Olympiad problems

1985 IMO Longlists, 23

Let $\mathbb N = {1, 2, 3, . . .}$. For real $x, y$, set $S(x, y) = \{s | s = [nx+y], n \in \mathbb N\}$. Prove that if $r > 1$ is a rational number, there exist real numbers $u$ and $v$ such that \[S(r, 0) \cap S(u, v) = \emptyset, S(r, 0) \cup S(u, v) = \mathbb N.\]

2009 Tuymaada Olympiad, 4

Tags: floor function , algebra , function , inequalities

The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50. [i]Proposed by A. Khabrov[/i]

2007 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , floor function

Find all nonzero subsets $A$ of the set $\left\{2,3,4,5,\cdots\right\}$ such that $\forall n\in A$, we have that $n^{2}+4$ and $\left\lfloor{\sqrt{n}\right\rfloor}+1$ are both in $A$.

2014 Harvard-MIT Mathematics Tournament, 4

Tags: function , floor function

Compute \[\sum_{k=0}^{100}\left\lfloor\dfrac{2^{100}}{2^{50}+2^k}\right\rfloor.\] (Here, if $x$ is a real number, then $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.)

2007 Nicolae Coculescu, 4

Tags: number theory , set , floor function

Let be a natural number $ n\ge 2. $ Prove that there exists an unique bipartition $ \left( A,B \right) $ of the set $ \{ 1,2\ldots ,n \} $ such that $ \lfloor \sqrt x \rfloor\neq y , $ for any $ x,y\in A , $ and $ \lfloor \sqrt z \rfloor\neq t , $ for any $ z,t\in B. $ [i]Costin Bădică[/i]

2020 BMT Fall, 10

Tags: algebra , floor function

For $k\ge 1$, define $a_k=2^k$. Let $$S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right).$$ Compute $\lfloor 100S\rfloor$.

2017 BMT Spring, 12

Tags: floor function , geometry

Square $S$ is the unit square with vertices at $(0, 0)$, $(0, 1)$, $(1, 0)$ and $(1, 1)$. We choose a random point $(x, y)$ inside $S$ and construct a rectangle with length $x$ and width $y$. What is the average of $\lfloor p \rfloor$ where $p$ is the perimeter of the rectangle? $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.

PEN A Problems, 25

Tags: modular arithmetic , logarithm , least common multiple , function , inequalities , floor function , number theory

Show that ${2n \choose n} \; \vert \; \text{lcm}(1,2, \cdots, 2n)$ for all positive integers $n$.

2010 Serbia National Math Olympiad, 1

Tags: inequalities , combinatorics , floor function

Some of $n$ towns are connected by two-way airlines. There are $m$ airlines in total. For $i = 1, 2, \cdots, n$, let $d_i$ be the number of airlines going from town $i$. If $1\le d_i \le 2010$ for each $i = 1, 2,\cdots, 2010$, prove that \[\displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n\] Find all $n$ for which equality can be attained. [i]Proposed by Aleksandar Ilic[/i]

1998 Moldova Team Selection Test, 2

Tags: floor function

Determine the natural numbers that cannot be written as $\lfloor n + \sqrt{n} + \frac{1}{2} \rfloor$ for any $n \in \mathbb{N}$.

2012 Waseda University Entrance Examination, 2

Tags: gauss , algebra , induction , floor function

Consider a sequence $\{a_n\}_{n\geq 0}$ such that $a_{n+1}=a_n-\lfloor{\sqrt{a_n}}\rfloor\ (n\geq 0),\ a_0\geq 0$. (1) If $a_0=24$, then find the smallest $n$ such that $a_n=0$. (2) If $a_0=m^2\ (m=2,\ 3,\ \cdots)$, then for $j$ with $1\leq j\leq m$, express $a_{2j-1},\ a_{2j}$ in terms of $j,\ m$. (3) Let $m\geq 2$ be integer and for integer $p$ with $1\leq p\leq m-1$, let $a\0=m^2-p$. Find $k$ such that $a_k=(m-p)^2$, then find the smallest $n$ such that $a_n=0$.

2001 APMO, 2

Tags: counting , number theory , floor function

Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).

