Olympiad problems

2008 ITest, 29

Find the number of ordered triplets $(a,b,c)$ of positive integers such that $abc=2008$ (the product of $a$, $b$, and $c$ is $2008$).

2015 China Team Selection Test, 3

Tags: combinatorics

Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.

2012 May Olympiad, 3

Tags: geometry , area , paper

From a paper quadrilateral like the one in the figure, you have to cut out a new quadrilateral whose area is equal to half the area of the original quadrilateral.You can only bend one or more times and cut by some of the lines of the folds. Describe the folds and cuts and justify that the area is half. [img]https://2.bp.blogspot.com/-btvafZuTvlk/XNY8nba0BmI/AAAAAAAAKLo/nm4c21A1hAIK3PKleEwt6F9cd6zv4XffwCK4BGAYYCw/s400/may%2B2012%2Bl1.png[/img]

2023 Baltic Way, 9

Tags: combinatorics

Determine if there exists a triangle that can be cut into $101$ congruent triangles.

2005 AMC 10, 24

Tags: polynomial , algebra

For each positive integer $ m > 1$, let $ P(m)$ denote the greatest prime factor of $ m$. For how many positive integers $ n$ is it true that both $ P(n) \equal{} \sqrt{n}$ and $ P(n \plus{} 48) \equal{} \sqrt{n \plus{} 48}$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$

2008 HMNT, 4

Tags:

How many numbers between $1$ and $1,000,000$ are perfect squares but not perfect cubes?

2019 USA TSTST, 8

Tags: combinatorics , plane geometry , geometry , combinatorial geometry

Let $\mathcal S$ be a set of $16$ points in the plane, no three collinear. Let $\chi(S)$ denote the number of ways to draw $8$ lines with endpoints in $\mathcal S$, such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of $\chi(\mathcal S)$ across all such $\mathcal S$. [i]Ankan Bhattacharya[/i]

1999 Argentina National Olympiad, 2

Tags: tangent , segment , circles , geometry

Let $C_1$ and $C_2$ be the outer circumferences of centers $O_1$ and $O_2$, respectively. The two tangents to the circumference $C_2$ are drawn by $O_1$, intersecting $C_1$ at $P$ and $P'$. The two tangents to the circumference $C_1$ are drawn by $O_2$, intersecting $C_2$ at $Q$ and $Q'$. Prove that the segment $PP'$ is equal to the segment $QQ'$.

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.