[b]p1.[/b] Suppose $a_1 \cdot 2 = a_2 \cdot 3 = a_3$ and $a_1 + a_2 + a_3 = 66$. What is $a_3$? [b]p2.[/b] Ankit buys a see-through plastic cylindrical water bottle. However, in coming home, he accidentally hits the bottle against a wall and dents the top portion of the bottle (above the $7$ cm mark). Ankit now wants to determine the volume of the bottle. The area of the base of the bottle is $20$ cm$^2$ . He fills the bottle with water up to the $5$ cm mark. After flipping the bottle upside down, he notices that the height of the empty space is at the $7$ cm mark. Find the total volume (in cm$^3$) of this bottle. [img]https://cdn.artofproblemsolving.com/attachments/1/9/f5735c77b056aaf31b337ea1b777a591807819.png[/img] [b]p3.[/b] If $P$ is a quadratic polynomial with leading coefficient $ 1$ such that $P(1) = 1$, $P(2) = 2$, what is $P(10)$? [b]p4.[/b] Let ABC be a triangle with $AB = 1$, $AC = 3$, and $BC = 3$. Let $D$ be a point on $BC$ such that $BD =\frac13$ . What is the ratio of the area of $BAD$ to the area of $CAD$? [b]p5.[/b] A coin is flipped $ 12$ times. What is the probability that the total number of heads equals the total number of tails? Express your answer as a common fraction in lowest terms. [b]p6.[/b] Moor pours $3$ ounces of ginger ale and $ 1$ ounce of lime juice in cup $A$, $3$ ounces of lime juice and $ 1$ ounce of ginger ale in cup $B$, and mixes each cup well. Then he pours $ 1$ ounce of cup $A$ into cup $B$, mixes it well, and pours $ 1$ ounce of cup $B$ into cup $A$. What proportion of cup $A$ is now ginger ale? Express your answer as a common fraction in lowest terms. [b]p7.[/b] Determine the maximum possible area of a right triangle with hypotenuse $7$. Express your answer as a common fraction in lowest terms. [b]p8.[/b] Debbie has six Pusheens: $2$ pink ones, $2$ gray ones, and $2$ blue ones, where Pusheens of the same color are indistinguishable. She sells two Pusheens each to Alice, Bob, and Eve. How many ways are there for her to do so? [b]p9.[/b] How many nonnegative integer pairs $(a, b)$ are there that satisfy $ab = 90 - a - b$? [b]p10.[/b] What is the smallest positive integer $a_1...a_n$ (where $a_1, ... , a_n$ are its digits) such that $9 \cdot a_1 ... a_n = a_n ... a_1$, where $a_1$, $a_n \ne 0$? [b]p11.[/b] Justin is growing three types of Japanese vegetables: wasabi root, daikon and matsutake mushrooms. Wasabi root needs $2$ square meters of land and $4$ gallons of spring water to grow, matsutake mushrooms need $3$ square meters of land and $3$ gallons of spring water, and daikon need $ 1$ square meter of land and $ 1$ gallon of spring water to grow. Wasabi sell for $60$ per root, matsutake mushrooms sell for $60$ per mushroom, and daikon sell for $2$ per root. If Justin has $500$ gallons of spring water and $400$ square meters of land, what is the maximum amount of money, in dollars, he can make? [b]p12.[/b] A [i]prim [/i] number is a number that is prime if its last digit is removed. A [i]rime [/i] number is a number that is prime if its first digit is removed. Determine how many numbers between $100$ and $999$ inclusive are both prim and rime numbers. [b]p13.[/b] Consider a cube. Each corner is the intersection of three edges; slice off each of these corners through the midpoints of the edges, obtaining the shape below. If we start with a $2\times 2\times 2$ cube, what is the volume of the resulting solid? [img]https://cdn.artofproblemsolving.com/attachments/4/8/856814bf99e6f28844514158344477f6435a3a.png[/img] [b]p14.[/b] If a parallelogram with perimeter $14$ and area $ 12$ is inscribed in a circle, what is the radius of the circle? [b]p15.[/b] Take a square $ABCD$ of side length $1$, and draw $\overline{AC}$. Point $E$ lies on $\overline{BC}$ such that $\overline{AE}$ bisects $\angle BAC$. What is the length of $BE$? [b]p16.[/b] How many integer solutions does $f(x) = (x^2 + 1)(x^2 + 2) + (x^2 + 3)(x + 4) = 2017$ have? [b]p17.[/b] Alice, Bob, Carol, and Dave stand in a circle. Simultaneously, each player selects another player at random and points at that person, who must then sit down. What is the probability that Alice is the only person who remains standing? [b]p18.[/b] Let $x$ be a positive integer with a remainder of $2$ when divided by $3$, $3$ when divided by $4$, $4$ when divided by $5$, and $5$ when divided by $6$. What is the smallest possible such $x$? [b]p19[/b]. A circle is inscribed in an isosceles trapezoid such that all four sides of the trapezoid are tangent to the circle. If the radius of the circle is $ 1$, and the upper base of the trapezoid is $ 1$, what is the area of the trapezoid? [b]p20.[/b] Ray is blindfolded and standing $ 1$ step away from an ice cream stand. Every second, he has a $1/4$ probability of walking $ 1$ step towards the ice cream stand, and a $3/4$ probability of walking $ 1$ step away from the ice cream stand. When he is $0$ steps away from the ice cream stand, he wins. What is the probability that Ray eventually wins? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For which complex numbers $\alpha$ does there exist a completely multiplicative, complex-valued arithmetic function $f$ such that \[ \sum_{n<x}f(n)=\alpha x+O(1)\,\,? \]

