Vishal starts with $n$ copies of the number $1$ written on the board. Every minute, he takes two numbers $a, b$ and replaces them with either $a+b$ or $\min(a^2, b^2)$. After $n-1$ there is $1$ number on the board. Let the maximal possible value of this number be $f(n)$. Prove $2^{n/3}<f(n)\leq 3^{n/3}$.

Let \( A \) be a \( 2 \times 2 \) matrix with integer entries and \(\det A \neq 0\). If the sequence \(\operatorname{tr}(A^n)\), for \( n = 1, 2, 3, \ldots \), is bounded, show that \[ A^{12} = I \quad \text{or} \quad (A^2 - I)^2 = O. \] Here, \( I \) and \( O \) denote the identity and zero matrices, respectively, and \(\operatorname{tr}\) denotes the trace of the matrix (the sum of the elements on the main diagonal).

Let $a=256$. Find the unique real number $x>a^2$ such that \[\log_a \log_a \log_a x = \log_{a^2} \log_{a^2} \log_{a^2} x.\] [i]Proposed by James Lin.[/i]

$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that $$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$ [i]Proposed by Ali Partofard[/i]

The best part of grandma’s $18$ cm $\times 36$ cm rectangle-shaped cake is the chocolate covering on the edges. Her three grandchildren would like to split the cake between each other so that everyone gets the same amount (of the area) of the cake, and they all get the same amount of the delicious perimeter too. a) Can they cut the cake into three convex pieces like that? b) The next time grandma baked this cake, the whole family wanted to try it so they had to cut the cake into six convex pieces this way. Is this possible? c) Soon the entire neighbourhood has heard of the delicious cake. Can the cake be cut into $12$ convex pieces with the same conditions?

Let $ G$ be a connected graph with $ n$ vertices and $ m$ edges such that each edge is contained in at least one triangle. Find the minimum value of $ m$.

Given an integer $k>1$, show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$, all greater than $1$, such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$-th powers of some integers. (Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$.)

On the plane are given $ k\plus{}n$ distinct lines , where $ k>1$ is integer and $ n$ is integer as well.Any three of these lines do not pass through the same point . Among these lines exactly $ k$ are parallel and all the other $ n$ lines intersect each other.All $ k\plus{}n$ lines define on the plane a partition of triangular , polygonic or not bounded regions. Two regions are colled different, if the have not common points or if they have common points only on their boundary.A regions is called ''good'' if it contained in a zone between two parallel lines . If in a such given configuration the minimum number of ''good'' regionrs is $ 176$ and the maximum number of these regions is $ 221$, find $ k$ and $ n$. Babis

Let $n$ and $k$ be natural numbers such that $n = 3k +2$. Show that the sum of all factors of $n$ is divisible by $3$.

Ten points are marked in the plane so that no three of them lie on a line. Each pair of points is connected with a segment. Each of these segments is painted with one of $k$ colors, in such a way that for any $k$ of the ten points, there are $k$ segments each joining two of them and no two being painted with the same color. Determine all integers $k$, $1\leq k\leq 10$, for which this is possible.

Prove that the number $ 9^{(a_1 + a_2)(a_2 + a_3)(a_3 + a_4)...(a_{98} + a_{99})(a_{99} + a_1)}$ − $1$ is divisible by $10$, for any choice of positive integers $a_1, a_2, a_3, . . . , a_{99}$.

Let $\Gamma_1$, $\Gamma_2$ be two circles, where$ \Gamma_1$ has a smaller radius, intersect at two points $A$ and $B$. Points $C, D$ lie on $\Gamma_1$, $\Gamma_2$ respectively so that the point $A$ is the midpoint of the segment $CD$ . Line$ CB$ intersects the circle $\Gamma_2$ for the second time at the point $F$, line $DB$ intersects the circle $\Gamma_1$ for the second time at the point $E$. The perpendicular bisectors of the segments $CD$ and $EF$ intersect at a point $P$. Knowing that $CA =12$ and $PE = 5$ , find $AP$.

