Prove that for every positive integer $n$, the number $A_n = 5^n + 2 \cdot 3^{n-1} + 1$ is a multiple of $8$.

Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Determine all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ that satisfy: [list] [*]$f(mn)+1=f(m)+f(n)$ for all positive integers $m$ and $n$; [*]$f(2024)=1$; [*]$f(n)=1$ for all positive $n\equiv22\pmod{23}$. [/list]

Let $N$ be the number of sequences of positive integers greater than $ 1$ where the product of all of the terms of the sequence is $12^{64}$. If $N$ can be expressed as $a(2^b)$ ), where $a$ is an odd positive integer, determine $b$.

For positive integers $n$, let $f_2(n)$ denote the number of divisors of $n$ which are perfect squares, and $f_3(n)$ denotes the number of positive divisors which are perfect cubes. Prove that for each positive integer $k$ there exists a positive integer $n$ for which $\frac{f_2(n)}{f_3(n)}=k$.

Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.

Find the least possible value for the fraction $$\frac{lcm(a,b)+lcm(b,c)+lcm(c,a)}{gcd(a,b)+gcd(b,c)+gcd(c,a)}$$ over all distinct positive integers $a, b, c$. By $lcm(x, y)$ we mean the least common multiple of $x, y$ and by $gcd(x, y)$ we mean the greatest common divisor of $x, y$.

Let $m, n$ be integers such that $n \geq m \geq 1$. Prove that $\frac{\text{gcd} (m,n)}{n} \binom{n}{m}$ is an integer. Here $\text{gcd}$ denotes greatest common divisor and $\binom{n}{m} = \frac{n!}{m!(n-m)!}$ denotes the binomial coefficient.

[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].

Find all pairs of integers $(x, y)$ such that $x^2 + 5y^2 = 2021y$.

What is the sum of all numbers between $0$ and $511$ inclusive that have an even number of $1$s when written in binary?

Positive integers are written on the blackboard one after another. The next integer $a_{n+1}$ (to be written after $a_1$,$a_2$,$...$,$a_n$) is an arbitrary integer not representable as a sum of several previous integers taken one or more times (i.e. $a_{n+1}$ is not of the form $k_1 *a_i + k_2 *a_2 + ... + k_n *a_n$ where$ k_1$, $k_2$,$..$, $k_n$ are non-negative integers). Prove that the process of writing cannot be infinite. (A Belov)

Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$.

Show that for every natural number $n$ there are $n$ natural numbers $ x_1 < x_2 < ... < x_n $ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}-\frac{1}{x_1x_2...x_n}\in \mathbb{N}\cup {0}$$ (15 points )

Let the integers $a, b, c, d$ be relatively prime to $$m = ad - bc.$$ Prove that the pairs of integers $(x,y)$ for which $ax+by$ is a multiple of $m$ are identical with those for which $cx + dy$ is a multiple of $m$.

This problem consists of four parts. 1. Show that for any nonzero integers $m,n,$ and prime $p$, we have $v_p(mn)=v_p(m)+v_p(n).$ 2. Show that if an off prime $p$, a positive integer $k$ and integers $a,b$ satisfy $p \nmid ~^\text{'}~p|a-b$ and $p\nmid k$, then $v_p(a^k-b^k)=v_p(a-b).$ 3. Show that if $p$ is an off prime with $p|a-b$ and $p\nmid a,b$, then $v_p(a^p-b^p)=v_p(a-b)+1)$. 4. Show that if an odd prime $p$, a positive integer $k$ and integers $a,b$ satisfy $p\nmid a,b ~^\text{'}~ p|a-b$, then $v_p(a^k-b^k)=v_p(a-b)$. Proposed by LTE.

Find all positive integers $n \ge 2$ such that equality $i+j \equiv C_{n}^{i} + C_{n}^{j}$ (mod $2$) is true for arbitrary $0 \le i \le j \le n$.

Arnaldo selects a nonnegative integer $a$ and Bernaldo selects a nonnegative integer $b$. Both of them secretly tell their number to Cernaldo, who writes the numbers $5$, $8$, and $15$ on the board, one of them being the sum $a+b$. Cernaldo rings a bell and Arnaldo and Bernaldo, individually, write on different slips of paper whether they know or not which of the numbers on the board is the sum $a+b$ and they turn them in to Cernaldo. If both of the papers say NO, Cernaldo rings the bell again and the process is repeated. It is known that both Arnaldo and Bernaldo are honest and intelligent. What is the maximum number of times that the bell can be rung until one of them knows the sum? Personal note: They really phoned it in with the names there…

Let $x<y$ be positive integers and $P=\frac{x^3-y}{1+xy}$. Find all integer values that $P$ can take.

A function $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, where $\mathbb{Z}_+$ is the set of positive integers, is non-decreasing and satisfies $f(mn) = f(m)f(n)$ for all relatively prime positive integers $m$ and $n$. Prove that $f(8)f(13) \ge (f(10))^2$.

The sequence $a_n (n\geq 1)$ of natural numbers is defined as $a_{n+1}=a_n+b_n,$ where $b_n$ is the number that has the same digits as $a_n$ but in the opposite order ($b_n$ can start with $0$). For example, if $a_1=180,$ then $a_2=261, a_3=423.$ a) Decide if $a_1$ can be chosen so that $a_7$ is prime. b) Decide if $a_1$ can be chosen so that $a_5$ is prime.

Let $A=\{1,2,3,\cdots ,17\}$. A mapping $f:A\rightarrow A$ is defined as follows: $f^{[1]}(x)=f(x), f^{[k+1]}(x)=f(f^{[k]}(x))$ for $k\in\mathbb{N}$. Suppose that $f$ is bijective and that there exists a natural number $M$ such that: i) when $m<M$ and $1\le i\le 16$, we have $f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}$ and $f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}$; ii) when $1\le i\le 16$, we have $f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}$ and $f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}$. Find the maximal value of $M$.

For each positive integer $n$ whose canonical decomposition is $n = p_1^{a_1} \cdot p_2^{a_2} \cdot\cdot\cdot p_k^{a_k}$, we define $t(n) = (p_1 + 1) \cdot (p_2 + 1) \cdot\cdot\cdot (p_k + 1)$. For example, $t(20) = t(2^2\cdot 5^1) = (2 + 1) (5 + 1) = 18$, $t(30) = t(2^1\cdot 3^1\cdot 5^1) = (2 + 1) (3 + 1) (5 + 1) = 72$ and $t(125) = t(5^3) = (5 + 1) = 6$ . We say that a positive integer $n$ is [i]special [/i]if $t(n)$ is a divisor of $n$. How many positive divisors of the number $54610$ are special?

Find all prime numbers $p, q, r$ such that \[15p+7pq+qr=pqr.\]

A set $S$ of $n-1$ natural numbers is given ($n\ge 3$). There exist at least at least two elements in this set whose difference is not divisible by $n$. Prove that it is possible to choose a non-empty subset of $S$ so that the sum of its elements is divisible by $n$.

Alice has an integer $N > 1$ on the blackboard. Each minute, she deletes the current number $x$ on the blackboard and writes $2x+1$ if $x$ is not the cube of an integer, or the cube root of $x$ otherwise. Prove that at some point of time, she writes a number larger than $10^{100}$. [i]Proposed by Anant Mudgal and Rohan Goyal[/i]