tags: combinatorial geometry , geometry , pentagon

$ABCBE$ is a regular pentagon. Point $B'$ is symmetric to point $B$ wrt line $AC$ (see figure). Is it possible to pave the plane with pentagons equal to $AB'CBE$? (S. Markelov) [img]https://cdn.artofproblemsolving.com/attachments/9/2/cbb5756517e85e56c4a931e761a6b4da8fe547.png[/img]

2006 Estonia Math Open Junior Contests, 9

Tags: number theory

A computer outputs the values of the expression $ (n\plus{}1) . 2^n$ for $ n \equal{} 1, n \equal{} 2, n \equal{} 3$, etc. What is the largest number of consecutive values that are perfect squares?

2008 Iran MO (3rd Round), 5

Tags: polynomial , modular arithmetic , function , number theory , algebra

Find all polynomials $ f\in\mathbb Z[x]$ such that for each $ a,b,x\in\mathbb N$ \[ a\plus{}b\plus{}c|f(a)\plus{}f(b)\plus{}f(c)\]

2004 Bulgaria Team Selection Test, 2

Tags: inequalities , algebra , inequality

Prove that if $a,b,c \ge 1$ and $a+b+c=9$, then $\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}$

2013 Albania Team Selection Test, 5

Tags: abstract algebra , number theory

Let $k$ be a natural number.Find all the couples of natural numbers $(n,m)$ such that : $(2^k)!=2^n*m$

2001 Spain Mathematical Olympiad, Problem 4

Tags: magic , number theory

The integers between $1$ and $9$ inclusive are distributed in the units of a $3$ x $3$ table. You sum six numbers of three digits: three that are read in the rows from left to right, and three that are read in the columns from top to bottom. Is there any such distribution for which the value of this sum is equal to $2001$?

2003 Tournament Of Towns, 2

Tags: geometry

In $7$-gon $A_1A_2A_3A_4A_5A_6A_7$ diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_7, A_6A_1$ and $A_7A_2$ are congruent to each other and diagonals $A_1A_4, A_2A_5, A_3A_6, A_4A_7, A_5A_1, A_6A_2$ and $A_7A_3$ are also congruent to each other. Is the polygon necessarily regular?

1992 Miklós Schweitzer, 7

Tags: topology

Prove that in a topological space X , if all discrete subspaces have compact closure , then X is compact.

2001 Estonia National Olympiad, 4

Tags: composite , sum of powers , prime , number theory

Prove that for any integer $a > 1$ there is a prime $p$ for which $1+a+a^2+...+ a^{p-1}$ is composite.

2023 Malaysian IMO Training Camp, 2

Tags: combinatorics

Ivan is playing Lego with $4n^2$ $1 \times 2$ blocks. First, he places $2n^2$ $1 \times 2$ blocks to fit a $2n \times 2n$ square as the bottom layer. Then he builds the top layer on top of the bottom layer using the remaining $2n^2$ $1 \times 2$ blocks. Note that the blocks in the bottom layer are connected to the blocks above it in the top layer, just like real Lego blocks. He wants the whole two-layered building to be connected and not in seperate pieces. Prove that if he can do so, then the four $1\times 2$ blocks connecting the four corners of the bottom layer, must be all placed horizontally or all vertically. [i]Proposed by Ivan Chan Kai Chin[/i]

2019 India IMO Training Camp, P1

Tags: number theory , inequalities

Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that \[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\] Prove that \[5m+12n\le 581.\]

Novosibirsk Oral Geo Oly VII, 2021.2

Tags: geometry , angle

The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.

1968 Swedish Mathematical Competition, 3

Tags: geometry , geometric inequalities

Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?

2024 Assara - South Russian Girl's MO, 8

Tags: combinatorics

There are $15$ boys and $15$ girls in the class. The first girl is friends with $4$ boys, the second with $5$, the third with $6$, . . . , the $11$th with $14$, and each of the other four girls is friends with all the boys. It turned out that there are exactly $3 \cdot 2^{25}$ ways to split the entire class into pairs, so that each pair has a boy and a girl who are friends. Prove that any of the friends of the first girl are friends with all the other girls too. [i]G.M.Sharafetdinova[/i]

2022 USEMO, 5

Tags: algebra , polynomial , number theory

Let $\tau(n)$ denote the number of positive integer divisors of a positive integer $n$ (for example, $\tau(2022) = 8$). Given a polynomial $P(X)$ with integer coefficients, we define a sequence $a_1, a_2,\ldots$ of nonnegative integers by setting \[a_n =\begin{cases}\gcd(P(n), \tau (P(n)))&\text{if }P(n) > 0\\0 &\text{if }P(n) \leq0\end{cases}\] for each positive integer $n$. We then say the sequence [i]has limit infinity[/i] if every integer occurs in this sequence only finitely many times (possibly not at all). Does there exist a choice of $P(X)$ for which the sequence $a_1$, $a_2$, . . . has limit infinity? [i]Jovan Vuković[/i]

LMT Team Rounds 2021+, B6

Tags: combinatorics

Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $ 1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $ 1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$. Proposed by Audrey Chun