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Somya has a football game $4$ days from today. If the day before yesterday was Wednesday, what day of the week is the game? [b]p2.[/b] Sammy writes the following equation: $$\frac{2 + 2}{8 + 8}=\frac{x}{8}.$$ What is the value of $x$ in Sammy's equation? [b]p3.[/b] On $\pi$ day, Peter buys $7$ pies. The pies costed $\$3$, $\$1$, $\$4$, $\$1$, $\$5$, $\$9$, and $\$2$. What was the median price of Peter's $7$ pies in dollars? [b]p4.[/b] Antonio draws a line on the coordinate plane. If the line passes through the points ($1, 3$) and ($-1,-1$), what is slope of the line? [b]p5.[/b] Professor Varun has $25$ students in his science class. He divides his students into the maximum possible number of groups of $4$, but $x$ students are left over. What is $x$? [b]p6.[/b] Evaluate the following: $$4 \times 5 \div 6 \times 3 \div \frac47$$ [b]p7.[/b] Jonny, a geometry expert, draws many rectangles with perimeter $16$. What is the area of the largest possible rectangle he can draw? [b]p8.[/b] David always drives at $60$ miles per hour. Today, he begins his trip to MIT by driving $60$ miles. He stops to take a $20$ minute lunch break and then drives for another $30$ miles to reach the campus. What is the total time in minutes he spends getting to MIT? [b]p9.[/b] Richard has $5$ hats: blue, green, orange, red, and purple. Richard also has 5 shirts of the same colors: blue, green, orange, red, and purple. If Richard needs a shirt and a hat of different colors, how many outts can he wear? [b]p10.[/b] Poonam has $9$ numbers in her bag: $1, 2, 3, 4, 5, 6, 7, 8, 9$. Eric runs by with the number $36$. How many of Poonam's numbers evenly divide Eric's number? [b]p11.[/b] Serena drives at $45$ miles per hour. If her car runs at $6$ miles per gallon, and each gallon of gas costs $2$ dollars, how many dollars does she spend on gas for a $135$ mile trip? [b]p12.[/b] Grace is thinking of two integers. Emmie observes that the sum of the two numbers is $56$ but the difference of the two numbers is $30$. What is the sum of the squares of Grace's two numbers? [b]p13.[/b] Chang stands at the point ($3,-3$). Fang stands at ($-3, 3$). Wang stands in-between Chang and Fang; Wang is twice as close to Fang as to Chang. What is the ordered pair that Wang stands at? [b]p14.[/b] Nithin has a right triangle. The longest side has length $37$ inches. If one of the shorter sides has length $12$ inches, what is the perimeter of the triangle in inches? [b]p15.[/b] Dora has $2$ red socks, $2$ blue socks, $2$ green socks, $2$ purple socks, $3$ black socks, and $4$ gray socks. After a long snowstorm, her family loses electricity. She picks socks one-by-one from the drawer in the dark. How many socks does she have to pick to guarantee a pair of socks that are the same color? [b]p16.