Found problems: 5802
Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]
Let $p$ be a prime number and let $X$ be a finite set containing at least $p$ elements. A collection of pairwise mutually disjoint $p$-element subsets of $X$ is called a $p$-family. (In particular, the empty collection is a $p$-family.) Let $A$(respectively, $B$) denote the number of $p$-families having an even (respectively, odd) number of $p$-element subsets of $X$. Prove that $A$ and $B$ differ by a multiple of $p$.
We call an even positive integer $n$ [i]nice[/i] if the set $\{1, 2, \dots, n\}$ can be partitioned into $\frac{n}{2}$ two-element subsets, such that the sum of the elements in each subset is a power of $3$. For example, $6$ is nice, because the set $\{1, 2, 3, 4, 5, 6\}$ can be partitioned into subsets $\{1, 2\}$, $\{3, 6\}$, $\{4, 5\}$. Find the number of nice positive integers which are smaller than $3^{2022}$.
Let $n$ be positive integer and fix $2n$ distinct points on a circle. Determine the number of ways to connect the points with $n$ arrows (oriented line segments) such that all of the following conditions hold: [list] [*]each of the $2n$ points is a startpoint or endpoint of an arrow; [*]no two arrows intersect; and [*]there are no two arrows $\overrightarrow{AB}$ and $\overrightarrow{CD}$ such that $A$, $B$, $C$ and $D$ appear in clockwise order around the circle (not necessarily consecutively). [/list]
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.
Call a real-valued function $ f$ [i]very convex[/i] if
\[ \frac {f(x) \plus{} f(y)}{2} \ge f\left(\frac {x \plus{} y}{2}\right) \plus{} |x \minus{} y|
\]
holds for all real numbers $ x$ and $ y$. Prove that no very convex function exists.
Let $a$ be a real number. Let $(f_n(x))_{n\ge 0}$ be a sequence of polynomials such that $f_0(x)=1$ and $f_{n+1}(x)=xf_n(x)+f_n(ax)$ for all non-negative integers $n$.
a) Prove that $f_n(x)=x^nf_n\left(x^{-1}\right)$ for all non-negative integers $n$.
b) Find an explicit expression for $f_n(x)$.
Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ \minus{} 1$. Prove that if \[ a_{0} \plus{} 2a_{1} \plus{} 2^{2}a_{2} \plus{} \cdots \plus{} 2^{n}a_{n} \equal{} 0,\] then $ a_{0} \plus{} a_{1} \plus{} \cdots \plus{} a_{n} > 0$.
We are given $2001$ balloons and a positive integer $k$. Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most $k$ balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of $k$ such that, whatever the initial sizes are, it is possible to make all the balloons have equal size after a finite number of steps.
Let $n,k \in \mathbb{Z}^{+}$ with $k \leq n$ and let $S$ be a set containing $n$ distinct real numbers. Let $T$ be a set of all real numbers of the form $x_1 + x_2 + \ldots + x_k$ where $x_1, x_2, \ldots, x_k$ are distinct elements of $S.$ Prove that $T$ contains at least $k(n-k)+1$ distinct elements.
Let $X \neq \varnothing$ be a finite set and let $f: X \to X$ be a function such that for every $x \in X$ and a fixed prime $p$ we have $f^p(x)=x.$ Let $Y=\{x \in X | f(x) \neq x\}.$ Prove that the number of the members of the set $Y$ is divisible by $p.$
[i]Note.[/i] ${f^p(x)=x = \underbrace{f(f(f(\cdots ((f}_{ p \text{ times}}(x) ) \cdots )))} .$
Show that for all $n\ge 3$ there are $n$ different positive integers $x_1,x_2, ...,x_n$ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= 1.$$
Several distinct real numbers are written on a blackboard. Peter wants to create an algebraic expression such that among its values there would be these and only these numbers. He may use any real numbers, brackets, signs $+, -, \times$ and a special sign $\pm$. Usage of $\pm$ is equivalent to usage of $+$ and $-$ in all possible combinations. For instance, the expression $5 \pm 1$ results in $\{4, 6\}$, while $(2 \pm 0.5) \pm 0.5$ results in $\{1, 2, 3\}$.
Can Peter construct an expression if the numbers on the blackboard are :
(a) $1, 2, 4$ ? [i]($2$ points)[/i]
(b) any $100$ distinct real numbers ? [i]($6$ points)[/i]
A [i]lattice point[/i] is a point in the plane whose two coordinates are both integers. A [i]lattice line[/i] is a line in the plane that contains at least two lattice points. Is it possible to color every lattice point red or blue such that every lattice line contains exactly 2017 red lattice points? Prove that your answer is correct.
