Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$. [i]Proposed by Mahyar Sefidgaran, Iran[/i]

What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?

For positive integers $a_1,a_2 ,\ldots,a_{2006}$ such that $\frac{a_1}{a_2},\frac{a_2}{a_3},\ldots,\frac{a_{2005}}{a_{2006}}$ are pairwise distinct, find the minimum possible amount of distinct positive integers in the set$\{a_1,a_2,...,a_{2006}\}$.

Let $C_k=\frac{1}{k+1}\binom{2k}{k}$ denote the $k^{\text{th}}$ Catalan number and $p$ be an odd prime. Prove that exactly half of the numbers in the set \[\left\{\sum_{k=1}^{p-1}C_kn^k\,\middle\vert\, n\in\{1,2,\ldots,p-1\}\right\}\] are divisible by $p$. [i]Tristan Shin[/i]

A sequence of integers is constructed as follows: $a_1$ is an arbitrary three-digit number, $a_2$ is the sum of squares of the digits of $a_1, a_3$ is the sum of squares of the digits of $a_2$, etc. Prove that either $1$ or $4$ must occur in the sequence $a_1, a_2, a_3, ....$

Determine all three primes $(a, b, c)$ that satisfied the equality $a^2+ab+b^2=c^2+3$.

For a positive number $n$, let $g(n)$ be the product of all $1 \le k \le n$ such that gcd $(k, n) =1$, and say that $n > 1$ is reckless if $n$ is odd and $g(n) \equiv -1$ (mod $n$). Find the number of reckless numbers less than $50$.

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

Prove that $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a} <1$$ where $a, b$ and $c$ are positive numbers such that $abc = 1$. (G Galperin)

Can the numbers $1,2,3,\ldots,100$ be covered with $12$ geometric progressions? [i]A. Golovanov[/i]

Prove that $ p $ divides $ \varphi (1+a^p) , $ where $ a\ge 2 $ is a natural number, $ p $ is a prime, and $ \varphi $ is Euler's totient. [i]Cristinel Mortici[/i]

Let $p$ be an odd prime and $x$ be an integer such that $p \mid x^3 - 1$ but $p \nmid x - 1$. Prove that \[ p \mid (p - 1)!\left(x - \frac {x^2}{2} + \frac {x^3}{3} - \cdots - \frac {x^{p - 1}}{p - 1}\right).\][i]John Berman[/i]

Consider the function $f_k:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying \[f_k(x)=x+k\varphi(x)\] where $\varphi(x)$ is Euler's totient function, that is, the number of positive integers up to $x$ coprime to $x$. We define a sequence $a_1,a_2,...,a_{10}$ with [list] [*] $a_1=c$, and [*] $a_n=f_k(a_{n-1}) \text{ }\forall \text{ } 2\le n\le 10$ [/list] Is it possible to choose the initial value $c\ne 1$ such that each term is a multiple of the previous, if (a) $k=2025$ ? (b) $k=2065$ ? [i]Proposed by chorn[/i]

[b]p1.[/b] An organization decides to raise funds by holding a $\$60$ a plate dinner. They get prices from two caterers. The first caterer charges $\$50$ a plate. The second caterer charges according to the following schedule: $\$500$ set-up fee plus $\$40$ a plate for up to and including $61$ plates, and $\$2500$ $\log_{10}\left(\frac{p}{4}\right)$ for $p > 61$ plates. a) For what number of plates $N$ does it become at least as cheap to use the second caterer as the first? b) Let $N$ be the number you found in a). For what number of plates $X$ is the second caterer's price exactly double the price for $N$ plates? c) Let $X$ be the number you found in b). When X people appear for the dinner, how much profit does the organization raise for itself by using the second caterer? [b]p2.[/b] Let $N$ be a positive integer. Prove the following: a) If $N$ is divisible by $4$, then $N$ can be expressed as the sum of two or more consecutive odd integers. b) If $N$ is a prime number, then $N$ cannot be expressed as the sum of two or more consecutive odd integers. c) If $N$ is twice some odd integer, then $N$ cannot be expressed as the sum of two or more consecutive odd integers. [b]p3.[/b] Let $S =\frac{1}{1^2} +\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...$ a) Find, in terms of $S$, the value of $S =\frac{1}{2^2} +\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...$ b) Find, in terms of $S$, the value of$S =\frac{1}{1^2} +\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+...$ c) Find, in terms of $S$, the value of$S =\frac{1}{1^2} -\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+...$ [b]p4.[/b] Let $\{P_1, P_2, P_3, ...\}$ be an infinite set of points on the $x$-axis having positive integer coordinates, and let $Q$ be an arbitrary point in the plane not on the $x$-axis. Prove that infinitely many of the distances $|P_iQ|$ are not integers. a) Draw a relevant picture. b) Provide a proof. [b]p5.[/b] Point $P$ is an arbitrary point inside triangle $ABC$. Points $X$, $Y$ , and $Z$ are constructed to make segments $PX$, $PY$ , and $PZ$ perpendicular to $AB$, $BC$, and $CA$, respectively. Let $x$, $y$, and $z$ denote the lengths of the segments $PX$, $PY$ , and $PZ$, respectively. a) If triangle $ABC$ is an equilateral triangle, prove that $x + y + z$ does not change regardless of the location of $P$ inside triangle ABC. b) If triangle $ABC$ is an isosceles triangle with $|BC| = |CA|$, prove that $x + y + z$ does not change when $P$ moves along a line parallel to $AB$. c) Now suppose that triangle $ABC$ is scalene (i.e., $|AB|$, $|BC|$, and $|CA|$ are all different). Prove that there exists a line for which $x+y+z$ does not change when $P$ moves along this line. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ be a prime number and $n > 2$ an integer. Find all integer solutions of the equation $x^n +ay^n = a^2z^n$ .

