Let $n$ be a positive integer. Initially the sequence $0,0,\cdots,0$ ($n$ times) is written on the board. In each round, Ananya choses an integer $t$ and a subset of the numbers written on the board and adds $t$ to all of them. What is the minimum number of rounds in which Ananya can make the sequence on the board strictly increasing? Proposed by Shantanu Nene

Find the biggest $ n < 2007 $ such that there exists a partition of the integers from $1$ to $n$ into two sets the sums of the squares of whose elements are equal.

The base three representation of $x$ is \[ 12112211122211112222. \]The first digit (on the left) of the base nine representation of $x$ is $\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 5$

[b]p1.[/b] To every face of a given cube a new cube of the same size is glued. The resulting solid has how many faces? [b]p2.[/b] A father and his son returned from a fishing trip. To make their catches equal the father gave to his son some of his fish. If, instead, the son had given his father the same number of fish, then father would have had twice as many fish as his son. What percent more is the father's catch more than his son's? [b]p3.[/b] A radio transmitter has $4$ buttons. Each button controls its own switch: if the switch is OFF the button turns it ON and vice versa. The initial state of switches in unknown. The transmitter sends a signal if at least $3$ switches are ON. What is the minimal number of times you have to push the button to guarantee the signal is sent? [b]p4.[/b] $19$ matches are placed on a table to show the incorrect equation: $XXX + XIV = XV$. Move exactly one match to change this into a correct equation. [b]p5.[/b] Cut the grid shown into two parts of equal area by cutting along the lines of the grid. [img]https://cdn.artofproblemsolving.com/attachments/c/1/7f2f284acf3709c2f6b1bea08835d2fb409c44.png[/img] [b]p6.[/b] A family of funny dwarfs consists of a dad, a mom, and a child. Their names are: $A$, $R$, and $C$ (not in order). During lunch, $C$ made the statements: “$R$ and $A$ have different genders” and “$R$ and $A$ are my parents”, and $A$ made the statements “I am $C$'s dad” and “I am $R$'s daughter.” In fact, each dwarf told truth once and told a lie once. What is the name of the dad, what is the name of the child, and is the child a son or a daughter? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The polynomial P(x,y) is such that for every integer n >= 0 each of the polynomials P(x,n) and P(n,y) either is a constant zero or has a degree not greater than n. Is it possible that P(x,x) has an odd degree?

Triangle $ABC$ is given. On its sides $AB$, $BC$ and $CA$, respectively, points $X, Y, Z$ are chosen so that $$AX : XB =BY : YC = CZ : ZA = 2:1.$$ It turned out that the triangle $XYZ$ is equilateral. Prove that the original triangle $ABC$ is also equilateral.

In $\triangle ABC$, let its incircle touch $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$ respectively. Let its $A$-excircle have center $I_A$ and touch $\overline{BC}$ at $K$. Let $P$ and $Q$ be points so that $BFPI_A$ and $CEQI_A$ are parallelograms. If $\overline{AI_A}$ and $\overline{PQ}$ intersect at $X$, prove $\overline{KX} \perp \overline{PQ}$.

Let $ \mathcal{P}$ be a convex polygon with $ n$ sides, $ n\ge3$. Any set of $ n \minus{} 3$ diagonals of $ \mathcal{P}$ that do not intersect in the interior of the polygon determine a [i]triangulation[/i] of $ \mathcal{P}$ into $ n \minus{} 2$ triangles. If $ \mathcal{P}$ is regular and there is a triangulation of $ \mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $ n$.

Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$ [i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]

We consider an integer $n > 1$ with the following property: for every positive divisor $d$ of $n$ we have that $d + 1$ is a divisor of$ n + 1$. Prove that $n$ is a prime number.

Let $x$, $y$ and $z$ be nonnegative integers. Find all numbers in form $\overline{13xy45z}$ divisible with $792$, where $x$, $y$ and $z$ are digits.

Determine the least integer $k$ for which the following story could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games.

A convex quadrilateral $ABCD$ is such that $\angle B = \angle D$ and are both acute angles. $E$ is on $AB$ such that $CB = CE$ and $F$ is on $AD$ such that $CF = CD$. If the circumcenter of $CEF$ is $O_1$ and the circumcenter of $ABD$ is $O_2$. Prove that $C,O_1,O_2$ are collinear. [i]Proposed by Kapil Pause[/i]

Let $ABC$ be a triangle with incenter $I$. The line $BI$ meets $AC$ at $D$. Let $P$ be a point on $CI$ such that $DI=DP$ ($P\ne I$), $E$ the second intersection point of segment $BC$ with the circumcircle of $ABD$ and $Q$ the second intersection point of line $EP$ with the circumcircle of $AEC$. Prove that $\angle PDQ=90^\circ$. [i]Proposed by Ariel García[/i]

In the convex pentagon $ABCDE$ all interior angles have the same measure. Prove that the perpendicular bisector of segment $EA$, the perpendicular bisector of segment $BC$ and the angle bisector of $\angle CDE$ intersect in one point.

Each of the cells of a table 2014 x 2014 is colored in white or black. It is known that each square 2 x 2 contains an even number of black cells and each cross (3 x 3 square without its corner cells) contains an odd number of black cells. Prove that the 4 corner cells of the table are in the same color.

Find all ordered pairs of real numbers $(x, y)$ such that $x^2y = 3$ and $x + xy = 4$.

Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$. She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$. Mary wants to prevent Pat from winning the game. Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.

In the plane there are $n$ $(n\geq4)$ marked points. There are at least $n+1$ pairs of marked points such that the distance between each pair of points is $1$. Find the greatest integer $k$ such that there is a marked point that is the center of the circle with radius $1$ on which there are at least $k$ of the marked points.

Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that $\angle PAB+\angle PDC \leq 90^{\circ}$ and $\angle PBA+\angle PCD \leq 90^{\circ}.$ Prove that $AB+CD\geq BC+AD.$

Compute $$\sum_{n=1}^{\infty} \frac{2^{n+1}}{8 \cdot 4^n - 6 \cdot 2^n +1}$$

Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that \[ \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.\]

Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$? $ \textbf{(A) }0 \qquad \textbf{(B) }2 \qquad \textbf{(C) }4 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad $

Let $a_1, a_2,...$ a sequence of real numbers. For each positive integer $n$, we denote $m_n =\frac{a_1 + a_2 +... + a_n}{n}$. It is known that there exists a real number $c$ such that for any different positive integers $i, j, k$: $(i - j) m_k + (j - k) m_i + (k - i) m_j = c$. Prove that the sequence $a_1, a_2,..$ is arithmetic

The quadrilateral $ABCD$ is inscribed in a circle and has perpendicular diagonals. Points $K,L,M,Q$ are the points of intersection of the altitudes of the triangles $ABD, ACD, BCD, ABC$, respectively. Prove that the quadrilateral $KLMQ$ is equal to the quadrilateral $ABCD$. (Rozhkova Maria)