Found problems: 1340
Prove that for all natural numbers $n$ greater than $1$ :
$$[\sqrt{n}] + [\sqrt[3]{n}] +...+[ \sqrt[n]{n}] = [\log_2 n] + [\log_3 n] + ... + [\log_n n]$$
(VV Kisil)
Let a function $g:\mathbb{N}_0\to\mathbb{N}_0$ satisfy $g(0)=0$ and $g(n)=n-g(g(n-1))$ for all $n\ge 1$. Prove that:
a) $g(k)\ge g(k-1)$ for any positive integer $k$.
b) There is no $k$ such that $g(k-1)=g(k)=g(k+1)$.
Compute the sum of the real solutions to $\lfloor x \rfloor \{x\} = 2020x$.
Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$, and$ \{x\} = x -\lfloor x \rfloor$.
$ 8$ students took part in exam that contains $ 8$ questions. If it is known that each question was solved by at least $ 5$ students, prove that we can always find $ 2$ students such that each of questions was solved by at least one of them.
Let $ \{a_n\}_{n \in \mathbb{N}_0}$ be a sequence defined as follows: $ a_1=0$, $ a_n=a_{[\frac{n}{2}]}+(-1)^{n(n+1)/2}$, where $ [x]$ denotes the floor function. For every $ k \ge 0$, find the number $ n(k)$ of positive integers $ n$ such that $ 2^k \le n < 2^{k+1}$ and $ a_n=0$.
Find the sum of all $\left\lfloor x\right\rfloor$ such that $x^2-15\left\lfloor x\right\rfloor+36=0$.
$\text{(A) }15\qquad\text{(B) }26\qquad\text{(C) }45\qquad\text{(D) }49\qquad\text{(E) }75$
Let $p$ be a prime such that $p \equiv 1 (\text {mod}4)$. Evaluate
\[\sum_{k=1}^{p-1} \left( \left \lfloor \frac{2k^2}{p}\right \rfloor - 2 \left \lfloor {\frac{k^2}{p}}\right \rfloor \right)\]
A shepherd left, as an inheritance, to his children a flock of $k$ sheep, distributed as follows: the oldest received $\left\lfloor\frac{k}{2}\right\rfloor$ sheep, the middle one $\left\lfloor\frac{k}{3}\right\rfloor$ sheep and the youngest $\left\lfloor\frac{k}{5}\right\rfloor$ sheep. Knowing that there are no sheep left, determine all possible values for $k$.
Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.
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]
Determine all functions $f:[0,\infty)\rightarrow\mathbb{R}$ such that $f(0)=0$ and \[f(x)=1+5f\left(\left\lfloor{\frac{x}{2}\right\rfloor}\right)-6f\left(\left\lfloor{\frac{x}{4}\right\rfloor}\right)\] for all $x>0$.
Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.
Show that for every $n\in\mathbb{N}$, $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}}$ and that for every $\epsilon >0$, there exists an $n\in\mathbb{N}$ such that $ n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon$.
All the positive divisors of a positive integer $n$ are stored into an increasing array. Mary is writing a programme which decides for an arbitrarily chosen divisor $d > 1$ whether it is a prime. Let $n$ have $k$ divisors not greater than $d$. Mary claims that it suffices to check divisibility of $d$ by the first $\left\lceil\frac{k}{2}\right\rceil$ divisors of $n$: $d$ is prime if and only if none of them but $1$ divides $d$.
Is Mary right?
Suppose the sequence of nonnegative integers $a_1, a_2, \ldots, a_{1997}$ satisfies
\[ a_i + a_j \leq a_{i+j} \leq a_i + a_j + 1 \]
for all $i,j \geq 1$ with $i + j \leq 1997$. Show that there exists a real number $x$ such that $a_n = \lfloor nx \rfloor$ (the greatest integer $\leq nx$) for all $1 \leq n \leq 1997$.
In the coordinate system in the plane we consider a convex polygon $W$ and lines given by equations $x = k, y = m$, where $k$ and $m$ are integers. The lines determine a tiling of the plane with unit squares. We say that the boundary of $W$ intersects a square if the boundary contains an interior point of the square. Prove that the boundary of $W$ intersects at most $4 \lceil d \rceil $ unit squares, where $d$ is the maximal distance of points belonging to $W$ (i.e., the diameter of $W$) and $ \lceil d \rceil$ is the least integer not less than $d.$
If $a,b,c>0$, what is the smallest possible value of $ \left\lfloor \dfrac {a+b}{c} \right\rfloor + \left\lfloor \dfrac {b+c}{a} \right\rfloor + \left\lfloor \dfrac {c+a}{b} \right\rfloor $? (Note that $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $x$.)
Let $n\geq2$ be a positive integer. Determine all possible values of the sum
\[ S=\left\lfloor x_{2}-x_{1}\right\rfloor +\left\lfloor x_{3}-x_{2}\right\rfloor+...+\left\lfloor x_{n}-x_{n-1}\right\rfloor \]
where $x_i\in \mathbb{R}$ satisfying $\lfloor{x_i}\rfloor=i$ for $i=1,2,\ldots n$.
For a positive real number $\alpha$, define \[S(\alpha)=\{ \lfloor n\alpha\rfloor \; \vert \; n=1,2,3,\cdots \}.\] Prove that $\mathbb{N}$ cannot be expressed as the disjoint union of three sets $S(\alpha)$, $S(\beta)$, and $S(\gamma)$.
At a tourist camp, each person has at least $50$ and at most $100$ friends among the other persons at the camp. Show that one can hand out a t-shirt to every person such that the t-shirts have (at most) $1331$ different colors, and any person has $20$ friends whose t-shirts all have pairwisely different colors.
Find all non-integers $x$ such that $x+\frac{13}{x}=[x]+\frac{13}{[x]} . $where$[x]$ mean the greatest integer $n$ , where $n\leq x.$
Let $P$ be the number of ways to partition $2013$ into an ordered tuple of prime numbers. What is $\log_2 (P)$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor\frac{125}2\left(\min\left(\frac CA,\frac AC\right)-\frac35\right)\right\rfloor$ or zero, whichever is larger.
Find the largest natural number $ n$ for which there exist different sets $ S_1,S_2,\ldots,S_n$ such that:
$ 1^\circ$ $ |S_i\cup S_j|\leq 2004$ for each two $ 1\leq i,j\le n$ and
$ 2^\circ$ $ S_i\cup S_j\cup S_k\equal{}\{1,2,\ldots,2008\}$ for each three integers $ 1\le i<j<k\le n$.
What is the largest possible number of subsets of the set $\{1, 2, \dots , 2n+1\}$ such that the intersection of any two subsets consists of one or several consecutive integers?
Let $c$ be a fixed positive integer, and $\{x_k\}^{\inf}_{k=1}$ be a sequence such
that $x_1=c$ and
$x_n=x_{n-1}+\lfloor \frac{2x_{n-1}-2}{n} \rfloor$
for $n \ge 2$. Determine the explicit formula of $x_n$ in terms of $n$ and $c$.
(Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)