Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real numbers $a, b, c$, we have \[ f(a+b+c)f(ab+bc+ca) - f(a)f(b)f(c) = f(a+b)f(b+c)f(c+a). \] [i]Proposed by Mainak Ghosh and Rijul Saini[/i]

Given a polynomial $P(x)=a_{d}x^{d}+ \ldots +a_{2}x^{2}+a_{0}$ with positive integers for coefficients and degree $d\geq 2$. Consider the sequence defined by $$b_{1}=a_{0} ,b_{n+1}=P(b_{n}) $$ for $n \geq 1$ . Prove that for all $n \geq 2$ there exists a prime $p$ such that $p$ divides $b_{n}$ but does not divide $b_{1}b_{2} \ldots b_{n-1}$.

Let f be a ternary operation on a set of at least four elements for which (1) $f ( x , x , y ) \equiv f ( x , y , x ) \equiv f( x , y , y ) \equiv x$ (2) $f ( x , y , z ) = f ( y , z , x ) = f ( y , x , z ) \in \{ x , y , z \}$ for pairwise distinct x,y,z. Prove that f is a nontrivial composition of g such that g is not a composition of f. (The n-variable operation g is trivial if $g(x_1, ..., x_n) \equiv x_i$ for some i ($1 \leq i \leq n$) )

We have $n$ points in the plane, no three on a line. We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon. Suppose that for a fixed $k$ the number of $k$ good points is $c_k$. Show that the following sum is independent of the structure of points and only depends on $n$ : \[ \sum_{i=3}^n (-1)^i c_i \]

A sequence $ \left( a_n \right)_{n\ge 1} $ has the property that it´s nondecreasing, nonconstant and, for every natural $ n, a_n\big| n^2. $ Show that at least one of the following affirmations are true. $ \text{(i)} $ There exists an index $ n_1 $ such that $ a_n=n, $ for all $ n\ge n_1. $ $ \text{(ii)} $ There exists an index $ n_2 $ such that $ a_n=n^2, $ for all $ n\ge n_2. $

In a game, Andi and a computer take turns. At the beginning, the computer shows a polynomial $x^2 + mx + n$ where $m,n \in \mathbb{Z}$, such that it doesn't have real roots. Andi then begins the game. On his turn, Andi may change a polynomial in the form $x^2 + ax + b$ into either $x^2 + (a+b)x + b$ or $x^2 + ax + (a+b)$. However, Andi may only choose a polynomial that has real roots. On the computer's turn, it simply switches the coefficient of $x$ and the constant of the polynomial. Andi loses if he can't continue to play. Find all $(m,n)$ such that Andi always loses (in finitely many turns).

A function $f$ satisfies $f(x)+x f(1-x) = x^2$ for all real $x$. Determine $f$ .

Let $f$ be a function defined on $[0, 2022]$, such that $f(0) = f(2022) = 2022$, and $$|f(x) - f(y)| \le 2|x -y|,$$ for all $x, y$ in $[0, 2022]$. Prove that for each $x, y$ in $[0, 2022]$, the distance between $f(x)$ and $f(y)$ does not exceed $2022$.

Let $x,y,z\in \mathbb{R}^+$ be such that $xy+yz+zx=x+y+z$. Prove the following inequality: $\frac{1}{x^2+y+1}+\frac{1}{y^2+z+1}+\frac{1}{z^2+x+1}\leq 1$.

$A$ gives $B$ as many cents as $B$ has and $C$ as many cents as $C$ has. Similarly, $B$ then gives $A$ and $C$ as many cents as each then has. $C$, similarly, then gives $A$ and $B$ as many cents as each then has. If each finally has $16$ cents, with how many cents does $A$ start? $\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26\qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$

Let $\alpha$ and $\beta$ be reals. Find the least possible value of $$(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2.$$

Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? [i]Nanang Susyanto, Jogjakarta[/i]

[b]p1.[/b] Zion, RJ, Cam, and Tre decide to start learning languages. The four most popular languages that Duke offers are Spanish, French, Latin, and Korean. If each friend wants to learn exactly three of these four languages, how many ways can they pick courses such that they all attend at least one course together? [b]p2. [/b] Suppose we wrote the integers between $0001$ and $2019$ on a blackboard as such: $$000100020003 · · · 20182019.$$ How many $0$’s did we write? [b]p3.[/b] Duke’s basketball team has made $x$ three-pointers, $y$ two-pointers, and $z$ one-point free throws, where $x, y, z$ are whole numbers. Given that $3|x$, $5|y$, and $7|z$, find the greatest number of points that Duke’s basketball team could not have scored. [b]p4.[/b] Find the minimum value of $x^2 + 2xy + 3y^2 + 4x + 8y + 12$, given that $x$ and $y$ are real numbers. Note: calculus is not required to solve this problem. [b]p5.[/b] Circles $C_1, C_2$ have radii $1, 2$ and are centered at $O_1, O_2$, respectively. They intersect at points $ A$ and $ B$, and convex quadrilateral $O_1AO_2B$ is cyclic. Find the length of $AB$. Express your answer as $x/\sqrt{y}$ , where $x, y$ are integers and $y$ is square-free. [b]p6.[/b] An infinite geometric sequence $\{a_n\}$ has sum $\sum_{n=0}^{\infty} a_n = 3$. Compute the maximum possible value of the sum $\sum_{n=0}^{\infty} a_{3n} $. [b]p7.[/b] Let there be a sequence of numbers $x_1, x_2, x_3,...$ such that for all $i$, $$x_i = \frac{49}{7^{\frac{i}{1010}} + 49}.$$ Find the largest value of $n$ such that $$\left\lfloor \sum_{i=1}{n} x_i \right\rfloor \le 2019.$$ [b]p8.[/b] Let $X$ be a $9$-digit integer that includes all the digits $1$ through $9$ exactly once, such that any $2$-digit number formed from adjacent digits of $X$ is divisible by $7$ or $13$. Find all possible values of $X$. [b]p9.[/b] Two $2025$-digit numbers, $428\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}571$ and $571\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}428$ , form the legs of a right triangle. Find the sum of the digits in the hypotenuse. [b]p10.[/b] Suppose that the side lengths of $\vartriangle ABC$ are positive integers and the perimeter of the triangle is $35$. Let $G$ the centroid and $I$ be the incenter of the triangle. Given that $\angle GIC = 90^o$ , what is the length of $AB$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $a, b, c, d$ are the solutions of the equation $x^4-kx-15=0$, find the equation whose solutions are $\frac{a+b+c}{d^2}, \frac{a+b+d}{c^2}, \frac{a+c+d}{b^2}, \frac{b+c+d}{a^2}$.

Let $p = \overline{abc}$ be the 3-digit prime number. Prove that the equation $ax^2 + bx + c = 0$ has no rational roots.

Let $x,y{}$ and $z{}$ be positive real numbers satisfying $x+y+z=1$. Prove that [list=a] [*]\[1-\frac{x^2-yz}{x^2+x}=\frac{(1-y)(1-z)}{x^2+x};\] [*]\[\frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leqslant 0.\] [/list]

Let $x, y, z$ be real numbers satisfying $x+y+z=0$ and $x^2+y^2+z^2=6$. Find the maximum value of \[ |(x-y)(y-z)(z-x) | \]

A finite set $\mathcal S$ of distinct positive real numbers is called [i]radiant[/i] if it satisfies the following property: if $a$ and $b$ are two distinct elements of $\mathcal S$, then $a^2 + b^2$ is also an element of $\mathcal S$. [list=a] [*]Does there exist a radiant set with a size greater than or equal to $4$? [*]Determine all radiant sets of size $2$ or $3$. [/list]

Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be a function such that (i) For all $x,y \in \Bbb{R}$, \[ f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y) \] (ii) For all $x \in [0,1)$, $f(0) \geq f(x)$, (iii) $-f(-1) = f(1) = 1$. Find all such functions $f$.

Let $x_0 = 5$ and define the sequence recursively as $x_{n+1} = x_n + \frac{1}{x_n}$. Prove that: \[ 45 < x_{1000} < 45.1. \]

