Does there exist a function $ f: {\Bbb N} \to {\Bbb N}$ such that $ f(f(n \minus{} 1)) \equal{} f(n \plus{} 1) \minus{} f(n)$ for all $ n > 2$.

The sequence $x_0, x_1, x_2, ...$ is defined by $x_0 = 0$, $x_{k+1} = [(n - \sum_0^k x_i)/2]$. Show that $x_k = 0$ for all sufficiently large $k$ and that the sum of the non-zero terms $x_k$ is $n-1$.

The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?

Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter such as TAMEMHET.

Let $n$ and $k$ be two positive integers. Consider a group of $k$ people such that, for each group of $n$ people, there is a $(n+1)$-th person that knows them all (if $A$ knows $B$ then $B$ knows $A$). 1) If $k=2n+1$, prove that there exists a person who knows all others. 2) If $k=2n+2$, give an example of such a group in which no-one knows all others.

Solve the system of equations: $$\begin{cases}\sqrt{\frac{y(t-y)}{t-x}-\frac4x}+\sqrt{\frac{z(t-z)}{t-x}-\frac4x}=\sqrt x\\\sqrt{\frac{z(t-z)}{t-y}-\frac4y}+\sqrt{\frac{x(t-x)}{t-y}-\frac4y}=\sqrt y\\\sqrt{\frac{x(t-x)}{t-z}-\frac4z}+\sqrt{\frac{y(t-y)}{t-z}-\frac4z}=\sqrt z\\x+y+z=2t\end{cases}$$ if the following conditions are satisfied: $0<x<t$, $0<y<t$, $0<z<t$. [i]H. Lesov[/i]

Prove that it is impossible to divide a square with side length $7$ into exactly $36$ squares with integer side lengths.

Inscribe the equilateral triangle of the largest perimeter in a given semicircle.

A right triangle $ABC$ with $\angle C=90^o$ is inscribed in a circle. Suppose that $K$ is the midpoint of the arc $BC$ that does not contain $A$. Let $N$ be the midpoint of the segment $AC$, and $M$ be the intersection point of the ray $KN$ and the circle.The tangents to the circle drawn at $A$ and $C$ meet at $E$. prove that $\angle EMK = 90^o$

Let $ABC$ be an equilateral triangle with the side of $20$ units. Amir divides this triangle into $400$ smaller equilateral triangles with the sides of $1$ unit. Reza then picks $4$ of the vertices of these smaller triangles. The vertices lie inside the triangle $ABC$ and form a parallelogram with sides parallel to the sides of the triangle $ABC.$ There are exactly $46$ smaller triangles that have at least one point in common with the sides of this parallelogram. Find all possible values for the area of this parallelogram. [asy] unitsize(150); defaultpen(linewidth(0.7)); int n = 20; /* # of vertical lines, including BC */ pair A = (0,0), B = dir(-30), C = dir(30); draw(A--B--C--cycle,linewidth(1)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0)); label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE); for(int i = 1; i < n; ++i) { draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n); draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n); draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n); }[/asy] [Thanks azjps for drawing the diagram.] [hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]

In 1772 Euler discovered the curious fact that $n^2 +n+41$ is prime when $n$ is any of $0,1,2, \cdots, 39$. Show that there exist $40$ consecutive integer values of $n$ for which this polynomial is not prime.

Let $a$ and $b$ be positive integers such that $1\leq a<b\leq 100$. If there exists a positive integer $k$ such that $ab|a^k+b^k$, we say that the pair $(a, b)$ is good. Determine the number of good pairs.

Let $p$ and $q$ be relatively prime positive integers. A subset $S\subseteq \mathbb{N}_0$ is called ideal if $0 \in S$ and, for each element $n \in S$, the integers $n+p$ and $n+q$ belong to $S$. Determine the number of ideal subsets of $\mathbb{N}_0$.

