There is $500$ dollars in a bank. Two bank operations are allowed: to withdraw $300$ dollars from the bank or to deposit $198$ dollars into the bank. These operations can be repeated as many times as necessary but only the money that was initially in the bank can be used. What is the largest amount of money that can be borrowed from the bank? How can this be done? (AK Tolpygo)

A circle cuts off four right-angled triangles from rectangle $ABCD$.Let $A_0, B_0, C_0$ and $D_0$ be the midpoints of the correspondent hypotenuses. Prove that $A_0C_0 = B_0D_0$ [i]Proosed by L.Shteingarts[/i]

For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i \equal{} 1} (A_i \plus{} k)$ is a power for each natural number $ k.$

A triangle whose all sides have length not smaller than $1$ is inscribed in a square of side length $1$. Prove that the center of the square lies inside the triangle or on its boundary.

$16$ chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given $1$ point, for a draw $0.5$ points, for defeat $0$ points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?

On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set. [i]Proposed by Tonći Kokan, Croatia[/i]

$\Delta ABC$ is right-angled at $B$, with $AB=1$ and $BC=3$. $E$ is the foot of perpendicular from $B$ to $AC$. $BA$ and $BE$ are produced to $D$ and $F$ respectively such that $D$, $F$, $C$ are collinear and $\angle DAF=\angle BAC$. Find the length of $AD$.

The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels. (Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.) [i]Proposed by Michael Ma[/i]

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Alisha has $6$ cupcakes and Tyrone has $10$ brownies. Tyrone gives some of his brownies to Alisha so that she has three times as many desserts as Tyrone. How many desserts did Tyrone give to Alisha? [b]p2.[/b] Bisky adds one to her favorite number. She then divides the result by $2$, and gets $56$. What is her favorite number? [b]p3.[/b] What is the maximum number of points at which a circle and a square can intersect? [b]p4.[/b] An integer $N$ leaves a remainder of 66 when divided by $120$. Find the remainder when $N$ is divided by $24$. [b]p5.[/b] $7$ people are chosen to run for student council. How many ways are there to pick $1$ president, $1$ vice president, and $1$ secretary? [b]p6.[/b] Anya, Beth, Chloe, and Dmitri are all close friends, and like to make group chats to talk. How many group chats can be made if Dmitri, the gossip, must always be in the group chat and Anya is never included in them? Group chats must have more than one person. [b]p7.[/b] There exists a telephone pole of height $24$ feet. From the top of this pole, there are two wires reaching the ground in opposite directions, with one wire $25$ feet, and the other wire 40 feet. What is the distance (in feet) between the places where the wires hit the ground? [b]p8.[/b] Tarik is dressing up for a job-interview. He can wear a chill, business, or casual outfit. If he wears a chill oufit, he must wear a t-shirt, shorts, and flip-flops. He has eight of the first, seven of the second, and three of the third. If he wears a business outfit, he must wear a blazer, a tie, and khakis; he has two of the first, six of the second, and five of the third; finally, he can also choose the casual style, for which he has three hoodies, nine jeans, and two pairs of sneakers. How many different combinations are there for his interview? [b]p9.[/b] If a non-degenerate triangle has sides $11$ and $13$, what is the sum of all possibilities for the third side length, given that the third side has integral length? [b]p10.[/b] An unknown disease is spreading fast. For every person who has the this illness, it is spread on to $3$ new people each day. If Mary is the only person with this illness at the start of Monday, how many people will have contracted the illness at the end of Thursday? [b]p11.[/b] Gob the giant takes a walk around the equator on Mars, completing one lap around Mars. If Gob’s head is $\frac{13}{\pi}$ meters above his feet, how much farther (in meters) did his head travel than his feet? [b]p12.[/b] $2022$ leaves a remainder of $2$, $6$, $9$, and $7$ when divided by $4$, $7$, $11$, and $13$ respectively. What is the next positive integer which has the same remainders to these divisors? [b]p13.[/b] In triangle $ABC$, $AB = 20$, $BC = 21$, and $AC = 29$. Let D be a point on $AC$ such that $\angle ABD = 45^o$. If the length of $AD$ can be represented as $\frac{a}{b}$ , what is $a + b$? [b]p14.[/b] Find the number of primes less than $100$ such that when $1$ is added to the prime, the resulting number has $3$ divisors. [b]p15.[/b] What is the coefficient of the term $a^4z^3$ in the expanded form of $(z - 2a)^7$? [b]p16.[/b] Let $\ell$ and $m$ be lines with slopes $-2$, $1$ respectively. Compute $|s_1 \cdot s_2|$ if $s_1$, $s_2$ represent the slopes of the two distinct angle bisectors of $\ell$ and $m$. [b]p17.[/b] R1D2, Lord Byron, and Ryon are creatures from various planets. They are collecting monkeys for King Avanish, who only understands octal (base $8$). R1D2 only understands binary (base $2$), Lord Byron only understands quarternary (base $4$), and Ryon only understands decimal (base $10$). R1D2 says he has $101010101$ monkeys and adds his monkey to the pile. Lord Byron says he has $3231$ monkeys and adds them to the pile. Ryon says he has $576$ monkeys and adds them to the pile. If King Avanish says he has $x$ monkeys, what is the value of $x$? [b]p18.[/b] A quadrilateral is defined by the origin, $(3, 0)$, $(0, 10)$, and the vertex of the graph of $y = x^2 -8x+22$. What is the area of this quadrilateral? [b]p19.[/b] There is a sphere-container, filled to the brim with fruit punch, of diameter $6$. The contents of this container are poured into a rectangular prism container, again filled to the brim, of dimensions $2\pi$ by $4$ by $3$. However, there is an excess amount in the original container. If all the excess drink is poured into conical containers with diameter $4$ and height $3$, how many containers will be used? [b]p20.[/b] Brian is shooting arrows at a target, made of concurrent circles of radius $1$, $2$, $3$, and $4$. He gets $10$ points for hitting the innermost circle, $8$ for hitting between the smallest and second smallest circles, $5$ for between the second and third smallest circles, $2$ points for between the third smallest and outermost circle, and no points for missing the target. Assume for each shot he takes, there is a $20\%$ chance Brian will miss the target, but otherwise the chances of hitting each target are proportional to the area of the region. The chance that after three shots, Brian will have scored $15$ points can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p21.[/b] What is the largest possible integer value of $n$ such that $\frac{2n^3+n^2+7n-15}{2n+1}$ is an integer? [b]p22.[/b] Let $f(x, y) = x^3 + x^2y + xy^2 + y^3$. Compute $f(0, 2) + f(1, 3) +... f(9, 11).$ [b]p23.[/b] Let $\vartriangle ABC$ be a triangle. Let $AM$ be a median from $A$. Let the perpendicular bisector of segment $\overline{AM}$ meet $AB$ and $AC$ at $D$, $E$ respectively. Given that $AE = 7$, $ME = MC$, and $BDEC$ is cyclic, then compute $AM^2$. [b]p24.[/b] Compute the number of ordered triples of positive integers $(a, b, c)$ such that $a \le 10$, $b \le 11$, $c \le 12$ and $a > b - 1$ and $b > c - 1$. [b]p25.[/b] For a positive integer $n$, denote by $\sigma (n)$ the the sum of the positive integer divisors of $n$. Given that $n + \sigma (n)$ is odd, how many possible values of $n$ are there from $1$ to $2022$, inclusive? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For which integers $n\geq 2$, the number $(n-1)^{n^{n+1}}+(n+1)^{n^{n-1}}$ is divisible by $n^{n}$ ?

Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\] in which $a,~ b,~ c$, and $d$ vary over the set of positive integers. (Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)

A random permutation $a = (a_1, a_2,...,a_{40})$ of $(1, 2,...,40)$ is chosen, with all permutations being equally likely. William writes down a $20 \times 20$ grid of numbers $b_{ij}$ such that $b_{ij} = \max (a_i, a_{j+20})$ for all $1 \le i, j \le 20$, but then forgets the original permutation $a$. Compute the probability that, given the values of $b_{ij}$ alone, there are exactly $2$ permutations $a$ consistent with the grid.

For which $n{}$ can we partition a regular $n{}$-gon into finitely many triangles such that no two triangles share a side? [i]Based on a problem of the 2017 Miklós Schweitzer competition[/i]

In the trapezoid $ABCD$ we know that $CD \perp BC, $ and $CD \perp AD .$ Circle $w$ with diameter $AB$ intersects $AD$ in points $A$ and $P,$ tangent from $P$ to $w$ intersects $CD$ at $M.$ The second tangent from $M$ to $w$ touches $w$ at $Q.$ Prove that midpoint of $CD$ lies on $BQ.$

Given a positive integer number $n$, determine the maximum number of edges a triangle-free Hamiltonian simple graph on $n$ vertices may have.

In triangle $ABC$, ($AB \ne AC$), let the orthocenter be $H$, circumcenter be $O$, and the midpoint of $BC$ be $M$. Let $HM \cap AO = D$. Let $P,Q,R,S$ be the midpoints of $AB,CD,AC,BD$. Let $X = PQ\cap RS$. Find $AH/OX$.