Determine all totally multiplicative and non-negative functions $f\colon\mathbb{Z}\rightarrow \mathbb{Z}$ with the property that if $a, b\in \mathbb{Z}$ and $b\neq 0$, then there exist integers $q$ and $r$ such that $a-qb+r$ and $f(r)<f(b)$.

Find the greatest integer which doesn't exceed $\frac{3^{100}+2^{100}}{3^{96}+2^{96}}$ (A)$81$ (B)$80$ (C)$79$ (D)$82$

Perpendiculars at their centers of gravity (points of intersection of medians) are restored to the faces of the tetrahedron. Prove that the projections of the three perpendiculars to the fourth face intersect at one point.

Sphere $S$ with radius $100$ has diameter $\overline{AB}$ and center $C$. Four small spheres all with radius $17$ have centers that lie in a plane perpendicular to $\overline{AB}$ such that each of the four spheres is internally tangent to $S$ and externally tangent to two of the other small spheres. Find the radius of the smallest sphere that is both externally tangent to two of the four spheres with radius $17$ and internally tangent to $S$ at a point in the plane perpendicular to $\overline{AB}$ at $C$.

Five points $K_i, i = 1, 2, 3, 4$ and $P$ are chosen arbitrarily on the same circle. Denote by $P(i, j)$ the distance from $P$ to the line passing through $K_i$ and $K_j$ . Prove that $$P(1, 2)P(3, 4) = P(1, 4)P(2, 3) = P(1, 3)P(2, 4)$$ Bùi Quang Tuấn

Prove that for every integer $n$ ($n > 1$) there exist two positive integers $x$ and $y$ ($x \leq y$) such that $$\frac{1}{n} = \frac{1}{x(x+1)} + \frac{1}{(x+1)(x+2)} + \cdots + \frac{1}{y(y+1)}$$

The points $K$ and $M$ are the centres of the $AB$ and $CD$ sides of the convex quadrangle $ABCD$. The points $L$ and $M$ belong to two other sides and $KLMN$ is a rectangle. Prove that $KLMN$ area is a half of $ABCD$ area.

We say that a strictly increasing sequence of positive integers $n_1, n_2,\ldots$ is [i]non-decelerating[/i] if $n_{k+1}-n_k\le n_{k+2}-n_{k+1}$ holds for all positive integers $k$. We say that a strictly increasing sequence $n_1, n_2, \ldots$ is [i]convergence-inducing[/i], if the following statement is true for all real sequences $a_1, a_2, \ldots$: if subsequence $a_{m+n_1}, a_{m+n_2}, \ldots$ is convergent and tends to $0$ for all positive integers $m$, then sequence $a_1, a_2, \ldots$ is also convergent and tends to $0$. Prove that a non-decelerating sequence $n_1, n_2,\ldots$ is convergence-inducing if and only if sequence $n_2-n_1$, $n_3-n_2$, $\ldots$ is bounded from above. [i]Proposed by András Imolay[/i]

Find the kind of a triangle if $$\frac{a\cos\alpha+b\cos\beta+c\cos\gamma}{a\sin\alpha+b\sin\beta+c\sin\gamma}=\frac{2p}{9R}.$$ ($\alpha,\beta,\gamma$ are the measures of the angles, $a,b,c$ are the respective lengths of the sides, $p$ the semiperimeter, $R$ is the circumradius) [i]K. Petrov[/i]

You have three colors $\{\text{red}, \text{blue}, \text{green}\}$ with which you can color the faces of a regular octahedron (8 triangle sided polyhedron, which is two square based pyramids stuck together at their base), but you must do so in a way that avoids coloring adjacent pieces with the same color. How many different coloring schemes are possible? (Two coloring schemes are considered equivalent if one can be rotated to fit the other.)

Let $O$ be the centre of a parallelogram $ABCD$ and $P$ be any point in the plane. Let $M, N$ be the midpoints of $AP, BP$, respectively and $Q$ be the intersection of $MC$ and $ND$. Prove that $O, P$ and $Q$ are collinear.

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

We denote $N_{2010}=\{1,2,\cdots,2010\}$ [b](a)[/b]How many non empty subsets does this set have? [b](b)[/b]For every non empty subset of the set $N_{2010}$ we take the product of the elements of the subset. What is the sum of these products? [b](c)[/b]Same question as the [b](b)[/b] part for the set $-N_{2010}=\{-1,-2,\cdots,-2010\}$. Albanian National Mathematical Olympiad 2010---12 GRADE Question 2.

$(BEL 3)$ Construct the circle that is tangent to three given circles.

Dinesh has several squares and regular pentagons, all with side length $ 1$. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so? [img]https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg[/img]

A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.

The sum of all the roots of $ 4x^3\minus{}8x^2\minus{}63x\minus{}9\equal{}0$ is: $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \minus{}8 \qquad \textbf{(D)}\ \minus{}2 \qquad \textbf{(E)}\ 0$

a) We are playing the following game on this table: In each move we select a row or a column of the table, reduce two neighboring numbers in that row or column by $1$ and increase the third one by $1$. After some of these moves can we get to a table with all the same entries? b) This time we have the choice to arrange the integers from $1$ to $9$ in the$ 3 \times3$ table. Still using the same moves now our aim is to create a table with all the same entries, maximising the value of the entries. What is the highest possible number we can achieve?

For all non-zero real numbers $x$ and $y$ such that $x-y=xy$, $\frac{1}{x}-\frac{1}{y}$ equals $\textbf{(A) }\frac{1}{xy}\qquad\textbf{(B) }\frac{1}{x-y}\qquad\textbf{(C) }0\qquad\textbf{(D) }-1\qquad\textbf{(E) }y-x$

The function $f$ has the property that, for each real number $x,$ \[ f(x)+f(x-1) = x^2. \] If $f(19)=94,$ what is the remainder when $f(94)$ is divided by 1000?

George is drawing a Christmas tree; he starts with an isosceles triangle $AB_0C_0$ with $AB_0=AC_0=41$ and $B_0C_0=18$. Then, he draws points $B_i$ and $C_i$ on sides $AB_0$ and $AC_0$, respectively, such that $B_iB_{i+1}=1$ and $C_iC_{i+1}=1$ ($B_{41}=C_{41}=A$). Finally, he uses a green crayon to color in triangles $B_iC_iC_{i+1}$ for $i$ from $0$ to $40$. What is the total area that he colors in?

The circle is inscribed in a triangle, inscribed in a semicircle. Find the marked angle $a$. [img]https://cdn.artofproblemsolving.com/attachments/8/e/334c8662377155086e9211da3589145f460b52.png[/img]

On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected respectively such that $NC=NP$. The point $Q$ is chosen on the segment $AN$ so that $\angle QPN = \angle NCB$. Prove that $2\angle BCQ = \angle AQP$.

Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game? $\textbf{(A) } \text{Bela will always win.}$ $\textbf{(B) } \text{Jenn will always win.} $ $\textbf{(C) } \text{Bela will win if and only if }n \text{ is odd.}$ $\textbf{(D) } \text{Jenn will win if and only if }n \text{ is odd.} $ $\textbf{(E) } \text{Jenn will win if and only if }n > 8.$

Consider all pairs of distinct points on the Cartesian plane $(A, B)$ with integer coordinates. Among these pairs of points, find all for which there exist two distinct points $(X, Y)$ with integer coordinates, such that the quadrilateral $AXBY$ is convex and inscribed. [i]Proposed by Anton Trygub[/i]