Find all pairs of natural numbers $(\alpha,\beta)$ for which, if $\delta$ is the greatest common divisor of $\alpha,\beta$, and $\varDelta$ is the least common multiple of $\alpha,\beta$, then \[ \delta + \Delta = 4(\alpha + \beta) + 2021\]

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.

Determine the smallest positive integer $n$ whose prime factors are all greater than $18$, and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$.

[u]Round 1[/u] [b]p1.[/b] Define the operation $\clubsuit$ so that $a \,\clubsuit \, b = a^b + b^a$. Then, if $2 \,\clubsuit \,b = 32$, what is $b$? [b]p2. [/b] A square is changed into a rectangle by increasing two of its sides by $p\%$ and decreasing the two other sides by $p\%$. The area is then reduced by $1\%$. What is the value of $p$? [b]p3.[/b] What is the sum, in degrees, of the internal angles of a heptagon? [b]p4.[/b] How many integers in between $\sqrt{47}$ and $\sqrt{8283}$ are divisible by $7$? [u]Round 2[/u] [b]p5.[/b] Some mutant green turkeys and pink elephants are grazing in a field. Mutant green turkeys have six legs and three heads. Pink elephants have $4$ legs and $1$ head. There are $100$ legs and $37$ heads in the field. How many animals are grazing? [b]p6.[/b] Let $A = (0, 0)$, $B = (6, 8)$, $C = (20, 8)$, $D = (14, 0)$, $E = (21, -10)$, and $F = (7, -10)$. Find the area of the hexagon $ABCDEF$. [b]p7.[/b] In Moscow, three men, Oleg, Igor, and Dima, are questioned on suspicion of stealing Vladimir Putin’s blankie. It is known that each man either always tells the truth or always lies. They make the following statements: (a) Oleg: I am innocent! (b) Igor: Dima stole the blankie! (c) Dima: I am innocent! (d) Igor: I am guilty! (e) Oleg: Yes, Igor is indeed guilty! If exactly one of Oleg, Igor, and Dima is guilty of the theft, who is the thief?? [b]p8.[/b] How many $11$-letter sequences of $E$’s and $M$’s have at least as many $E$’s as $M$’s? [u]Round 3[/u] [b]p9.[/b] John is entering the following summation $31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39$ in his calculator. However, he accidently leaves out a plus sign and the answer becomes $3582$. What is the number that comes before the missing plus sign? [b]p10.[/b] Two circles of radius $6$ intersect such that they share a common chord of length $6$. The total area covered may be expressed as $a\pi + \sqrt{b}$, where $a$ and $b$ are integers. What is $a + b$? [b]p11.[/b] Alice has a rectangular room with $6$ outlets lined up on one wall and $6$ lamps lined up on the opposite wall. She has $6$ distinct power cords (red, blue, green, purple, black, yellow). If the red and green power cords cannot cross, how many ways can she plug in all six lamps? [b]p12.[/b] Tracy wants to jump through a line of $12$ tiles on the floor by either jumping onto the next block, or jumping onto the block two steps ahead. An example of a path through the $12$ tiles may be: $1$ step, $2$ steps, $2$ steps, $2$ steps, $1$ step, $2$ steps, $2$ steps. In how many ways can Tracy jump through these $12$ tiles? PS. You should use hide for answers. Last rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784268p24464984]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let$ p$ be an odd prime number. Determine the number of tuples $(a_1, a_2, . . . , a_p)$ of natural numbers with the following properties: 1) $1 \le ai \le p$ for all $i = 1, . . . , p$. 2) $a_1 + a_2 + · · · + a_p$ is not divisible by $p$. 3) $a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1$ is divisible by $p$.

Finland is going to change the monetary system again and replace the Euro by the Finnish Mark. The Mark is divided into $100$ pennies. There shall be coins of three denominations only, and the number of coins a person has to carry in order to be able to pay for any purchase less than one mark should be minimal. Determine the coin denominations.

