[b]p1.[/b] One day, Granny Smith bought a certain number of apples at Horock’s Farm Market. When she returned the next day she found that the price of the apples was reduced by $20\%$. She could therefore buy more apples while spending the same amount as the previous day. How many percent more? [b]p2.[/b] You are asked to move several boxes. You know nothing about the boxes except that each box weighs no more than $10$ tons and their total weight is $100$ tons. You can rent several trucks, each of which can carry no more than $30$ tons. What is the minimal number of trucks you can rent and be sure you will be able to carry all the boxes at once? [b]p3.[/b] The five numbers $1, 2, 3, 4, 5$ are written on a piece of paper. You can select two numbers and increase them by $1$. Then you can again select two numbers and increase those by $1$. You can repeat this operation as many times as you wish. Is it possible to make all numbers equal? [b]p4.[/b] There are $15$ people in the room. Some of them are friends with others. Prove that there is a person who has an even number of friends in the room. [u]Bonus Problem [/u] [b]p5.[/b] Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For some positive integer \( n \), Katya wrote the numbers from \( 1 \) to \( 2^n \) in a row in increasing order. Oleksii rearranged Katya's numbers and wrote the new sequence directly below the first row. Then, they calculated the sum of the two numbers in each column. Katya calculated \( N \), the number of powers of two among the results, while Oleksii calculated \( K \), the number of distinct powers of two among the results. What is the maximum possible value of \( N + K \)? [i]Proposed by Oleksii Masalitin[/i]

Each edge and each diagonal of the convex $ n$-gon $ (n\geq 3)$ is colored in red or blue. Prove that the vertices of the $ n$-gon can be labeled as $ A_1,A_2,...,A_n$ in such a way that one of the following two conditions is satisfied: (a) all segments $ A_1A_2,A_2A_3,...,A_{n\minus{}1}A_n,A_nA_1$ are of the same colour. (b) there exists a number $ k, 1<k<n$ such that the segments $ A_1A_2,A_2A_3,...,A_{k\minus{}1}A_k$ are blue, and the segments $ A_kA_{k\plus{}1},...,A_{n\minus{}1}A_n,A_nA_1$ are red.

$8$ singers take part in a festival. The organiser wants to plan $m$ concerts. For every concert there are $4$ singers who go on stage, with the restriction that the times of which every two singers go on stage in a concert are all equal. Find a schedule that minimises $m$.

A sequence $a_0, a_1, \dots , a_n, \dots$ of positive integers is constructed as follows: [list] [*] If the last digit of $a_n$ is less than or equal to $5$, then this digit is deleted and $a_{n+1}$ is the number consisting of the remaining digits. (If $a_{n+1}$ contains no digits, the process stops.) [*] Otherwise, $a_{n+1}= 9a_n$. [/list] Can one choose $a_0$ so that this sequence is infinite?

Let $C$ be a circle, $A_1 , A_2,\ldots ,A_n$ be distinct points inside $C$ and $B_1 , B_2 ,\ldots ,B_n$ be distinct points on $C$ such that no two of the segments $A_1B_1 , A_2 B_2 ,\ldots ,A_n B_n$ intersect. A grasshopper can jump from $A_r$ to $A_s$ if the line segment $A_r A_s$ does not intersect any line segment $A_t B_t (t \neq r, s)$. Prove that after a certain number of jumps, the grasshopper can jump from any $A_u$ to any $A_v$ .

[u]Round 1[/u] [b]p1.[/b] A box contains $1$ ball labelledW, $1$ ball labelled $E$, $1$ ball labelled $L$, $1$ ball labelled $C$, $1$ ball labelled $O$, $8$ balls labelled $M$, and $1$ last ball labelled $E$. One ball is randomly drawn from the box. The probability that the ball is labelled $E$ is $\frac{1}{a}$ . Find $a$. [b]p2.[/b] Let $$G +E +N = 7$$ $$G +E +O = 15$$ $$N +T = 22.$$ Find the value of $T +O$. [b]p3.[/b] The area of $\vartriangle LMT$ is $22$. Given that $MT = 4$ and that there is a right angle at $M$, find the length of $LM$. [u]Round 2[/u] [b]p4.[/b] Kevin chooses a positive $2$-digit integer, then adds $6$ times its unit digit and subtracts $3$ times its tens digit from itself. Find the greatest common factor of all possible resulting numbers. [b]p5.[/b] Find the maximum possible number of times circle $D$ can intersect pentagon $GRASS'$ over all possible choices of points $G$, $R$, $A$, $S$, and $S'$. [b]p6.[/b] Find the sum of the digits of the integer solution to $(\log_2 x) \cdot (\log_4 \sqrt{x}) = 36$. [u]Round 3[/u] [b]p7.[/b] Given that $x$ and $y$ are positive real numbers such that $x^2 + y = 20$, the maximum possible value of $x + y$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p8.[/b] In $\vartriangle DRK$, $DR = 13$, $DK = 14$, and $RK = 15$. Let $E$ be the point such that $ED = ER = EK$. Find the value of $\lfloor DE +RE +KE \rfloor$. [b]p9.[/b] Subaru the frog lives on lily pad $1$. There is a line of lily pads, numbered $2$, $3$, $4$, $5$, $6$, and $7$. Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either $1$ or $2$ greater, chosen at random from valid possibilities. There are alligators on lily pads $2$ and $5$. If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number $1$. Find how many times Subaru is expected to die before he reaches pad $7$. [u]Round 4[/u] [b]p10.[/b] Find the sum of the following series: $$\sum^{\infty}_{i=1} = \frac{\sum^i_{j=1} j}{2^i}=\frac{1}{2^1}+\frac{1+2}{2^2}+\frac{1+2+3}{2^3}+\frac{1+2+3+4}{2^4}+... $$ [b]p11.[/b] Let $\phi (x)$ be the number of positive integers less than or equal to $x$ that are relatively prime to $x$. Find the sum of all $x$ such that $\phi (\phi(x)) = x -3$. Note that $1$ is relatively prime to every positive integer. [b]p12.[/b] On a piece of paper, Kevin draws a circle. Then, he draws two perpendicular lines. Finally, he draws two perpendicular rays originating from the same point (an $L$ shape). What is the maximum number of sections into which the lines and rays can split the circle? [u]Round 5 [/u] [b]p13.[/b] In quadrilateral $ABCD$, $\angle A = 90^o$, $\angle C = 60^o$, $\angle ABD = 25^o$, and $\angle BDC = 5^o$. Given that $AB = 4\sqrt3$, the area of quadrilateral $ABCD$ can be written as $a\sqrt{b}$. Find $10a +b$. [b]p14.[/b] The value of $$\sum^6_{n=2} \left( \frac{n^4 +1}{n^4 -1}\right) -2 \sum^6_{n=2}\left(\frac{n^3 -n^2+n}{n^4 -1}\right)$$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $100m+n$. [b]p15.[/b] Positive real numbers $x$ and $y$ satisfy the following $2$ equations. $$x^{1+x^{1+x^{1+...}}}= 8$$ $$\sqrt[24]{y +\sqrt[24]{y + \sqrt[24]{y +...}}} = x$$ Find the value of $\lfloor y \rfloor$. PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167130p28823260]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider all the possible subsets of the set $\{1,2,..., N\}$ which do not contain any consecutive numbers. Prove that the sum of the squares of the products of the numbers in these subsets is $(N + 1)! - 1$. (Based on idea of R.P. Stanley)