The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$? ${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $

Let $S = \frac{1}{a_1}+\frac{2}{a_2}+ ... +\frac{100}{a_{100}}$ where $a_1, a_2,..., a_{100}$ are positive integers. What are all the possible integer values that $S$ can take ?

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]Z15.[/b] Let $AOB$ be a quarter circle with center $O$ and radius $4$. Let $\omega_1$ and $\omega_2$ be semicircles inside $AOB$ with diameters $OA$ and $OB$, respectively. Find the area of the region within $AOB$ but outside of $\omega_1$ and $\omega_2$. [u]Set 4[/u] [b]Z16.[/b] Integers $a, b, c$ form a geometric sequence with an integer common ratio. If $c = a + 56$, find $b$. [b]Z17 / D24.[/b] In parallelogram $ABCD$, $\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o$ where all angles are in degrees. Find the value of $\angle C$. [b]Z18.[/b] Steven likes arranging his rocks. A mountain formation is where the sequence of rocks to the left of the tallest rock increase in height while the sequence of rocks to the right of the tallest rock decrease in height. If his rocks are $1, 2, . . . , 10$ inches in height, how many mountain formations are possible? For example: the sequences $(1-3-5-6-10-9-8-7-4-2)$ and $(1-2-3-4-5-6-7-8-9-10)$ are considered mountain formations. [b]Z19.[/b] Find the smallest $5$-digit multiple of $11$ whose sum of digits is $15$. [b]Z20.[/b] Two circles, $\omega_1$ and $\omega_2$, have radii of $2$ and $8$, respectively, and are externally tangent at point $P$. Line $\ell$ is tangent to the two circles, intersecting $\omega_1$ at $A$ and $\omega_2$ at $B$. Line $m$ passes through $P$ and is tangent to both circles. If line $m$ intersects line $\ell$ at point $Q$, calculate the length of $P Q$. [u]Set 5[/u] [b]Z21.[/b] Sen picks a random $1$ million digit integer. Each digit of the integer is placed into a list. The probability that the last digit of the integer is strictly greater than twice the median of the digit list is closest to $\frac{1}{a}$, for some integer $a$. What is $a$? [b]Z22.[/b] Let $6$ points be evenly spaced on a circle with center $O$, and let $S$ be a set of $7$ points: the $6$ points on the circle and $O$. How many equilateral polygons (not self-intersecting and not necessarily convex) can be formed using some subset of $S$ as vertices? [b]Z23.[/b] For a positive integer $n$, define $r_n$ recursively as follows: $r_n = r^2_{n-1} + r^2_{n-2} + ... + r^2_0$,where $r_0 = 1$. Find the greatest integer less than $$\frac{r_2}{r^2_1}+\frac{r_3}{r^2_2}+ ...+\frac{r_{2023}}{r^2_{2022}}.$$ [b]Z24.[/b] Arnav starts at $21$ on the number line. Every minute, if he was at $n$, he randomly teleports to $2n^2$, $n^2$, or $\frac{n^2}{4}$ with equal chance. What is the probability that Arnav only ever steps on integers? [b]Z25.[/b] Let $ABCD$ be a rectangle inscribed in circle $\omega$ with $AB = 10$. If $P$ is the intersection of the tangents to $\omega$ at $C$ and $D$, what is the minimum distance from $P$ to $AB$? PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here [/url]and D.16-30/Z.9-14, 17, 26-30 [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a<b<c$ be three positive integers. Prove that among any $2c$ consecutive positive integers there exist three different numbers $x,y,z$ such that $abc$ divides $xyz$.

Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

a) Prove that every sub-group $(A,+)$ of group $(\mathbb{Z},+)$ is in the form $A=n \cdot \mathbb{Z}$ for some $n \in \mathbb{Z}$ where $n \cdot \mathbb{Z}=\{n \cdot x/x\in\mathbb{Z}\}$. b) Using problem (a) , prove that the greatest common divisor $d$ of non zero integers $a_1, a_2,... ,a_n$ is given by relation $d=\lambda_1a_1+\lambda_2 a_2+...\lambda_n a_n$ with $\lambda_i\in\mathbb{Z}, \,\, i=1,2,...,n$

Find all pairs of positive integers $(x, y)$ so that $$(xy - 6)^2 | x^2 + y^2$$

For any positive integer $m\ge2$ define $A_m=\{m+1, 3m+2, 5m+3, 7m+4, \ldots, (2k-1)m + k, \ldots\}$. (1) For every $m\ge2$, prove that there exists a positive integer $a$ that satisfies $1\le a<m$ and $2^a\in A_m$ or $2^a+1\in A_m$. (2) For a certain $m\ge2$, let $a, b$ be positive integers that satisfy $2^a\in A_m$, $2^b+1\in A_m$. Let $a_0, b_0$ be the least such pair $a, b$. Find the relation between $a_0$ and $b_0$.

Consider the sequence given by $a_0 = 3$ and such that for $i \ge 1$, we have $ai = 2^{a_{i-1}} + 1$. Let $m$ be the smallest integer such that $a^3_3$ divides $a_m$. Let $m'$ the smallest integer such that $a^3_m$ divides $a_{m'}$ . Find the value of $m'$.

Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$. .

Find all positive integers $n$ with the folowing property: for all triples ($a$,$b$,$c$) of positive real there is a triple of non negative integers ($l$,$j$,$k$) such that $an^k$, $bn^j$ and $cn^l$ are sides of a non degenate triangle

Is there an integer $n$ such that $n^2 + n + 1$ is divisible by $1955$ ?

Find all positive primes of the form $4x^4 + 1$, for $x$ an integer.

Suppose $p < q < r < s$ are prime numbers such that $pqrs + 1 = 4^{p+q}$. Find $r + s$.

Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$. (H. Nestra)

Given $a, b, p$ arbitrary integers. Prove that there always exist relatively prime (i.e. that have no common divisor) $k$ and $l$, that $(ak + bl)$ is divisible by $p$.

Let $p > 3$ be a prime such that $p\equiv 3 \pmod 4.$ Given a positive integer $a_0$ define the sequence $a_0, a_1, \ldots $ of integers by $a_n = a^{2^n}_{n-1}$ for all $n = 1, 2,\ldots.$ Prove that it is possible to choose $a_0$ such that the subsequence $a_N , a_{N+1}, a_{N+2}, \ldots $ is not constant modulo $p$ for any positive integer $N.$