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$, \[ f(f(x)f(y)) + f(x+y) = f(xy). \]
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Suppose that $a$ is real number. Sequence $a_1,a_2,a_3,....$ satisfies
$$a_1=a, a_{n+1} =
\begin{cases}
a_n - \frac{1}{a_n}, & a_n\ne 0 \\
0, & a_n=0
\end{cases}
(n=1,2,3,..)$$
Find all possible values of $a$ such that $|a_n|<1$ for all positive integer $n$.
Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence.
Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths.
[asy]
size(4cm);
draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin);
draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin);
draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin);
draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin);
draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin);
draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin);
draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin);
[/asy]
[i]Proposed by Zixiang Zhou, Canada[/i]
Define a function $f:\mathbb{N}\rightarrow\mathbb{N}_0$ by $f(1)=0$ and
\[f(n)=\max_j\{ f(j)+f(n-j)+j\}\quad\forall\, n\ge 2 \]
Determine $f(2000)$.
We are given $n$ intervals $[l_1,r_1],[l_2,r_2],[l_3,r_3],\dots, [l_n,r_n]$ in the number line. We can divide the intervals into two sets such that no two intervals in the same set have overlaps. Prove that there are at most $n-1$ pairs of overlapping intervals.
Let $ m,n$ be two natural nonzero numbers and sets $ A \equal{} \{ 1,2,...,n\}, B \equal{} \{1,2,...,m\}$. We say that subset $ S$ of Cartesian product $ A \times B$ has property $ (j)$ if $ (a \minus{} x)(b \minus{} y)\le 0$ for each pairs $ (a,b),(x,y) \in S$. Prove that every set $ S$ with propery $ (j)$ has at most $ m \plus{} n \minus{} 1$ elements.
[color=#FF0000]The statement was edited, in order to reflect the actual problem asked. The sign of the inequality was inadvertently reversed into $ (a \minus{} x)(b \minus{} y)\ge 0$, and that accounts for the following two posts.[/color]
There are 2019 cards in a box. Each card has a number written on one of its sides and a letter on the other side. Amy and Ben play the following game: in the beginning Amy takes all the cards, places them on a line and then she flips as many cards as she wishes. Each time Ben touches a card he has to flip it and its neighboring cards. Ben is allowed to have as many as 2019 touches. Ben wins if all the cards are on the numbers' side, otherwise Amy wins. Determine who has a winning strategy.
In how many ways can we color exactly $k$ vertices of an $n$-gon in red such that any $2$ consecutive vertices are not both red. (Vertices are considered to be labeled)
For any finite sets $X$ and $Y$ of positive integers, denote by $f_X(k)$ the $k^{\text{th}}$ smallest positive integer not in $X$, and let $$X*Y=X\cup \{ f_X(y):y\in Y\}.$$Let $A$ be a set of $a>0$ positive integers and let $B$ be a set of $b>0$ positive integers. Prove that if $A*B=B*A$, then $$\underbrace{A*(A*\cdots (A*(A*A))\cdots )}_{\text{ A appears $b$ times}}=\underbrace{B*(B*\cdots (B*(B*B))\cdots )}_{\text{ B appears $a$ times}}.$$
[i]Proposed by Alex Zhai, United States[/i]
Let $S$ be a subset of $\{0,1,2,\dots ,9\}$. Suppose there is a positive integer $N$ such that for any integer $n>N$, one can find positive integers $a,b$ so that $n=a+b$ and all the digits in the decimal representations of $a,b$ (expressed without leading zeros) are in $S$. Find the smallest possible value of $|S|$.
[i]Proposed by Sutanay Bhattacharya[/i]
[hide=Original Wording]
As pointed out by Wizard_32, the original wording is:
Let $X=\{0,1,2,\dots,9\}.$ Let $S \subset X$ be such that any positive integer $n$ can be written as $p+q$ where the non-negative integers $p, q$ have all their digits in $S.$ Find the smallest possible number of elements in $S.$
[/hide]
Given the polynomial sequence $(p_{n}(x))$ satisfying $p_{1}(x)=x^{2}-1$, $p_{2}(x)=2x(x^{2}-1)$, and $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$, for $n\geq 2$, let $s_{n}$ be the sum of the absolute values of the coefficients of $p_{n}(x)$. For each $n$, find a non-negative integer $k_{n}$ such that $2^{-k_{n}}s_n$ is odd.