Olympiad problems

2011 Hanoi Open Mathematics Competitions, 3

What is the largest integer less than to $\sqrt[3]{(2011)^3 + 3 \times (2011)^2 + 4 \times 2011+ 5}$ ? (A) $2010$, (B) $2011$, (C) $2012$, (D) $2013$, (E) None of the above.

2014 District Olympiad, 4

Tags: floor function , ceiling function , number theory

Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\displaystyle2\leq\frac{a}{b}\leq5$.

2010 Princeton University Math Competition, 2

Tags: floor function

Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.

2009 China Team Selection Test, 1

Tags: function , floor function , combinatorics , logarithm , ceiling function , ratio

Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$

2012 May Olympiad, 5

Tags: combinatorics , floor function

There are 12 people such that for every person A and person B there exists a person C that is a friend to both of them. Determine the minimum number of pairs of friends and construct a graph where the edges represent friendships.

2002 Singapore Team Selection Test, 2

Tags: number theory , floor function , integer , divide , divisible

For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.

2007 Balkan MO Shortlist, C1

Tags: floor function , geometry , combinatorics , combinatorial geometry , geometric transformation , rotation

For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that $C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.

2011 IFYM, Sozopol, 1

Tags: floor function , algebra , polynomial

Let $n$ be a positive integer. Find the number of all polynomials $P$ with coefficients from the set $\{0,1,2,3\}$ and for which $P(2)=n$.

1998 Argentina National Olympiad, 4

Tags: floor function , algebra , number theory

Determine all possible values of the expression$$x-\left [\frac{x}{2}\right ]-\left [\frac{x}{3}\right ]-\left [\frac{x} {6}\right ]$$by varying $x$ in the real numbers. Clarification: The brackets indicate the integer part of the number they enclose.

2002 Irish Math Olympiad, 4

Tags: floor function , algebra

Let $ \alpha\equal{}2\plus{}\sqrt{3}$. Prove that $ \alpha^n\minus{}[\alpha^n]\equal{}1\minus{}\alpha^{\minus{}n}$ for all $ n \in \mathbb{N}_0$.

2007 Brazil National Olympiad, 6

Tags: floor function , combinatorics , induction

Given real numbers $ x_1 < x_2 < \ldots < x_n$ such that every real number occurs at most two times among the differences $ x_j \minus{} x_i$, $ 1\leq i < j \leq n$, prove that there exists at least $ \lfloor n/2\rfloor$ real numbers that occurs exactly one time among such differences.

1976 IMO, 3

Tags: algebra , floor function , sequence , recurrence relation , power of 2

A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where $[x]$ denotes the smallest integer $\leq x)$