2010 Postal Coaching, 6

Tags: number theory , floor function

Solve the equation for positive integers $m, n$: \[\left \lfloor \frac{m^2}n \right \rfloor + \left \lfloor \frac{n^2}m \right \rfloor = \left \lfloor \frac mn + \frac nm \right \rfloor +mn\]

2022 Brazil Team Selection Test, 2

Tags: algebra , summation , number theory , floor function

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

2009 Turkey MO (2nd round), 3

Tags: number theory , modular arithmetic , floor function

If $1<k_1<k_2<...<k_n$ and $a_1,a_2,...,a_n$ are integers such that for every integer $N,$ $k_i \mid N-a_i$ for some $1 \leq i \leq n,$ find the smallest possible value of $n.$

2021 Balkan MO Shortlist, N5

Tags: shortlist , number theory , floor function

A natural number $n$ is given. Determine all $(n - 1)$-tuples of nonnegative integers $a_1, a_2, ..., a_{n - 1}$ such that $$\lfloor \frac{m}{2^n - 1}\rfloor + \lfloor \frac{2m + a_1}{2^n - 1}\rfloor + \lfloor \frac{2^2m + a_2}{2^n - 1}\rfloor + \lfloor \frac{2^3m + a_3}{2^n - 1}\rfloor + ... + \lfloor \frac{2^{n - 1}m + a_{n - 1}}{2^n - 1}\rfloor = m$$ holds for all $m \in \mathbb{Z}$.

2010 District Olympiad, 3

Tags: floor function , algebra

For any real number $ x$ prove that: \[ x\in \mathbb{Z}\Leftrightarrow \lfloor x\rfloor \plus{}\lfloor 2x\rfloor\plus{}\lfloor 3x\rfloor\plus{}...\plus{}\lfloor nx\rfloor\equal{}\frac{n(\lfloor x\rfloor\plus{}\lfloor nx\rfloor)}{2}\ ,\ (\forall)n\in \mathbb{N}^*\]

2013 Princeton University Math Competition, 6

Tags: modular arithmetic , trig identities , function , trigonometry , induction , floor function

Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.

2002 Tuymaada Olympiad, 4

Tags: pigeonhole principle , integration , calculus , floor function , number theory

A real number $a$ is given. The sequence $n_{1}< n_{2}< n_{3}< ...$ consists of all the positive integral $n$ such that $\{na\}< \frac{1}{10}$. Prove that there are at most three different numbers among the numbers $n_{2}-n_{1}$, $n_{3}-n_{2}$, $n_{4}-n_{3}$, $\ldots$. [i]A corollary of a theorem from ergodic theory[/i]

PEN I Problems, 11

Tags: quadratic , function , floor function

Let $p$ be a prime number of the form $4k+1$. Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]

2000 AIME Problems, 14

Tags: rhombus , floor function , trapezoid , trigonometry , geometry

In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP=PQ=QB=BC.$ Angle $ACB$ is $r$ times as large as angle $APQ,$ where $r$ is a positive real number. Find the greatest integer that does not exceed $1000r.$

2011 Romania Team Selection Test, 3

Tags: ratio , modular arithmetic , floor function , number theory

Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

2012 Brazil Team Selection Test, 1

Tags: function , golden ratio , ratio , algebra , floor function

Let $\phi = \frac{1+\sqrt5}{2}$. Prove that a positive integer appears in the list $$\lfloor \phi \rfloor , \lfloor 2 \phi \rfloor, \lfloor 3\phi \rfloor ,... , \lfloor n\phi \rfloor , ... $$ if and only if it appears exactly twice in the list $$\lfloor 1/ \phi \rfloor , \lfloor 2/ \phi \rfloor, \lfloor 3/\phi \rfloor , ... ,\lfloor n/\phi \rfloor , ... $$

2013 Benelux, 1

Tags: combinatorics , floor function

Let $n \ge 3$ be an integer. A frog is to jump along the real axis, starting at the point $0$ and making $n$ jumps: one of length $1$, one of length $2$, $\dots$ , one of length $n$. It may perform these $n$ jumps in any order. If at some point the frog is sitting on a number $a \le 0$, its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number $a > 0$, its next jump must be to the left (towards the negative numbers). Find the largest positive integer $k$ for which the frog can perform its jumps in such an order that it never lands on any of the numbers $1, 2, \dots , k$.

2012 Balkan MO, 3

Tags: induction , logarithm , inequalities , combinatorics , floor function

Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$ Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$