Alice is playing a game with $2018$ boxes, numbered $1 - 2018$, and a number of balls. At the beginning, boxes $1 - 2017$ have one ball each, and box $2018$ has $2018n$ balls. Every turn, Alice chooses $i$ and $j$ with $i > j$, and moves exactly $i$ balls from box $i$ to box $j$. Alice wins if all balls end up in box $1$. What is the minimum value of n so that Alice can win this game?

Given is an ellipse $ e$ in the plane. Find the locus of all points $ P$ in space such that the cone of apex $ P$ and directrix $ e$ is a right circular cone.

State consists of $2021$ cities, between some of them there are direct flights. Each pair of cities has not more than one flight, every flight belongs to one of $2021$ companies. Call a group of cities [i]incomplete[/i], if at least one company doesn't have any flights between cities of the group. Find the maximum positive integer $m$, so that one can always find an incomplete group of $m$ cities.

Ana and Bob are playing the following game. [list] [*] First, Bob draws triangle $ABC$ and a point $P$ inside it. [*] Then Ana and Bob alternate, starting with Ana, choosing three different permutations $\sigma_1$, $\sigma_2$ and $\sigma_3$ of $\{A, B, C\}$. [*] Finally, Ana draw a triangle $V_1V_2V_3$. [/list] For $i=1,2,3$, let $\psi_i$ be the similarity transformation which takes $\sigma_i(A), \sigma_i(B)$ and $\sigma_i(C)$ to $V_i, V_{i+1}$ and $ X_i$ respectively (here $V_4=V_1$) where triangle $\Delta V_iV_{i+1}X_i$ lies on the outside of triangle $V_1V_2V_3$. Finally, let $Q_i=\psi_i(P)$. Ana wins if triangles $Q_1Q_2Q_3$ and $ABC$ are similar (in some order of vertices) and Bob wins otherwise. Determine who has the winning strategy.

How many positive integers $n$ are there such that $n$ is an exact divisors of at least one of the numbers $10^{40}$ and $20^{30}$?

Jack has a quadrilateral that consists of four sticks. It turned out that Jack can form three different triangles from those sticks. Prove that he can form a fourth triangle that is different from the others.

Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy $$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$ for any nonnegative integer $ n. $ [i]Mihai Piticari[/i]

[b]Q14.[/b] Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$. Suppose that $IH$ is perpendicular to $BC$ ($H$ belongs to $BC$). If $HB=5 \text{cm}, \; HC=8 \text{cm}$, compute the area of $\triangle ABC$.

Given are two monic quadratics $f(x), g(x)$ such that $f, g, f+g$ have two distinct real roots. Suppose that the difference of the roots of $f$ is equal to the difference of the roots of $g$. Prove that the difference of the roots of $f+g$ is not bigger than the above common difference.

Non-zero real numbers $a, b$ and $c$ are given such that $ab+bc+ac=0$. Prove that numbers $a+b+c$ and $\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{a+c}$ are either both positive or both negative. [i]Proposed by Mykhailo Shtandenko[/i]

At the beginning, three boxes contain $m$, $n$, and $k$ pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece will win the game. Ayşe will be the first player. They are playing the game once for each $(m,n,k)=(1,2012,2014)$, $(2011,2011,2012)$, $(2011,2012,2013)$, $(2011,2012,2014)$, $(2011,2013,2013)$. In how many of them can Ayşe guarantee to win the game? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Determine all integer $n$ such that a surface of an $n \times n \times n$ grid cube can be pasted in one layer by paper $1 \times 2$ rectangles so that each rectangle has exactly five neighbors (by a line segment). (A.Shapovalov)

Let $a_1,a_2,a_3,\ldots,a_{250}$ be real numbers such that $a_1=2$ and $$a_{n+1}=a_n+\frac{1}{a_n^2}$$ for every $n=1,2, \ldots, 249$. Let $x$ be the greatest integer which is less than $$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_{250}}$$ How many digits does $x$ have? [i]Proposed by Miroslav Marinov, Bulgaria[/i]

Let $ABC$ be an acute-angled triangle with $AC>AB$, and $\Omega$ is your circumcircle. Let $P$ be the midpoint of the arc $BC$ of $\Omega$ (not containing $A$) and $Q$ be the midpoint of the arc $BC$ of $\Omega$(containing the point $A$). Let $M$ be the foot of perpendicular of $Q$ on the line $AC$. Prove that the circumcircle of $\triangle AMB$ cut the segment $AP$ in your midpoint.