Let $Q(n)$ denote the sum of the digits of $n$ in its decimal representation. Prove that for every positive integer $k$, there exists a multiple $n$ of $k$ such that $Q(n)=Q(n^2)$.

Does there exist a sequence of $2017$ consecutive integers which contains exactly $17$ primes?

Find all positive integers $ n\in\{1,2,3,\ldots,2009\}$ such that \[ 4n^6 \plus{} n^3 \plus{} 5\] is divisible by $ 7$.

For all positive integers $n$, the function $\gamma: \mathbb{Z}^+ \to \mathbb{Z}_{\geq 0}$ is defined as, $\gamma(1) = 0$ and for all $n > 1$, if the prime factorization of $n$ is $n = p_1^{\alpha_1} p_2^{\alpha_2} \dots p_k^{\alpha_k},$ then $\gamma(n) = \alpha_1 + \alpha_2 + \dots + \alpha_k$. We have an arithmetic sequence $X = \{x_i\}_{i=1}^{\infty}$. If for a positive integer $a > 1$, the sequence $\{ \gamma(a^{x_i} -1) \}$ is also an arithmetic sequence, show that the sequence $X$ has to be constant.

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Compute the sum $2019 + 201 + 20 + 2$. [b]p2.[/b] The sequence $100, 102, 104,..., 996$ and $998$ is the sequence of all three-digit even numbers. How many three digit even numbers are there? [b]p3.[/b] Find the units digit of $25\times 37\times 113\times 22$. [b]p4.[/b] Samuel has a number in his head. He adds $4$ to the number and then divides the result by $2$. After doing this, he ends up with the same number he had originally. What is his original number? [b]p5.[/b] According to Shay's Magazine, every third president is terrible (so the third, sixth, ninth president and so on were all terrible presidents). If there have been $44$ presidents, how many terrible presidents have there been in total? [b]p6.[/b] In the game Tic-Tac-Toe, a player wins by getting three of his or her pieces in the same row, column, or diagonal of a $3\times 3$ square. How many configurations of $3$ pieces are winning? Rotations and reflections are considered distinct. [b]p7.[/b] Eddie is a sad man. Eddie is cursed to break his arm $4$ times every $20$ years. How many times would he break his arm by the time he reaches age $100$? [b]p8. [/b]The figure below is made from $5$ congruent squares. If the figure has perimeter $24$, what is its area? [img]https://cdn.artofproblemsolving.com/attachments/1/9/6295b26b1b09cacf0c32bf9d3ba3ce76ddb658.png[/img] [b]p9.[/b] Sancho Panza loves eating nachos. If he eats $3$ nachos during the first minute, $4$ nachos during the second, $5$ nachos during the third, how many nachos will he have eaten in total after $15$ minutes? [b]p10.[/b] If the day after the day after the day before Wednesday was two days ago, then what day will it be tomorrow? [b]p11.[/b] Neetin the Rabbit and Poonam the Meerkat are in a race. Poonam can run at $10$ miles per hour, while Neetin can only hop at $2$ miles per hour. If Neetin starts the race $2$ miles ahead of Poonam, how many minutes will it take for Poonam to catch up with him? [b]p12.[/b] Dylan has a closet with t-shirts: $3$ gray, $4$ blue, $2$ orange, $7$ pink, and $2$ black. Dylan picks one shirt at random from his closet. What is the probability that Dylan picks a pink or a gray t-shirt? [b]p13.[/b] Serena's brain is $200\%$ the size of Eric's brain, and Eric's brain is $200\%$ the size of Carlson's. The size of Carlson's brain is what percent the size of Serena's? [b]p14.[/b] Find the sum of the coecients of $(2x + 1)^3$ when it is fully expanded. [b]p15. [/b]Antonio loves to cook. However, his pans are weird. Specifically, the pans are rectangular prisms without a top. What is the surface area of the outside of one of Antonio's pans if their volume is $210$, and their length and width are $6$ and $5$, respectively? [b]p16.[/b] A lattice point is a point on the coordinate plane with $2$ integer coordinates. For example, $(3, 4)$ is a lattice point since $3$ and $4$ are both integers, but $(1.5, 2)$ is not since $1.5$ is not an integer. How many lattice points are on the graph of the equation $x^2 + y^2 = 625$? [b]p17.[/b] Jonny has a beaker containing $60$ liters of $50\%$ saltwater ($50\%$ salt and $50\%$ water). Jonny then spills the beaker and $45$ liters pour out. If Jonny adds $45$ liters of pure water back into the beaker, what percent of the new mixture is salt? [b]p18.[/b] There are exactly 25 prime numbers in the set of positive integers between $1$ and $100$, inclusive. If two not necessarily distinct integers are randomly chosen from the set of positive integers from $1$ to $100$, inclusive, what is the probability that at least one of them is prime? [b]p19.[/b] How many consecutive zeroes are at the end of $12!$ when it is expressed in base $6$? [b]p20.[/b] Consider the following figure. How many triangles with vertices and edges from the following figure contain exactly $1$ black triangle? [img]https://cdn.artofproblemsolving.com/attachments/f/2/a1c400ff7d06b583c1906adf8848370e480895.png[/img] [b]p21.[/b] After Akshay got kicked o the school bus for rowdy behavior, he worked out a way to get home from school with his dad. School ends at $2:18$ pm, but since Akshay walks slowly he doesn't get to the front door until $2:30$. His dad doesn't like to waste time, so he leaves home everyday such that he reaches the high school at exactly $2:30$ pm, instantly picks up Akshay and turns around, then drives home. They usually get home at $3:30$ pm. However, one day Akshay left school early at exactly $2:00$ pm because he was expelled. Trying to delay telling his dad for as long as possible, Akshay starts jogging home. His dad left home at the regular time, saw Akshay on the way, picked him up and turned around instantly. They then drove home while Akshay's dad yelled at him for being a disgrace. They reached home at $3:10$ pm. How long had Akshay been walking before his dad picked him up? [b]p22.[/b] In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Then $\angle BOC = \angle BCD$, $\angle COD =\angle BAD$, $AB = 4$, $DC = 6$, and $BD = 5$. What is the length of $BO$? [b]p23.[/b] A standard six-sided die is rolled. The number that comes up first determines the number of additional times the die will be rolled (so if the first number is $3$, then the die will be rolled $3$ more times). Each time the die is rolled, its value is recorded. What is the expected value of the sum of all the rolls? [b]p24.[/b] Dora has a peculiar calculator that can only perform $2$ operations: either adding $1$ to the current number or squaring the current number. Each minute, Dora randomly chooses an operation to apply to her number. She starts with $0$. What is the expected number of minutes it takes Dora's number to become greater than or equal to $10$? [b]p25.[/b] Let $\vartriangle ABC$ be such that $AB = 2$, $BC = 1$, and $\angle ACB = 90^o$. Let points $D$ and $E$ be such that $\vartriangle ADE$ is equilateral, $D$ is on segment $\overline{BC}$, and $D$ and $E$ are not on the same side of $\overline{AC}$. Segment $\overline{BE}$ intersects the circumcircle of $\vartriangle ADE$ at a second point $F$. If $BE =\sqrt{6}$, find the length of $\overline{BF}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all pairs $(p,q)$ of prime numbers such that \[m^{3pq} \equiv m \pmod{3pq} \qquad \forall m \in \mathbb Z.\]

[b]p1.[/b] Find the unknown angle $a$ of the triangle inscribed in the square. [img]https://cdn.artofproblemsolving.com/attachments/b/1/4aab5079dea41637f2fa22851984f886f034df.png[/img] [b]p2.[/b] Draw a polygon in the plane and a point outside of it with the following property: no edge of the polygon is completely visible from that point (in other words, the view is obstructed by some other edge). [b]p3.[/b] This problem has two parts. In each part, $2022$ real numbers are given, with some additional property. (a) Suppose that the sum of any three of the given numbers is an integer. Show that the total sum of the $2022$ numbers is also an integer. (b) Suppose that the sum of any five of the given numbers is an integer. Show that 5 times the total sum of the $2022$ numbers is also an integer, but the sum itself is not necessarily an integer. [b]p4.[/b] Replace stars with digits so that the long multiplication in the example below is correct. [img]https://cdn.artofproblemsolving.com/attachments/9/7/229315886b5f122dc0675f6d578624e83fc4e0.png[/img] [b]p5.[/b] Five nodes of a square grid paper are marked (called marked points). Show that there are at least two marked points such that the middle point of the interval connecting them is also a node of the square grid paper [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=\dfrac{8}{3} \\ \dfrac{yz}{y+z}=\dfrac{12}{5} \\\dfrac{xz}{x+z}=\dfrac{24}{7} \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A number is said to be [i]balanced [/i] if one of its digits is average of the others. How many [i]balanced [/i]$3$-digit numbers are there?

Is it possible to complete the following square knowning that each row and column make an aritmetic progression?

For a natural number $p$, one can move between two points with integer coordinates if the distance between them equals $p$. Find all prime numbers $p$ for which it is possible to reach the point $(2002,38)$ starting from the origin $(0,0)$.