[u]Set 1[/u] [b]1.1[/b] How many positive integer divisors are there of $2^2 \cdot 3^3 \cdot 5^4$? [b]1.2[/b] Let $T$ be the answer from the previous problem. For how many integers $n$ between $1$ and $T$ (inclusive) is $\frac{(n)(n - 1)(n - 2)}{12}$ an integer? [b]1.3[/b] Let $T$ be the answer from the previous problem. Find $\frac{lcm(T, 36)}{gcd(T, 36)}$. PS. You should use hide for answers.

Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose \[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\] Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$

Find the smallest positive value taken by $a^3 + b^3 + c^3 - 3abc$ for positive integers $a$, $b$, $c$ . Find all $a$, $b$, $c$ which give the smallest value

Given prime numbers $p$ and $q$ with $p<q$, determine all pairs $(x,y)$ of positive integers such that $$\frac1x+\frac1y=\frac1p-\frac1q.$$

The two squares shown share the same center $O$ and have sides of length 1. The length of $\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ [asy] real alpha = 25; pair W=dir(225), X=dir(315), Y=dir(45), Z=dir(135), O=origin; pair w=dir(alpha)*W, x=dir(alpha)*X, y=dir(alpha)*Y, z=dir(alpha)*Z; draw(W--X--Y--Z--cycle^^w--x--y--z--cycle); pair A=intersectionpoint(Y--Z, y--z), C=intersectionpoint(Y--X, y--x), E=intersectionpoint(W--X, w--x), G=intersectionpoint(W--Z, w--z), B=intersectionpoint(Y--Z, y--x), D=intersectionpoint(Y--X, w--x), F=intersectionpoint(W--X, w--z), H=intersectionpoint(W--Z, y--z); dot(O); label("$O$", O, SE); label("$A$", A, dir(O--A)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); label("$D$", D, dir(O--D)); label("$E$", E, dir(O--E)); label("$F$", F, dir(O--F)); label("$G$", G, dir(O--G)); label("$H$", H, dir(O--H));[/asy]

Find all triples $(a, b, c)$ of positive integers such that: $$a + b + c = 24$$ $$a^2 + b^2 + c^2 = 210$$ $$abc = 440$$

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [u] Set 1[/u] [b]R1.1 / P1.1[/b] Find $291 + 503 - 91 + 492 - 103 - 392$. [b]R1.2[/b] Let the operation $a$ & $b$ be defined to be $\frac{a-b}{a+b}$. What is $3$ & $-2$? [b]R1.3[/b]. Joe can trade $5$ apples for $3$ oranges, and trade $6$ oranges for $5$ bananas. If he has $20$ apples, what is the largest number of bananas he can trade for? [b]R1.4[/b] A cone has a base with radius $3$ and a height of $5$. What is its volume? Express your answer in terms of $\pi$. [b]R1.5[/b] Guang brought dumplings to school for lunch, but by the time his lunch period comes around, he only has two dumplings left! He tries to remember what happened to the dumplings. He first traded $\frac34$ of his dumplings for Arman’s samosas, then he gave $3$ dumplings to Anish, and lastly he gave David $\frac12$ of the dumplings he had left. How many dumplings did Guang bring to school? [u]Set 2[/u] [b]R2.6 / P1.3[/b] In the recording studio, Kanye has $10$ different beats, $9$ different manuscripts, and 8 different samples. If he must choose $1$ beat, $1$ manuscript, and $1$ sample for his new song, how many selections can he make? [b]R2.7[/b] How many lines of symmetry does a regular dodecagon (a polygon with $12$ sides) have? [b]R2.8[/b] Let there be numbers $a, b, c$ such that $ab = 3$ and $abc = 9$. What is the value of $c$? [b]R2.9[/b] How many odd composite numbers are there between $1$ and $20$? [b]R2.10[/b] Consider the line given by the equation $3x - 5y = 2$. David is looking at another line of the form ax - 15y = 5, where a is a real number. What is the value of a such that the two lines do not intersect at any point? [u]Set 3[/u] [b]R3.11[/b] Let $ABCD$ be a rectangle such that $AB = 4$ and $BC = 3$. What is the length of BD? [b]R3.12[/b] Daniel is walking at a constant rate on a $100$-meter long moving walkway. The walkway moves at $3$ m/s. If it takes Daniel $20$ seconds to traverse the walkway, find his walking speed (excluding the speed of the walkway) in m/s. [b]R3.13 / P1.3[/b] Pratik has a $6$ sided die with the numbers $1, 2, 3, 4, 6$, and $12$ on the faces. He rolls the die twice and records the two numbers that turn up on top. What is the probability that the product of the two numbers is less than or equal to $12$? [b]R3.14 / P1.5[/b] Find the two-digit number such that the sum of its digits is twice the product of its digits. [b]R3.15[/b] If $a^2 + 2a = 120$, what is the value of $2a^2 + 4a + 1$? PS. You should use hide for answers. R16-30 /P6-10/ P26-30 have been posted [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here[/url], and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a circumference at some points sit $12$ grasshoppers. The points divide the circumference into $12$ arcs. By a signal each grasshopper jumps from its point to the midpoint of its arc (in clockwise direction). In such way new arcs are created. The process repeats for a number of times. Can it happen that at least one of the grasshoppers returns to its initial point after a) $12$ jumps? (4) a) $13$ jumps? (3)