[/b] Justin selects a random positive $2$-digit integer. What is the probability that the sum of the two digits of Justin's number equals $11$? [b]p17.[/b] Eddie correctly computes $1! + 2! + .. + 9! + 10!$. What is the remainder when Eddie's sum is divided by $80$? [b]p18.[/b] $\vartriangle PQR$ is drawn such that the distance from $P$ to $\overline{QR}$ is $3$, the distance from $Q$ to $\overline{PR}$ is $4$, and the distance from $R$ to $\overline{PQ}$ is $5$. The angle bisector of $\angle PQR$ and the angle bisector of $\angle PRQ$ intersect at $I$. What is the distance from $I$ to $\overline{PR}$? [b]p19.[/b] Maxwell graphs the quadrilateral $|x - 2| + |y + 2| = 6$. What is the area of the quadrilateral? [b]p20.[/b] Uncle Gowri hits a speed bump on his way to the hospital. At the hospital, patients who get a rare disease are given the option to choose treatment $A$ or treatment $B$. Treatment $A$ will cure the disease $\frac34$ of the time, but since the treatment is more expensive, only $\frac{8}{25}$ of the patients will choose this treatment. Treatment $B$ will only cure the disease $\frac{1}{2}$ of the time, but since it is much more aordable, $\frac{17}{25}$ of the patients will end up selecting this treatment. Given that a patient was cured, what is the probability that the patient selected treatment $A$? [b]p21.[/b] In convex quadrilateral $ABCD$, $AC = 28$ and $BD = 15$. Let $P, Q, R, S$ be the midpoints of $AB$, $BC$, $CD$ and $AD$ respectively. Compute $PR^2 + QS^2$. [b]p22.[/b] Charlotte writes the polynomial $p(x) = x^{24} - 6x + 5$. Let its roots be $r_1$, $r_2$, $...$, $r_{24}$. Compute $r^{24}_1 +r^{24}_2 + r^{24}_3 + ... + r^{24}_24$. [b]p23.[/b] In rectangle $ABCD$, $AB = 6$ and $BC = 4$. Let $E$ be a point on $CD$, and let $F$ be the point on $AB$ which lies on the bisector of $\angle BED$. If $FD^2 + EF^2 = 52$, what is the length of $BE$? [b]p24.[/b] In $\vartriangle ABC$, the measure of $\angle A$ is $60^o$ and the measure of $\angle B$ is $45^o$. Let $O$ be the center of the circle that circumscribes $\vartriangle ABC$. Let $I$ be the center of the circle that is inscribed in $\vartriangle ABC$. Finally, let $H$ be the intersection of the $3$ altitudes of the triangle. What is the angle measure of $\angle OIH$ in degrees? [b]p25.[/b] Kaitlyn fully expands the polynomial $(x^2 + x + 1)^{2018}$. How many of the coecients are not divisible by $3$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given acute triangle $\triangle ABC$ in plane $P$, a point $Q$ in space is defined such that $\angle AQB = \angle BQC = \angle CQA = 90^\circ.$ Point $X$ is the point in plane $P$ such that $QX$ is perpendicular to plane $P$. Given $\angle ABC = 40^\circ$ and $\angle ACB = 75^\circ,$ find $\angle AXC.$

