Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base $10$) positive integers $\underline{a}\, \underline{b} \, \underline{c}$, if $\underline{a} \, \underline{b} \, \underline{c}$ is a multiple of $x$, then the three-digit (base $10$) number $\underline{b} \, \underline{c} \, \underline{a}$ is also a multiple of $x$.

Let $a_1, a_2,\cdots
,a_n$ be positive integers and let $a$ be an integer greater than $1$ and divisible by the product $a_1a_2\cdots a_n$. Prove that $a^{n+1} + a-1$ is not divisible by the product $(a + a_1 - 1)(a + a_2 - 1) \cdots
(a + a_n - 1)$.

Let $ x \equal{} \sqrt[3]{\frac{4}{25}}\,$. There is a unique value of $ y$ such that $ 0 < y < x$ and $ x^x \equal{} y^y$. What is the value of $ y$? Express your answer in the form $ \sqrt[c]{\frac{a}{b}}\,$, where $ a$ and $ b$ are relatively prime positive integers and $ c$ is a prime number.

Let $\sigma$ be a permutation of the set $S := \{1, 2, \ldots , 100\},$ such that $\sigma(a+b) \equiv \sigma(a)+\sigma(b) \pmod{100}$ whenever $a, b, a + b \in S.$ Denote by $f(s)$ the sum of the distinct values $\sigma(s)$ can take over all possible $\sigma$s satisfying the given condition. What is the nonnegative difference between the maximum and minimum value $f$ takes on when ranging over all $s \in S$?

Consider $p$ a prime number and $p$ consecutive positive integers $m_{1}, m_{2}, \ldots, m_{p}$. Choose a permutation $\sigma$ of $1, 2, \ldots, p$. Show that there exist two different numbers $k,l \in \{1,2, \ldots, p\}$ such that $m_{k}m_{\sigma(k)}-m_{l}m_{\sigma(l)}$ is divisible by $p$.

For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$. (b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.

For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have \[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \] and determine when equality holds. [i]Wenyu Cao.[/i]

[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].

Find all primes that can be written both as a sum of two primes and as a difference of two primes.

Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p\in P}p\right)-1$ are also in $S$. Determine all possible such sets $S$.

Given positive integers $x$ and $y$ such that $xy^2$ is a perfect cube, prove that $x^2y$ is also a perfect cube.

We will say that two positive integers $a$ and $b$ form a [i]suitable pair[/i] if $a+b$ divides $ab$ (its sum divides its multiplication). Find $24$ positive integers that can be distribute into $12$ suitable pairs, and so that each integer number appears in only one pair and the largest of the $24$ numbers is as small as possible.

For a given odd prime number $p$, define $f(n)$ the remainder of $d$ divided by $p$, where $d$ is the biggest divisor of $n$ which is not a multiple of $p$. For example when $p=5$, $f(6)=1, f(35)=2, f(75)=3$. Define the sequence $a_1, a_2, \ldots, a_n, \ldots$ of integers as the followings: [list] [*]$a_1=1$ [*]$a_{n+1}=a_n+(-1)^{f(n)+1}$ for all positive integers $n$. [/list] Determine all integers $m$, such that there exist infinitely many positive integers $k$ such that $m=a_k$.

