Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies (i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$. (ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.

Let $\Gamma_1,\Gamma_2$ be two circles centred at the points $(a,0),(b,0);0<a<b$ and having radii $a,b$ respectively.Let $\Gamma$ be the circle touching $\Gamma_1$ externally and $\Gamma_2$ internally. Find the locus of the centre of of $\Gamma$

Elisa has $2023$ treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules: [list=disc] [*]if more than one chests are unlocked, it locks one of them, or [*]if there is only one unlocked chest, it unlocks all the chests. [/list] Given that this process goes on forever, prove that there is a constant $C$ with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds $C$, regardless of how the fairy chooses the chests to unlock.

Let $P$ be a polynomial with integer coefficients. Suppose that for $n=1,2,3,\ldots ,1998$ the number $P(n)$ is a three-digit positive integer. Prove that the polynomial $P$ has no integer roots.

On a chess board ($8*8$) there are written the numbers $1$ to $64$: on the first line, from left to right, there are the numbers $1, 2, 3, ... , 8$; on the second line, from left to right, there are the numbers $9, 10, 11, ... , 16$;etc. The $\"+\"$ and $\"-\"$ signs are put to each number such that, in each line and in each column, there are $4$ $\"+\"$ signs and $4$ $\"-\"$ signs. Then, the $64$ numbers are added. Find all the possible values of this sum.

Let $x,y,a,b$ be positive real numbers such that $x\not= y$, $x\not= 2y$, $y\not= 2x$, $a\not=3b$ and $\frac{2x-y}{2y-x}=\frac{a+3b}{a-3b}$. Prove that $\frac{x^2+y^2}{x^2-y^2}\ge 1$.

In $\triangle{ADE}$ points $B$ and $C$ are on side $AD$ and points $F$ and $G$ are on side $AE$ so that $BG \parallel CF \parallel DE$, as shown. The area of $\triangle{ABG}$ is $36$, the area of trapezoid $CFED$ is $144$, and $AB = CD$. Find the area of trapezoid $BGFC$. [center][img]https://snag.gy/SIuOLB.jpg[/img][/center]

The absolute value of the sum of the elements of a real orthogonal matrix is at most the order of the matrix.

What is the area of the region in the complex plane consisting of all points $z$ satisfying both $|\tfrac{1}{z}-1|<1$ and $|z-1|<1$? ($|z|$ denotes the magnitude of a complex number, i.e. $|a+bi|=\sqrt{a^2+b^2}$.)

[b]p1.[/b] There are $5$ weights of masses $1,2,3,5$, and $10$ grams. One of the weights is counterfeit (its weight is different from what is written, it is unknown if the weight is heavier or lighter). How to find the counterfeit weight using simple balance scales only twice? [b]p2.[/b] There are $998$ candies and chocolate bars and $499$ bags. Each bag may contain two items (either two candies, or two chocolate bars, or one candy and one chocolate bar). Ann distributed candies and chocolate bars in such a way that half of the candies share a bag with a chocolate bar. Helen wants to redistribute items in the same bags in such a way that half of the chocolate bars would share a bag with a candy. Is it possible to achieve that? [b]p3.[/b] Insert in sequence $2222222222$ arithmetic operations and brackets to get the number $999$ (For instance, from the sequence $22222$ one can get the number $45$: $22*2+2/2 = 45$). [b]p4.[/b] Put numbers from $15$ to $23$ in a $ 3\times 3$ table in such a way to make all sums of numbers in two neighboring cells distinct (neighboring cells share one common side). [b]p5.[/b] All integers from $1$ to $200$ are colored in white and black colors. Integers $1$ and $200$ are black, $11$ and $20$ are white. Prove that there are two black and two white numbers whose sums are equal. [b]p6.[/b] Show that $38$ is the sum of few positive integers (not necessarily, distinct), the sum of whose reciprocals is equal to $1$. (For instance, $11=6+3+2$, $1/16+1/13+1/12=1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the sides $AB$ and $CD$ of the parallelogram $ABCD$ mark points $E$ and $F$, respectively. On the diagonals $AC$ and $BD$ chose the points $M$ and $N$ so that $EM\parallel BD$ and $FN\parallel AC$. Prove that the lines $AF, DE$ and $MN$ intersect at one point. (B. Rublev)

$m, n$ are relatively prime. We have three jugs which contain $m$, $n$ and $m+n$ liters. Initially the largest jug is full of water. Show that for any $k$ in $\{1, 2, ... , m+n\}$ we can get exactly $k$ liters into one of the jugs.

Every point on a line is painted either red or blue. Prove that there always exist three points $A,B,C$ that are painted the same color and are such that the point $B$ is the midpoint of the segment $AC$.

For which commutative finite groups is the product of all elements equal to the unit element?

Let $ABCD$ be a convex quadrilateral, let $E$ and $F$ be the midpoints of the sides $AD$ and $BC$, respectively. The segment $CE$ meets $DF$ in $O$. Show that if the lines $AO$ and $BO$ divide the side $CD$ in 3 equal parts, then $ABCD$ is a parallelogram.

For $a,b,c,d \geq 0$ with $a + b = c + d = 2$, prove \[(a^2 + c^2)(a^2 + d^2)(b^2 + c^2)(b^2 + d^2) \leq 25\]

Let $Y$ and $Z$ be the feet of the altitudes of a triangle $ABC$ drawn from angles $B$ and $C$, respectively. Let $U$ and $V$ be the feet of the perpendiculars from $Y$ and $Z$ on the straight line $BC$. The straight lines $YV$ and $ZU$ intersect at a point $L$. Prove that $AL \perp BC$.

Let $a,b,c$ be the length sides of a given triangle and $x,y,z$ be the sides length of bisectrices, respectively. Prove the following inequality $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Let $a,b$ be two integers and suppose that $n$ is a positive integer for which the set $\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\}$ is finite. Prove that $n=1$.

Ella lays out $16$ coins heads up in a $4\times 4$ grid as shown. [img]https://cdn.artofproblemsolving.com/attachments/3/3/a728be9c51b27f442109cc8613ef50d61182a0.png[/img] On a move, Ella can flip all the coins in any row, column, or diagonal (including small diagonals such as $H_1$ & $H_4$). If rotations are considered distinct, how many distinct grids of coins can she create in a finite number of moves?

The numbers from 1 to 100 are written in an unknown order. One may ask about any 50 numbers and find out their relative order. What is the fewest questions needed to find the order of all 100 numbers? [i]S. Tokarev[/i]

Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $ \$$1 each, begonias $ \$$1.50 each, cannas $ \$$2 each, dahlias $ \$$2.50 each, and Easter lilies $ \$$3 each. What is the least possible cost, in dollars, for her garden? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((6,0)--(0,0)--(0,1)--(6,1)); draw((0,1)--(0,6)--(4,6)--(4,1)); draw((4,6)--(11,6)--(11,3)--(4,3)); draw((11,3)--(11,0)--(6,0)--(6,3)); label("1",(0,0.5),W); label("5",(0,3.5),W); label("3",(11,1.5),E); label("3",(11,4.5),E); label("4",(2,6),N); label("7",(7.5,6),N); label("6",(3,0),S); label("5",(8.5,0),S);[/asy]$ \textbf{(A)}\ 108 \qquad \textbf{(B)}\ 115 \qquad \textbf{(C)}\ 132 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 156$

$\frac{a^3}{b^3}$+$\frac{a^3+1}{b^3+1}$+...+$\frac{a^3+2015}{b^3+2015}$=2016 b - positive integer, b can't be 0 a - real Find $\frac{a^3}{b^3}$*$\frac{a^3+1}{b^3+1}$*...*$\frac{a^3+2015}{b^3+2015}$

Let us divide by straight lines a quadrangle of unit area into $ n$ subpolygons and draw a circle into each subpolygon. Show that the sum of the perimeters of the circles is at most $ \pi \sqrt{n}$ (the lines are not allowed to cut the interior of a subpolygon). [i]G. and L. Fejes-Toth[/i]

A rectangle of size $ m \times n$ has been filled completely by trominoes (a tromino is an L-shape consisting of 3 unit squares). There are four ways to place a tromino 1st way: let the "corner" of the L be on top left 2nd way: let the "corner" of the L be on top right 3rd way: let the "corner" of the L be on bottom left 4th way: let the "corner" of the L be on bottom right Prove that the difference between the number of trominoes placed in the 1st and the 4th way is divisible by $ 3$.