In a lake, there are 2024 leaves arranged in a row. Two frogs are positioned, one on the first leaf and the other on the second leaf. Every minute, both frogs jump simultaneously. Each time a frog jumps, it decides whether to jump to the next leaf or to the leaf that is three positions ahead. Is it possible for each leaf to be visited exactly once by exactly one of the frogs?

For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.

Alberto and Barbara are sitting one next to each other in front of a table onto which they arranged in a line $15$ chocolates. Some of them are milk chocolates, while the others are dark chocolates. Starting from Alberto, they play the following game: during their turn, each player eats a positive number of consecutive chocolates, starting from the leftmost of the remaining ones, so that the number of chocolates eaten that are of the same type as the first one is odd (for example, if after some turns the sequence of the remaining chocolates is $\text{MMDMD},$ where $\text{M}$ stands for $\emph{milk}$ and $\text{D}$ for $\emph{dark},$ the player could either eat the first chocolate, the first $4$ chocolates or all $5$ of them). The player eating the last chocolate wins. Among all $2^{15}$ possible initial sequences of chocolates, how many of them allow Barbara to have a winning strategy?

Which one of the following combinations of given parts does not determine the indicated triangle? $ \textbf{(A)}\ \text{base angle and vertex angle; isosceles triangle}$ $ \textbf{(B)}\ \text{vertex angle and the base; isosceles triangle}$ $ \textbf{(C)}\ \text{the radius of the circumscribed circle; equilateral triangle}$ $ \textbf{(D)}\ \text{one arm and the radius of the inscribed circle; right triangle}$ $ \textbf{(E)}\ \text{two angles and a side opposite one of them; scalene triangle}$

In the acute isosceles triangle $ABC$ the altitudes $BB_1$ and $CC_1$ are drawn, which intersect at the point $H$. Let $L_1$ and $L_2$ be the feet of the angle bisectors of the triangles $B_1AC_1$ and $B_1HC_1$ drawn from vertices $A$ and $H$, respectively. The circumscribed circles of triangles $AHL_1$ and $AHL_2$ intersects the line $B_1C_1$ for the second time at points $P$ and $Q$, respectively. Prove that points $B, C, P$ and $Q$ lie on the same circle. (M. Plotnikov, D. Hilko)

Let $n$ be a positive odd integer. $C$ is a set consists of integral points on a plane, which is defined by \[ C = \{(i, j): i, j = 0, 1, \dots, 2n-1\} \] and forms a $2n \times 2n$ array. On every point there is a Guinea pig, which is facing toward one of the following directions: [i]positive/negative $x$-axis[/i], or [i]positive/negative $y$-axis[/i]. Jeff wants to keep $n^2+1$ of the Guinea pigs on the plane and remove all the others. After that, the Guinea pigs on the plane will move as the following: 1. In every round, the Guinea pigs move toward by an unit, and keep facing the same direction. 2. If a Guinea pig move to a point $(i, j)$ which is [i]not[/i] in $C$, it will further move to another point $(p, q)$ in $C$, such that $p \equiv i \pmod {2n}$ and $q \equiv j \pmod {2n}$. [i](For example, if a Guinea pig move from $(2, 0)$ to $(2, -1)$, it will then further move to $(2, 2n-1)$.)[/i] The next round begins after all the Guinea pigs settle up. Jeff's goal is to keep the appropriate Guinea pigs on the plane, so that in every single round, any two Guinea pigs will never move to the same endpoint, and will never move to the startpoints[i](in that round)[/i] of each other simultaneously. Prove that Jeff can always succeed wherever the Guinea pigs initially face. [i]Proposed by Weijiun Kao[/i] Edit: By the way, it can be proven that the number $n^2+1$ is optimal, i.e. if the Guinea pigs face appropriately, Jeff can only keep at most $n^2+1$ of them on the plane to avoid any collision.

For $n \in \mathbb{N}$, let $P(n)$ denote the product of distinct prime factors of $n$, with $P(1) = 1$. Show that for any $a_0 \in \mathbb{N}$, if we define a sequence $a_{k+1} = a_k + P(a_k)$ for $k \ge 0$, there exists some $k \in \mathbb{N}$ with $a_k/P(a_k) = 2015$.

Points $A$, $B$, and $C$ lie in the plane such that $AB=13$, $BC=14$, and $CA=15$. A peculiar laser is fired from $A$ perpendicular to $\overline{BC}$. After bouncing off $BC$, it travels in a direction perpendicular to $CA$. When it hits $CA$, it travels in a direction perpendicular to $AB$, and after hitting $AB$ its new direction is perpendicular to $BC$ again. If this process is continued indefinitely, the laser path will eventually approach some finite polygonal shape $T_\infty$. What is the ratio of the perimeter of $T_\infty$ to the perimeter of $\triangle ABC$?

We call a finite plane set $S$ consisting of points with integer coefficients a two-neighbour set, if for each point $(p, q)$ of $S$ exactly two of the points $(p +1, q), (p, q +1), (p-1, q), (p, q-1)$ belong to $S$. For which integers $n$ there exists a two-neighbour set which contains exactly $n$ points?

Inside a square with side lengths $1$ a broken line of length $>1000$ without selfintersection is drawn. Show that there is a line parallel to a side of the square that intersects the broken line in at least $501$ points.

How many a) bishops b) horses can be positioned on a chessboard at most, so that no one threatens another?