Found problems: 85335
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$,
$$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$
A quadrilateral $ABCD$ with $AB \perp BC$ , $AD \perp DC$, $E$ is a point that is on the line $BD$ with $EC=CA$ , $F$, $G$ is on the line $AB$ $AD$ such that $EF\perp AC $ and $EG\perp AC$ ,let $X Y$ be the midpoint of segment $AF AG $ , let $Z W$ be the midpoint of segment $BE DE $ , try to proof that $(WBX)$ is tangent to $(ZDY)$
For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2y}$. The sum of the solutions to the equation
\[4096f(f(x, x), x) = x^{13}\]
can be written in simplest form as $\tfrac{m}{n}$. Compute $m + n$.
The New York Times Mini Crossword is a $5\times5$ grid with the top left and bottom right corners shaded. Each row and column has a clue given (so that there are $10$ clues total). Jeffrey has a $\frac{1}{2}$ chance of knowing the answer to each clue. What is the probability that he can fill in every unshaded square in the crossword?
[asy]
size(4cm);
for (int i = 0; i < 6; ++i) {draw((i,0)--(i,5));
draw((0,i)--(5,i));}
fill((0,4)--(1,4)--(1,5)--(0,5)--cycle, black);
fill((5,0)--(5,1)--(4,1)--(4,0)--cycle, black);
[/asy]
[i]2018 CCA Math Bonanza Individual Round #8[/i]
In convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ meet at the point $P$. We know that $\angle DAC = 90^o$ and $2 \angle ADB = \angle ACB$. If we have $ \angle DBC + 2 \angle ADC = 180^o$ prove that $2AP = BP$.
Proposed by Iman Maghsoudi
Let $ABC$ be an equilateral triangle, and let $D,E$ and $F$ be points on $BC,BA$ and $AB$ respectively. Let $\angle BAD= \alpha, \angle CBE=\beta$ and $\angle ACF =\gamma$. Prove that if $\alpha+\beta+\gamma \geq 120^\circ$, then the union of the triangular regions $BAD,CBE,ACF$ covers the triangle $ABC$.
Find $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\ _n C_k.$
A ghost leg is a game with some vertical lines and some horizontal lines. A player starts at the top of the vertical line and go downwards, and always walkthrough a horizontal line if he encounters one. We define a layer is some horizontal line with the same height and has no duplicated endpoints. Find the smallest number of layers needed to grant that you can walk from $(1, 2, \ldots , n)$ on the top to any permutation $(\sigma_1, \sigma_2, \ldots, \sigma_n)$ on the bottom.
Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that
\[
ab\mid (cd)^{max(a,b)}
\]
Find all plane triangles whose sides have integer length and whose incircles have unit radius.
Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.
Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$
Let $p$ be a prime number and $n$ a natural one. How many natural numbers are between $1$ and $p^n$ that are relatively prime with $p^n$?
Determine all the functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \]
for all real numbers $x$ and $y$.
The equal segments $AB$ and $CD$ intersect at the point $O$ and divide it by the relation $AO: OB = CO: OD = 1: 2 $. The lines $AD$ and $BC$ intersect at the point $M$. Prove that $DM = MB$.
The circle $\omega_0$ touches the line at point A. Let $R$ be a given positive number. We consider various circles $\omega$ of radius $R$ that touch a line $\ell$ and have two different points in common with the circle $\omega_0$. Let $D$ be the touchpoint of the circle $\omega_0$ with the line $\ell$, and the points of intersection of the circles $\omega$ and $\omega_0$ are denoted by $B$ and $C$ (Assume that the distance from point $B$ to the line $\ell$ is greater than the distance from point $C$ to this line). Find the locus of the centers of the circumscribed circles of all such triangles $ABD$.
Find all natural numbers $ k\ge 1 $ and $ n\ge 2, $ which have the property that there exist two matrices $ A,B\in M_n\left(\mathbb{Z}\right) $ such that $ A^3=O_n $ and $ A^kB +BA=I_n. $
Let $a$, $b$, and $c$ be positive integers such that the product $ab$ divides the product $c(c^2-c+1)$ and the sum $a+b$ is divisible the number $c^2+1$. Prove that the sets ${a,b}$ and ${c,c^2-c+1}$ coincide.
Two player are playing in an $100 \times 100$ grid. Initially the whole board is black. On $A$'s move, he selects $4 \times 4$ subgrid and color it white. On $B$'s move, he selects a $3 \times 3$ subgrid and colors it black. $A$ wants to make the whole board white. Can he do it?
[i]Proposed by S M A Nahian[/i]
$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.
Prove that polynomial $x^8+98x^4+1$ can be factorized in $Z[X]$.
Let $S$ be the set of all non-increasing sequences of numbers $a_1 \geq a_2 \geq \ldots \geq a_{101}$ such that $a_i \in \{ 0,1,\ldots ,101 \}$ for all $1 \leq i \leq 101$
For every sequence $s \in S$ let $$f(s)=\lceil \frac{a_1}{2} \rceil+\lfloor \frac{a_2}{2} \rfloor + \lceil \frac{a_3}{2} \rceil + \ldots + \lfloor \frac{a_{100}}{2} \rfloor + \lceil \frac{a_{101}}{2} \rceil$$
where $\lfloor x \rfloor$ is the greatest integer, not exceeding $x$, and $\lceil x \rceil$ is the least integer at least $x$.
Prove that the number of sequences $s \in S$ for which $f(s)$ is even is the same, as the number of sequences $s$ for which $f(s)$ is odd
[i]M. Zorka[/i]
Show that there is a digit unequal to 2 in the decimal represesentation of $ \sqrt [3]{3}$ between the $ 1000000$-th und $ 3141592$-th position after decimal point.
$ABCD$ is a scalene tetrahedron and let $G$ be its baricentre. A plane $\alpha$ passes through $G$ such that it intersects neither the interior of $\Delta BCD$ nor its perimeter. Prove that
$$\textnormal{dist}(A,\alpha)=\textnormal{dist}(B,\alpha)+\textnormal{dist}(C,\alpha)+\textnormal{dist}(D,\alpha).$$
[i] (Adapted from folklore)[/i]
Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations:
[list=1]
[*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell.
[*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell.
[/list]
At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$.
[i]Proposed by Warut Suksompong, Thailand[/i]