Find the largest possible number of intersection points of the diagonals of a convex $n$-gon.

The ancient One-Dimensional Empire was located along a straight line. Initially, there were no cities. A total of $n$ different point-like cities were founded one by one; from the second onwards, each newly founded city and the nearest existing city (the older one, if there were two) were declared sister cities. The surviving map of the empire shows the cities and the distances between them, but not the order in which they were founded. Historians have tried to deduce from the map that each city had at most 41 sister cities. [list=a] [*] For $n=10^6$, give a map from which this deduction can be made. [*] Prove that for $n=10^{13}$, this conclusion cannot be drawn from any map. [/list]

Two ants crawl along the sides of the $49$ squares of a $7 * 7$ board. Each ant passes through all $64$ vertices exactly once and returns to its starting point. What is the smallest possible number of sides covered by both ants?

Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.

On a circle are arranged $100$ baskets, each containing at least one candy. The total number of candies is $780$. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad takes all the candies from $9$ consecutive non-empty baskets, while Sevinch takes all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least $700$ candies, regardless of the initial distribution of candies and Sevinch's actions. [i] Shubin Yakov, Russia [/i]

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Five points are given in space, no four of which are coplanar. Each of the segments connecting two of them is painted in white, green or red, so that all the colors are used and no three segments of the same color form a triangle. Prove that among these five points there is one at which segments of all the three colors meet.

On each edge of a graph is written a real number,such that for every even tour of this graph,sum the edges with signs alternatively positive and negative is zero.prove that one can assign to each of the vertices of the graph a real number such that sum of the numbers on two adjacent vertices is the number on the edge between them.(tour is a closed path from the edges of the graph that may have repeated edges or vertices)

For a positive integer $n$, consider a square cake which is divided into $n \times n$ pieces with at most one strawberry on each piece. We say that such a cake is [i]delicious[/i] if both diagonals are fully occupied, and each row and each column has an odd number of strawberries. Find all positive integers $n$ such that there is an $n \times n$ delicious cake with exactly $\left\lceil\frac{n^2}{2}\right\rceil$ strawberries on it.

Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

[list] [*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this? [*] Previous problem for the grid of $7\times 7$ lattice.[/list]

Let $A = \{a_1, a_2, a_3, a_4, a_5\}$ be a set of $5$ positive integers. Show that for any rearrangement of $A$, $a_{i1}$, $a_{i2}$, $a_{i3}$, $a_{i4}$, $a_{i5}$, the product $$(a_{i1} -a_1) (a_{i2} -a_2) (a_{i3} -a_3) (a_{i4} -a_4) (a_{i5} -a_5)$$ is always even.

Numbers $1,2,\ldots,1997^2$ are written in the cells of a $1997\times1997$ table. It is allowed to apply the following transformations: exchange places of any two rows or any two columns, or reverse a row or column. (When a row or column is reversed, the first and last entry exchange their positions, so do the second and second last, etc.) Is it possible that, after finitely many such transformations, arbitrary two numbers exchange their positions and no other number changes its position?

Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins. $(a)$ Determine which player has a winning strategy for $N=2005$. $(b)$ Determine which player has a winning strategy for $N=2004$. $(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

In a school six different courses are taught: mathematics, physics, biology, music, history, geography. The students were required to rank these courses according to their preferences, where equal preferences were allowed. It turned out that: [list] [*][b](i)[/b] mathematics was ranked among the most preferred courses by all students; [*][b](ii)[/b] no student ranked music among the least preferred ones; [*][b](iii) [/b]all students preferred history to geography and physics to biology; and [*][b](iv)[/b] no two rankings were the same. [/list] Find the greatest possible value for the number of students in this school.

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom