Olympiad problems

2007 Princeton University Math Competition, 2

Tags: floor function

Suppose that $A$ is a set of positive integers less than $N$ and that no two distinct elements of $A$ sum to a perfect square. That is, if $a_1, a_2 \in A$ and $a_1 \neq a_2$ then $|a_1+a_2|$ is not a square of an integer. Prove that the maximum number of elements in $A$ is at least $\left\lfloor\frac{11}{32}N\right\rfloor$ .

2008 ITest, 34

Tags: floor function , probability

While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\lfloor 1000a\rfloor$.

PEN A Problems, 28

Tags: function , algebra , floor function , number theory , greatest common divisor , divisibility theory

Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.

1980 AMC 12/AHSME, 1

Tags: floor function

The largest whole number such that seven times the number is less than 100 is $\text{(A)} \ 12 \qquad \text{(B)} \ 13 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 \qquad \text{(E)} \ 16$

2005 MOP Homework, 2

Tags: floor function , algebra , inequalities

Find all real numbers $x$ such that $\lfloor x^2-2x \rfloor+2\lfloor x \rfloor=\lfloor x \rfloor^2$. (For a real number $x$, $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$.)

1985 Dutch Mathematical Olympiad, 2

Tags: number theory , floor function , modular arithmetic

Among the numbers $ 11n \plus{} 10^{10}$, where $ 1 \le n \le 10^{10}$ is an integer, how many are squares?

2018 Mathematical Talent Reward Programme, MCQ: P 2

Tags: algebra , floor function , function

$\lim _{x \rightarrow 0^{+}} \frac{[x]}{\tan x}$ where $[x]$ is the greatest integer function [list=1] [*] -1 [*] 0 [*] 1 [*] Does not exists [/list]

PEN M Problems, 24

Tags: floor function , induction , recursive sequences

Let $k$ be a given positive integer. The sequence $x_n$ is defined as follows: $x_1 =1$ and $x_{n+1}$ is the least positive integer which is not in $\{x_{1}, x_{2},..., x_{n}, x_{1}+k, x_{2}+2k,..., x_{n}+nk \}$. Show that there exist real number $a$ such that $x_n = \lfloor an\rfloor$ for all positive integer $n$.

1998 AMC 12/AHSME, 30

Tags: number theory , floor function , prime factorization , modular arithmetic

For each positive integer $n$, let \[a_n = \frac {(n + 9)!}{(n - 1)!}.\] Let $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 9$

2012 Belarus Team Selection Test, 2

Tags: floor function , integer , odd , algebra

Given $\lambda^3 - 2\lambda^2- 1 = 0$ for some real $\lambda$ prove that $[\lambda[\lambda[\lambda n]]] - n$ is odd for any positive integer $n$ . (I Voronovich)

2012 Postal Coaching, 3

Tags: algebra , function , floor function , integer part , logarithm

Given an integer $n\ge 2$, prove that \[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\]. [hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]

2020 Regional Olympiad of Mexico Northeast, 1

Tags: recurrence relation , algebra , floor function , sequence

Let $a_1=2020$ and let $a_{n+1}=\sqrt{2020+a_n}$ for $n\ge 1$. How much is $\left\lfloor a_{2020}\right\rfloor$? Note: $\lfloor x\rfloor$ denotes the integer part of a number, that is that is, the immediate integer less than $x$. For example, $\lfloor 2.71\rfloor=2$ and $\lfloor \pi\rfloor=3$.

2017 Harvard-MIT Mathematics Tournament, 8

Tags: floor function

Does there exist an irrational number $\alpha > 1$ such that \[\lfloor \alpha^n \rfloor \equiv 0 \pmod{2017}\] for all integers $n \ge 1$?

2008 Postal Coaching, 2

Tags: floor function , power of 2 , number theory

Show that if $n \ge 4, n \in N$ and $\big [ \frac{2^n}{n} ]$ is a power of $2$, then $n$ is a power of $2$.

2006 USAMO, 1

Tags: number theory , floor function , inequalities , modular arithmetic , pigeonhole principle

Let $p$ be a prime number and let $s$ be an integer with $0 < s < p.$ Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and \[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \] if and only if $s$ is not a divisor of $p-1$. Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.

2005 Taiwan National Olympiad, 2

Tags: function , floor function , algebra

Find all reals $x$ satisfying $0 \le x \le 5$ and $\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor$.

2018 Peru Iberoamerican Team Selection Test, P6

Tags: algebra , floor function

Find all real numbers $a$ such that there exist $f:\mathbb{R} \to \mathbb{R}$ with$$f(x+f(y))=f(x)+a\lfloor y \rfloor $$for all $x,y\in \mathbb{R}$

2007 ITest, 46

Tags: algebra , floor function , system of equations

Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$, find the modulo $2007$ residue of $m$.

2012 Iran MO (3rd Round), 2

Tags: floor function , number theory

Prove that there exists infinitely many pairs of rational numbers $(\frac{p_1}{q},\frac{p_2}{q})$ with $p_1,p_2,q\in \mathbb N$ with the following condition: \[|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.\] [i]Proposed by Mohammad Gharakhani[/i]

2015 Belarus Team Selection Test, 3

Tags: floor function , sequence , parity , number theory

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2006 Pan African, 4

Tags: number theory , floor function

For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$. For every positive integer $n$ determine $a(1)+a(2)+\cdots+a(2^n)$.

1976 IMO Longlists, 38

Tags: number theory , floor function

Let $x =\sqrt{a}+\sqrt{b}$, where $a$ and $b$ are natural numbers, $x$ is not an integer, and $x < 1976$. Prove that the fractional part of $x$ exceeds $10^{-19.76}$.

2002 All-Russian Olympiad, 4

Tags: inequalities , pigeonhole principle , number theory , floor function

From the interval $(2^{2n},2^{3n})$ are selected $2^{2n-1}+1$ odd numbers. Prove that there are two among the selected numbers, none of which divides the square of the other.

PEN A Problems, 32

Tags: floor function , divisibility theory

Let $ a$ and $ b$ be natural numbers such that \[ \frac{a}{b}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{1318}+\frac{1}{1319}. \] Prove that $ a$ is divisible by $ 1979$.

2013 China Team Selection Test, 1

Tags: analytic geometry , combinatorics , induction , floor function

For a positive integer $k\ge 2$ define $\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}$ to be a collection of $k^2$ lattice points on the cartesian coordinate plane. Let $d_1(k)>d_2(k)>\cdots$ be the decreasing sequence of the distinct distances between any two points in $T_k$. Suppose $S_i(k)$ be the number of distances equal to $d_i(k)$. Prove that for any three positive integers $m>n>i$ we have $S_i(m)=S_i(n)$.