Found problems: 1340
Let $X$ be an $n$-element set and let $A_1,\ldots ,A_m$ be subsets of $X$ such that
i) $|A_i|=3$ for each $i=1,\ldots ,m$.
ii) $|A_i\cap A_j|\le 1$ for any two distinct indices $i,j$.
Show that there exists a subset of $X$ with at least $\lfloor\sqrt{2n}\rfloor$ elements which does not contain any of the $A_i$’s.
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$
[i]Andrew Carratu[/i]
We distribute $n\ge1$ labelled balls among nine persons $A,B,C, \dots , I$. How many ways are there to do this so that $A$ gets the same number of balls as $B,C,D$ and $E$ together?
Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs $5$ and $12$. Find the least integer greater than $L$.
$X$ has $n$ elements. $F$ is a family of subsets of $X$ each with three elements, such that any two of the subsets have at most one element in common. Show that there is a subset of $X$ with at least $\sqrt{2n}$ members which does not contain any members of $F$.
Let $x_1=1$ and $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$, for $n=1,2,3,\ldots $ where $x$ denotes the largest integer not greater than $x$. Determine $x_{1997}$.
Professor Piraldo takes part in soccer matches with a lot of goals and judges a match in his own peculiar way. A match with score of $m$ goals to $n$ goals, $m\geq n$, is [i]tough[/i] when $m\leq f(n)$, where $f(n)$ is defined by $f(0) = 0$ and, for $n \geq 1$, $f(n) = 2n-f(r)+r$, where $r$ is the largest integer such that $r < n$ and $f(r) \leq n$.
Let $\phi ={1+\sqrt 5\over 2}$. Prove that a match with score of $m$ goals to $n$, $m\geq n$, is tough if $m\leq \phi n$ and is not tough if $m \geq \phi n+1$.
Let $I = (0, 1]$ be the unit interval of the real line. For a given number $a \in (0, 1)$ we define a map $T : I \to I$ by the formula
if
\[ T (x, y) = \begin{cases} x + (1 - a),&\mbox{ if } 0< x \leq a,\\ \text{ } \\ x - a, & \mbox{ if } a < x \leq 1.\end{cases} \]
Show that for every interval $J \subset I$ there exists an integer $n > 0$ such that $T^n(J) \cap J \neq \emptyset.$
Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]
Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.
Let $m,n\ge 1$ and $a_1 < a_2 < \ldots < a_n$ be integers. Prove that there exists a subset $T$ of $\mathbb{N}$ such that
\[|T| \leq 1+ \frac{a_n-a_1}{2n+1}\]
and for every $i \in \{1,2,\ldots , m\}$, there exists $t \in T$ and $s \in [-n,n]$, such that $a_i=t+s$.
The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.
Find
\[\lim_{n\to\infty}\int _0^{2n} e^{-2x}\left|x-2\lfloor\frac{x+1}{2}\rfloor\right|\ dx.\]
[i]1985 Tohoku University entrance exam/Mathematics, Physics, Chemistry, Biology[/i]
What is the largest integer less than or equal to $$\frac{3^{31}+2^{31}}{3^{29}+2^{29}} \,\,\, ?$$
We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define \[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \] Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$.
[i]Proposed by Evan Chen[/i]
In convex quadrilateral $ABCD$, $\angle A \cong \angle C$, $AB = CD = 180$, and $AD \neq BC$. The perimeter of $ABCD$ is 640. Find $\lfloor 1000 \cos A \rfloor$. (The notation $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.)
Suppose $f : \mathbb{N} \longrightarrow \mathbb{N}$ is a function that satisfies $f(1) = 1$ and
$f(n + 1) =\{\begin{array}{cc} f(n)+2&\mbox{if}\ n=f(f(n)-n+1),\\f(n)+1& \mbox{Otherwise}\end {array}$
$(a)$ Prove that $f(f(n)-n+1)$ is either $n$ or $n+1$.
$(b)$ Determine$f$.
On a board there are $n$ nails, each two connected by a rope. Each rope is colored in one of $n$ given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors.
a) Can $n$ be $6$ ?
b) Can $n$ be $7$ ?
Let $\{a_n\}_{n\geq 0}$ be an arithmetic sequence with difference $d$ and $1\leq a_0\leq d$. Denote the sequence as $S_0$, and define $S_n$ recursively by two operations below:
Step $1$: Denote the first number of $S_n$ as $b_n$, and remove $b_n$.
Step $2$: Add $1$ to the first $b_n$ numbers to get $S_{n+1}$.
Prove that there exists a constant $c$ such that $b_n=[ca_n]$ for all $n\geq 0$, where $[]$ is the floor function.
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
Let $a\geq 2$ be a natural number. Prove that $\sum_{n=0}^\infty\frac1{a^{n^{2}}}$ is irrational.
A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$.
Prove that if the flea arrived twice at the point $n$, $n$ positive integer, then it performed at least $\lceil 2\sqrt n\rceil$ jumps.
Solve in $R$ equation $[x] \cdot \{x\} = 2001 x$, where$ [ .]$ and $\{ .\}$ represent respectively the floor and the integer functions.
Let $N$ be a natural number. Find (with prove) the number of solutions in the segment $[1,N]$ of the equation $x^2-[x^2]=(x-[x])^2$, where $[x]$ means the floor function of $x$.