An acute triangle $ABC$ is given. Let us denote $X$ for each of its inner points $X_a, X_b, X_c$ its images in axial symmetries sequentially along the lines $BC, CA, AB$. Prove that all $X_aX_bX_c$ triangles have a common interior point. (Josef Tkadlec)

Prove that the equation $ 2x^2 - 215y^2 = 1 $ has no integer solutions.

Find the area of the part bounded by two curves $y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $x$-axis. 1956 Tokyo Institute of Technology entrance exam

There is a rectangular field that measures $20\text{m}$ by $15\text{m}$. Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\text{m}$ sides such that he is $6\text{m}$ from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into two parts with equal area. How far in meters did Xiaoyu fly?

Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the segment $PQ$.

A massless string is wrapped around a frictionless pulley of mass $M$. The string is pulled down with a force of 50 N, so that the pulley rotates due to the pull. Consider a point $P$ on the rim of the pulley, which is a solid cylinder. The point has a constant linear (tangential) acceleration component equal to the acceleration of gravity on Earth, which is where this experiment is being held. What is the weight of the cylindrical pulley, in Newtons? [i](Proposed by Ahaan Rungta)[/i] [hide="Note"] This problem was not fully correct. Within friction, the pulley cannot rotate. So we responded: [quote]Excellent observation! This is very true. To submit, I'd say just submit as if it were rotating and ignore friction. In some effects such as these, I'm pretty sure it turns out that friction doesn't change the answer much anyway, but, yes, just submit as if it were rotating and you are just ignoring friction. [/quote]So do this problem imagining that the pulley does rotate somehow. [/hide]

In quadrilateral $ABCD$, angles $A, B, C, D$ form an increasing arithmetic sequence. Also, $\angle ACB = 90^o$ . If $CD = 14$ and the length of the altitude from $C$ to $AB$ is $9$, compute the area of $ABCD$.

Consider a system of inequalities \begin{align*}y-x&\ge|x+1|-|x-1|, \\ |y&-x|-y+x\ge2.\end{align*} Draw solutions of each inequality in the plane separately and highlight solution of the system.

Let $ ABCD$ be a convex quadrilateral. Square $ AB_1A_2B$ is constructed such that the two vertices $ A_2,B_1$ is located outside $ ABCD$. Similarly, we construct squares $ BC_1B_2C$, $ CD_1C_2D$, $ DA_1D_2A$. Let $ K$ be the intersection of $ AA_2$ and $ BB_1$, $ L$ be the intersection of $ BB_2$ and $ CC_1$, $ M$ be the intersection of $ CC_2$ and $ DD_1$, and $ N$ be the intersection of $ DD_2$ and $ AA_1$. Prove that $ KM$ is perpendicular to $ LN$.

Let $ x,y,z\in\mathbb R^{\plus{}}$ and $ x\plus{}y\plus{}z\equal{}3$. Prove that: \[ \frac{x^3}{y^3\plus{}8}\plus{}\frac{y^3}{z^3\plus{}8}\plus{}\frac{z^3}{x^3\plus{}8}\geq\frac19\plus{}\frac2{27}(xy\plus{}xz\plus{}yz)\]

A plant grows in the way we describe below. has a trunk which forks into two branches; each branch of the plant can, in turn, branch off into other two branches, or end in a bud. We will call the [i]load [/i] of a branch the total number of buds it bears, that is, the number of buds fed by the sap that passes by that branch; and we will call the [i]distance [/i] of a bud the number of bifurcations that it sap has to go through to get from the trunk to that bud. If n is the number of bifurcations that a certain plant of that type has, it is asks a) the number of branches of the plant, b) the number of buds, c) show that the sum of the charges of all the branches is equal to the sum of the clearances of all buds. Hint: You can proceed by induction, showing that if some results are correct for a given plant, they remain correct for the plant that is obtained substituting a bud in it for a pair of branches ending in individual buds.

Prove that for every positive integer $n$, the number $A_n = 5^n + 2 \cdot 3^{n-1} + 1$ is a multiple of $8$.