[b]p1.[/b] Sujay sees a shooting star go across the night sky, and took a picture of it. The shooting star consists of a star body, which is bounded by four quarter-circle arcs, and a triangular tail. Suppose $AB = 2$, $AC = 4$. Let the area of the shooting star be $X$. If $6X = a-b\pi$ for positive integers $a, b$, find $a + b$. [img]https://cdn.artofproblemsolving.com/attachments/0/f/f9c9ff23416565760df225c133330e795b9076.png[/img] [b]p2.[/b] Assuming that each distinct arrangement of the letters in $DISCUSSIONS$ is equally likely to occur, what is the probability that a random arrangement of the letters in $DISCUSSIONS$ has all the $S$’s together? [b]p3.[/b] Evaluate $$\frac{(1 + 2022)(1 + 2022^2)(1 + 2022^4) ... (1 + 2022^{2^{2022}})}{1 + 2022 + 2022^2 + ... + 2022^{2^{2023}-1}} .$$ [b]p4.[/b] Dr. Kraines has $27$ unit cubes, each of which has one side painted red while the other five are white. If he assembles his cubes into one $3 \times 3 \times 3$ cube by placing each unit cube in a random orientation, what is the probability that the entire surface of the cube will be white, with no red faces visible? If the answer is $2^a3^b5^c$ for integers $a$, $b$, $c$, find $|a + b + c|$. [b]p5.[/b] Let S be a subset of $\{1, 2, 3, ... , 1000, 1001\}$ such that no two elements of $S$ have a difference of $4$ or $7$. What is the largest number of elements $S$ can have? [b]p6.[/b] George writes the number $1$. At each iteration, he removes the number $x$ written and instead writes either $4x+1$ or $8x+1$. He does this until $x > 1000$, after which the game ends. What is the minimum possible value of the last number George writes? [b]p7.[/b] List all positive integer ordered pairs $(a, b)$ satisfying $a^4 + 4b^4 = 281 \cdot 61$. [b]p8.[/b] Karthik the farmer is trying to protect his crops from a wildfire. Karthik’s land is a $5 \times 6$ rectangle divided into $30$ smaller square plots. The $5$ plots on the left edge contain fire, the $5$ plots on the right edge contain blueberry trees, and the other $5 \times 4$ plots of land contain banana bushes. Fire will repeatedly spread to all squares with bushes or trees that share a side with a square with fire. How many ways can Karthik replace $5$ of his $20$ plots of banana bushes with firebreaks so that fire will not consume any of his prized blueberry trees? [b]p9.[/b] Find $a_0 \in R$ such that the sequence $\{a_n\}^{\infty}_{n=0}$ defined by $a_{n+1} = -3a_n + 2^n$ is strictly increasing. [b]p10.[/b] Jonathan is playing with his life savings. He lines up a penny, nickel, dime, quarter, and half-dollar from left to right. At each step, Jonathan takes the leftmost coin at position $1$ and uniformly chooses a position $2 \le k \le 5$. He then moves the coin to position $k$, shifting all coins at positions $2$ through $k$ leftward. What is the expected number of steps it takes for the half-dollar to leave and subsequently return to position $5$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S_n$ be the sum of the first $n$ prime numbers. For example, \[ S_5 = 2 + 3 + 5 + 7 + 11 = 28.\] Does there exist an integer $k$ such that $S_{2023} < k^2 < S_{2024}$?

The numbers $25$ and $76$ have the property that when squared in base 10, their squares also end in the same two digits. A positive integer that has at most $3$ digits when expressed in base 21 and also has the property that its base $21$ square ends in the same $3$ digits is called amazing. Find the sum of all amazing numbers. Express your answer in base $21$.

A three-digit natural number is called [i]tricubic [/i] if it is equal to the sum of the cubes of its digits. Find all pairs of consecutive numbers such that both are tricubic.

Find the number of positive integers $n < 100$ such that $\gcd(n^2,2023) \neq \gcd(n,2023^2).$

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

Determine the number of ordered triples $(a, b, c)$, with $0 \le a, b, c \le 10$ for which there exists $(x, y)$ such that $ax^2 + by^2 \equiv c$ (mod $11$)

p1. Four kite-shaped shapes as shown below ($a > b$, $a$ and $b$ are natural numbers less than $10$) arranged in such a way so that it forms a square with holes in the middle. The square hole in the middle has a perimeter of $16$ units of length. What is the possible perimeter of the outermost square formed if it is also known that $a$ and $b$ are numbers coprime? [img]https://cdn.artofproblemsolving.com/attachments/4/1/fa95f5f557aa0ca5afb9584d5cee74743dcb10.png[/img] p2. If $a = 3^p$, $b = 3^q$, $c = 3^r$, and $d = 3^s$ and if $p, q, r$, and $s$ are natural numbers, what is the smallest value of $p\cdot q\cdot r\cdot s$ that satisfies $a^2 + b^3 + c^5 = d^7$ 3. Ucok intends to compose a key code (password) consisting of 8 numbers and meet the following conditions: i. The numbers used are $1, 2, 3, 4, 5, 6, 7, 8$, and $9$. ii. The first number used is at least $1$, the second number is at least $2$, third digit-at least $3$, and so on. iii. The same number can be used multiple times. a) How many different passwords can Ucok compose? b) How many different passwords can Ucok make, if provision (iii) is replaced with: no numbers may be used more than once. p 4. For any integer $a, b$, and $c$ applies $a\times (b + c) = (a\times b) + (a\times c)$. a) Look for examples that show that $a + (b\times c)\ne (a + b)\times (a + c)$. b) Is it always true that $a + (b\times c) = (a + b)\times(a + c)$? Justify your answer. p5. The results of a survey of $N$ people with the question whether they maintain dogs, birds, or cats at home are as follows: $50$ people keep birds, $61$ people don't have dogs, $13$ people don't keep a cat, and there are at least $74$ people who keep the most a little two kinds of animals in the house. What is the maximum value and minimum of possible value of $N$ ?

Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$ is a perfect square.

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

Does there exist a natural number $n$ such that $n!$ ends with exactly $2021$ zeros?