Let $VABC$ be a regular triangular pyramid with base $ABC$, of center $O$. Points $I$ and $H$ are the center of the inscribed circle, respectively the orthocenter $\vartriangle VBC$. Knowing that $AH = 3 OI$, determine the measure of the angle between the lateral edge of the pyramid and the plane of the base.

Let $\Gamma_A$, $\Gamma_B$, $\Gamma_C$ be circles such that $\Gamma_A$ is tangent to $\Gamma_B$ and $\Gamma_B$ is tangent to $\Gamma_C$. All three circles are tangent to lines $L$ and $M$, with $A$, $B$, $C$ being the tangency points of $M$ with $\Gamma_A$, $\Gamma_B$, $\Gamma_C$, respectively. Given that $12=r_A<r_B<r_C=75$, calculate: a) the length of $r_B$. b) the distance between point $A$ and the point of intersection of lines $L$ and $M$.

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

Consider a triangle $ABC$ such that $\angle A = 90^o, \angle C =60^o$ and $|AC|= 6$. Three circles with centers $A, B$ and $C$ are pairwise tangent in points on the three sides of the triangle. Determine the area of the region enclosed by the three circles (the grey area in the figure). [asy] unitsize(0.2 cm); pair A, B, C; real[] r; A = (6,0); B = (6,6*sqrt(3)); C = (0,0); r[1] = 3*sqrt(3) - 3; r[2] = 3*sqrt(3) + 3; r[3] = 9 - 3*sqrt(3); fill(arc(A,r[1],180,90)--arc(B,r[2],270,240)--arc(C,r[3],60,0)--cycle, gray(0.7)); draw(A--B--C--cycle); draw(Circle(A,r[1])); draw(Circle(B,r[2])); draw(Circle(C,r[3])); dot("$A$", A, SE); dot("$B$", B, NE); dot("$C$", C, SW); [/asy]

Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$. [i]Remark[/i]. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.

In $2005$ Tycoon Tammy invested $\$100$ for two years. During the the first year her investment suffered a $15\%$ loss, but during the second year the remaining investment showed a $20\%$ gain. Over the two-year period, what was the change in Tammy's investment? $\textbf{(A)}\ 5\%\text{ loss}\qquad \textbf{(B)}\ 2\%\text{ loss}\qquad \textbf{(C)}\ 1\%\text{ gain}\qquad \textbf{(D)}\ 2\% \text{ gain} \qquad \textbf{(E)}\ 5\%\text{ gain}$

On the table, there are $1024$ marbles and two students, $A$ and $B$, alternatively take a positive number of marble(s). The student $A$ goes first, $B$ goes after that and so on. On the first move, $A$ takes $k$ marbles with $1 < k < 1024$. On the moves after that, $A$ and $B$ are not allowed to take more than $k$ marbles or $0$ marbles. The student that takes the last marble(s) from the table wins. Find all values of $k$ the student $A$ should choose to make sure that there is a strategy for him to win the game.

Compute the number of lattice points bounded by the quadrilateral formed by the points $(0, 0)$, $(0, 140)$, $(140, 0)$, and $(100, 100)$ (including the quadrilateral itself). A lattice point on the $xy$ -plane is a point $(x, y)$, where both $x$ and $y$ are integers.

Use $ \frac{d}{dx} \ln (2x\plus{}\sqrt{4x^2\plus{}1}),\ \frac{d}{dx}(x\sqrt{4x^2\plus{}1})$ to evaluate $ \int_0^1 \sqrt{4x^2\plus{}1}dx$.

In the figure, the rectangle is formed by $4$ smaller equal rectangles. If we count the total number of rectangles in the figure we find $10$. How many rectangles in total will there be in a rectangle that is formed by $n$ smaller equal rectangles?

Let $n$ be a natural number. Two numbers are called "unsociable" if their greatest common divisor is $1$. The numbers $\{1,2,...,2n\}$ are partitioned into $n$ pairs. What is the minimum number of "unsociable" pairs that are formed?