Determine all pairs $ (a,b)$ of positive integers, each integer having two decimal digits, such that $ 100a\plus{}b$ and $ 201a\plus{}b$ are both perfect squares.

If the positive integers $x$ and $y$ are such that $3x + 4y$ and $4x + 3y$ are both perfect squares, prove that both $x$ and $y$ are both divisible with $7$.

Let $a,b,c$ be positive integers such that the numbers $k=b^c+a, l=a^b+c, m=c^a+b$ to be prime numbers. Prove that at least two of the numbers $k,l,m$ are equal.

The decimal representation of an integer uses only two different digits. The number is at least $10$ digits long, and any two neighbouring digits are distinct. What is the greatest power of two that can divide this number?

Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$. Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.

Find, with argument, the integer solutions of the equation \[3z^2 = 2x^3 + 385x^2 + 256x - 58195.\]

A sequence of primes $p_1, p_2, \dots$ is given by two initial primes $p_1$ and $p_2$, and $p_{n+2}$ being the greatest prime divisor of $p_n + p_{n+1} + 2018$ for all $n \ge 1$. Prove that the sequence only contains finitely many primes for all possible values of $p_1$ and $p_2$.

Nirajan is trapped in a magical dungeon. He has infinitely many magical cards with arbitrary MPs(Mana Points) which is always an integer $\mathbb{Z}$. To escape, he must give the dungeon keeper some magical cards whose MPs add up to an integer with at least $2024$ divisors. Can Nirajan always escape? [i]( Proposed by Vlad Spǎtaru, Romania)[/i]

How many pairs $(a, b)$ for integers $a, b \ge 2$ which exist the sequence $x_1, x_2, . . . , x_{1000}$ which satisfy conditions as below? 1.Terms $x_1, x_2, . . . , x_{1000}$ are sorting of $1, 2, . . . , 1000$. 2.For each integers $1 \le i < 1000$, the sequence forms $x_{i+1} = x_i + a$ or $x_{i+1} = x_i - b$.

Let $n$ be a positive integer and let $A_n$ respectively $B_n$ be the set of nonnegative integers $k<n$ such that the number of distinct prime factors of $\gcd(n,k)$ is even (respectively odd). Show that $|A_n|=|B_n|$ if $n$ is even and $|A_n|>|B_n|$ if $n$ is odd. Example: $A_{10} = \left\{ 0,1,3,7,9 \right\}$, $B_{10} = \left\{ 2,4,5,6,8 \right\}$.