[b]p1.[/b] Ben and his dog are walking on a path around a lake. The path is a loop $500$ meters around. Suddenly the dog runs away with velocity $10$ km/hour. Ben runs after it with velocity $8$ km/hour. At the moment when the dog is $250$ meters ahead of him, Ben turns around and runs at the same speed in the opposite direction until he meets the dog. For how many minutes does Ben run? [b]p2.[/b] The six interior angles in two triangles are measured. One triangle is obtuse (i.e. has an angle larger than $90^o$) and the other is acute (all angles less than $90^o$). Four angles measure $120^o$, $80^o$, $55^o$ and $10^o$. What is the measure of the smallest angle of the acute triangle? [b]p3.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p4.[/b] Two three-digit whole numbers are called relatives if they are not the same, but are written using the same triple of digits. For instance, $244$ and $424$ are relatives. What is the minimal number of relatives that a three-digit whole number can have if the sum of its digits is $10$? [b]p5.[/b] Three girls, Ann, Kelly, and Kathy came to a birthday party. One of the girls wore a red dress, another wore a blue dress, and the last wore a white dress. When asked the next day, one girl said that Kelly wore a red dress, another said that Ann did not wear a red dress, the last said that Kathy did not wear a blue dress. One of the girls was truthful, while the other two lied. Which statement was true? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 5[/u] [b]p13.[/b] Pieck the Frog hops on Pascal’s Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p14.[/b] Maisy chooses a random set $(x, y)$ that satisfies $$x^2 + y^2 -26x -10y \le 482.$$ The probability that $y>0$ can be expressed as $\frac{A\pi -B\sqrt{C}}{D \pi}$. Find $A+B +C +D$. [color=#f00]Due to the problem having a typo, all teams who inputted answers received points[/color] [b]p15.[/b] $6$ points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). [u]Round 6[/u] [b]p16.[/b] Find the number of $3$ by $3$ grids such that each square in the grid is colored white or black and no two black squares share an edge. [b]p17.[/b] Let $ABC$ be a triangle with side lengths $AB = 20$, $BC = 25$, and $AC = 15$. Let $D$ be the point on BC such that $CD = 4$. Let $E$ be the foot of the altitude from $A$ to $BC$. Let $F$ be the intersection of $AE$ with the circle of radius $7$ centered at $A$ such that $F$ is outside of triangle $ABC$. $DF$ can be expressed as $\sqrt{m}$, where $m$ is a positive integer. Find $m$. [b]p18.[/b] Bill and Frank were arrested under suspicion for committing a crime and face the classic Prisoner’s Dilemma. They are both given the choice whether to rat out the other and walk away, leaving their partner to face a $9$ year prison sentence. Given that neither of them talk, they both face a $3$ year sentence. If both of them talk, they both will serve a $6$ year sentence. Both Bill and Frank talk or do not talk with the same probabilities. Given the probability that at least one of them talks is $\frac{11}{36}$ , find the expected duration of Bill’s sentence in months. [u]Round 7[/u] [b]p19.[/b] Rectangle $ABCD$ has point $E$ on side $\overline{CD}$. Point $F$ is the intersection of $\overline{AC}$ and $\overline{BE}$. Given that the area of $\vartriangle AFB$ is $175$ and the area of $\vartriangle CFE$ is $28$, find the area of $ADEF$. [b]p20.[/b] Real numbers $x, y$, and $z$ satisfy the system of equations $$5x+ 13y -z = 100,$$ $$25x^2 +169y^2 -z2 +130x y= 16000,$$ $$80x +208y-2z = 2020.$$ Find the value of $x yz$. [color=#f00]Due to the problem having infinitely many solutions, all teams who inputted answers received points. [/color] [b]p21.[/b] Bob is standing at the number $1$ on the number line. If Bob is standing at the number $n$, he can move to $n +1$, $n +2$, or $n +4$. In howmany different ways can he move to the number $10$? [u]Round 8[/u] [b]p22.[/b] A sequence $a_1,a_2,a_3, ...$ of positive integers is defined such that $a_1 = 4$, and for each integer $k \ge 2$, $$2(a_{k-1} +a_k +a_{k+1}) = a_ka_{k-1} +8.$$ Given that $a_6 = 488$, find $a_2 +a_3 +a_4 +a_5$. [b]p23.[/b] $\overline{PQ}$ is a diameter of circle $\omega$ with radius $1$ and center $O$. Let $A$ be a point such that $AP$ is tangent to $\omega$. Let $\gamma$ be a circle with diameter $AP$. Let $A'$ be where $AQ$ hits the circle with diameter $AP$ and $A''$ be where $AO$ hits the circle with diameter $OP$. Let $A'A''$ hit $PQ$ at $R$. Given that the value of the length $RA'$ is is always less than $k$ and $k$ is minimized, find the greatest integer less than or equal to $1000k$. [b]p24.[/b] You have cards numbered $1,2,3, ... ,100$ all in a line, in that order. You may swap any two adjacent cards at any time. Given that you make ${100 \choose 2}$ total swaps, where you swap each distinct pair of cards exactly once, and do not do any swaps simultaneously, find the total number of distinct possible final orderings of the cards. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a fixed prime number. Jomland has $p$ cities labelled $0,1,\dots,p-1$. Navi is a traveller and JomAirlines only has flights between two cities with labels $a$ and $b$ (flights are available in both directions) iff there exist positive integers $x$ and $y$ such that \[ \begin{cases} a \equiv x^2 + 2025xy + y^2\pmod{p}\\ b \equiv 20x^2 + xy + 25y^2\pmod{p} \end{cases} \] Prove that: i) There exist infinitely many primes $p$ such that there exist $2$ cities where Navi cannot start from one city and get to the other through a sequence of flights; ii) There exist infinitely many primes $p$ such that for any $2$ cities, Navi can start from one city and get to the other through a sequence of flights. [i](Proposed by Ivan Chan Guan Yu)[/i]

Given $a$ and $b$ distinct positive integers, show that the system of equations $x y +zw = a$ $xz + yw = b$ has only finitely many solutions in integers $x, y, z,w$.

An infinite increasing arithmetical progression consists of positive integers and contains a perfect cube. Prove that this progression also contains a term which is a perfect cube but not a perfect square.