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Given that $a + 19b = 3$ and $a + 1019b = 5$, what is $a + 2019b$? [b]p2.[/b] How many multiples of $3$ are there between $2019$ and $2119$, inclusive? [b]p3.[/b] What is the maximum number of quadrilaterals a $12$-sided regular polygon can be quadrangulated into? Here quadrangulate means to cut the polygon along lines from vertex to vertex, none of which intersect inside the polygon, to form pieces which all have exactly $4$ sides. [b]p4.[/b] What is the value of $|2\pi - 7| + |2\pi - 6|$, rounded to the nearest hundredth? [b]p5.[/b] In the town of EMCCxeter, there is a $30\%$ chance that it will snow on Saturday, and independently, a $40\%$ chance that it will snow on Sunday. What is the probability that it snows exactly once that weekend, as a percentage? [b]p6.[/b] Define $n!$ to be the product of all integers between $1$ and $n$ inclusive. Compute $\frac{2019!}{2017!} \times \frac{2016!}{2018!}$ . [b]p7.[/b] There are $2019$ people standing in a row, and they are given positions $1$, $2$, $3$, $...$, $2019$ from left to right. Next, everyone in an odd position simultaneously leaves the row, and the remaining people are assigned new positions from $1$ to $1009$, again from left to right. This process is then repeated until one person remains. What was this person's original position? [b]p8.[/b] The product $1234\times 4321$ contains exactly one digit not in the set $\{1, 2, 3, 4\}$. What is this digit? [b]p9.[/b] A quadrilateral with positive area has four integer side lengths, with shortest side $1$ and longest side $9$. How many possible perimeters can this quadrilateral have? [b]p10.[/b] Define $s(n)$ to be the sum of the digits of $n$ when expressed in base $10$, and let $\gamma (n)$ be the sum of $s(d)$ over all natural number divisors $d$ of $n$. For instance, $n = 11$ has two divisors, $1$ and $11$, so $\gamma (11) = s(1) + s(11) = 1 + (1 + 1) = 3$. Find the value of $\gamma (2019)$. [b]p11.[/b] How many five-digit positive integers are divisible by $9$ and have $3$ as the tens digit? [b]p12.[/b] Adam owns a large rectangular block of cheese, that has a square base of side length $15$ inches, and a height of $4$ inches. He wants to remove a cylindrical cheese chunk of height $4$, by making a circular hole that goes through the top and bottom faces, but he wants the surface area of the leftover cheese block to be the same as before. What should the diameter of his hole be, in inches? [i]Αddendum on 1/26/19: the hole must have non-zero diameter. [/i] [b]p13.[/b] Find the smallest prime that does not divide $20! + 19! + 2019!$. [b]p14.[/b] Convex pentagon $ABCDE$ has angles $\angle ABC = \angle BCD = \angle DEA = \angle EAB$ and angle $\angle CDE = 60^o$. Given that $BC = 3$, $CD = 4$, and $DE = 5$, find $EA$. [i]Addendum on 1/26/19: ABCDE is specified to be convex. [/i] [b]p15.[/b] Sophia has $3$ pairs of red socks, $4$ pairs of blue socks, and $5$ pairs of green socks. She picks out two individual socks at random: what is the probability she gets a pair with matching color? [b]p16.[/b] How many real roots does the function $f(x) = 2019^x - 2019x - 2019$ have? [b]p17.[/b] A $30-60-90$ triangle is placed on a coordinate plane with its short leg of length $6$ along the $x$-axis, and its long leg along the $y$-axis. It is then rotated $90$ degrees counterclockwise, so that the short leg now lies along the $y$-axis and long leg along the $x$-axis. What is the total area swept out by the triangle during this rotation? [b]p18.[/b] Find the number of ways to color the unit cells of a $2\times 4$ grid in four colors such that all four colors are used and every cell shares an edge with another cell of the same color. [b]p19.[/b] Triangle $\vartriangle ABC$ has centroid $G$, and $X, Y,Z$ are the centroids of triangles $\vartriangle BCG$, $\vartriangle ACG$, and $\vartriangle ABG$, respectively. Furthermore, for some points $D,E, F$, vertices $A,B,C$ are themselves the centroids of triangles $\vartriangle DBC$, $\vartriangle ECA$, and $\vartriangle FAB$, respectively. If the area of $\vartriangle XY Z = 7$, what is the area of $\vartriangle DEF$? [b]p20.[/b] Fhomas orders three $2$-gallon jugs of milk from EMCCBay for his breakfast omelette. However, every jug is actually shipped with a random amount of milk (not necessarily an integer), uniformly distributed between $0$ and $2$ gallons. If Fhomas needs $2$ gallons of milk for his breakfast omelette, what is the probability he will receive enough milk? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].