[b]p1.[/b] Evaluate $6^4 +5^4 +3^4 +2^4$. [b]p2.[/b] What digit is most frequent between $1$ and $1000$ inclusive? [b]p3.[/b] Let $n = gcd \, (2^2 \cdot 3^3 \cdot 4^4,2^4 \cdot 3^3 \cdot 4^2)$. Find the number of positive integer factors of $n$. [b]p4.[/b] Suppose $p$ and $q$ are prime numbers such that $13p +5q = 91$. Find $p +q$. [b]p5.[/b] Let $x = (5^3 -5)(4^3 -4)(3^3 -3)(2^3 -2)(1^3 -1)$. Evaluate $2018^x$ . [b]p6.[/b] Liszt the lister lists all $24$ four-digit integers that contain each of the digits $1,2,3,4$ exactly once in increasing order. What is the sum of the $20$th and $18$th numbers on Liszt’s list? [b]p7.[/b] Square $ABCD$ has center $O$. Suppose $M$ is the midpoint of $AB$ and $OM +1 =OA$. Find the area of square $ABCD$. [b]p8.[/b] How many positive $4$-digit integers have at most $3$ distinct digits? [b]p9.[/b] Find the sumof all distinct integers obtained by placing $+$ and $-$ signs in the following spaces $$2\_3\_4\_5$$ [b]p10.[/b] In triangle $ABC$, $\angle A = 2\angle B$. Let $I$ be the intersection of the angle bisectors of $B$ and $C$. Given that $AB = 12$, $BC = 14$,and $C A = 9$, find $AI$ . [b]p11.[/b] You have a $3\times 3\times 3$ cube in front of you. You are given a knife to cut the cube and you are allowed to move the pieces after each cut before cutting it again. What is the minimumnumber of cuts you need tomake in order to cut the cube into $27$ $1\times 1\times 1$ cubes? p12. How many ways can you choose $3$ distinct numbers fromthe set $\{1,2,3,...,20\}$ to create a geometric sequence? [b]p13.[/b] Find the sum of all multiples of $12$ that are less than $10^4$ and contain only $0$ and $4$ as digits. [b]p14.[/b] What is the smallest positive integer that has a different number of digits in each base from $2$ to $5$? [b]p15.[/b] Given $3$ real numbers $(a,b,c)$ such that $$\frac{a}{b +c}=\frac{b}{3a+3c}=\frac{c}{a+3b},$$ find all possible values of $\frac{a +b}{c}$. [b]p16.[/b] Let S be the set of lattice points $(x, y, z)$ in $R^3$ satisfying $0 \le x, y, z \le 2$. How many distinct triangles exist with all three vertices in $S$? [b]p17.[/b] Let $\oplus$ be an operator such that for any $2$ real numbers $a$ and $b$, $a \oplus b = 20ab -4a -4b +1$. Evaluate $$\frac{1}{10} \oplus \frac19 \oplus \frac18 \oplus \frac17 \oplus \frac16 \oplus \frac15 \oplus \frac14 \oplus \frac13 \oplus \frac12 \oplus 1.$$ [b]p18.[/b] A function $f :N \to N$ satisfies $f ( f (x)) = x$ and $f (2f (2x +16)) = f \left(\frac{1}{x+8} \right)$ for all positive integers $x$. Find $f (2018)$. [b]p19.[/b] There exists an integer divisor $d$ of $240100490001$ such that $490000 < d < 491000$. Find $d$. [b]p20.[/b] Let $a$ and $b$ be not necessarily distinct positive integers chosen independently and uniformly at random from the set $\{1,2, 3, ... ,511,512\}$. Let $x = \frac{a}{b}$ . Find the probability that $(-1)^x$ is a real number. [b]p21[/b]. In $\vartriangle ABC$ we have $AB = 4$, $BC = 6$, and $\angle ABC = 135^o$. $\angle ABC$ is trisected by rays $B_1$ and $B_2$. Ray $B_1$ intersects side $C A$ at point $F$, and ray $B_2$ intersects side $C A$ at point $G$. What is the area of $\vartriangle BFG$? [b]p22.[/b] A level number is a number which can be expressed as $x \cdot \lfloor x \rfloor \cdot \lceil x \rceil$ where $x$ is a real number. Find the number of positive integers less than or equal to $1000$ which are also level numbers. [b]p23.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, $C A = 15$ and circumcenter $O$. Let $D$ be the intersection of $AO$ and $BC$. Compute $BD/DC$. [b]p24.[/b] Let $f (x) = x^4 -3x^3 +2x^2 +5x -4$ be a quartic polynomial with roots $a,b,c,d$. Compute $$\left(a+1 +\frac{1}{a} \right)\left(b+1 +\frac{1}{b} \right)\left(c+1 +\frac{1}{c} \right)\left(d+1 +\frac{1}{d} \right).$$ [b]p25.[/b] Triangle $\vartriangle ABC$ has centroid $G$ and circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 2018$, $BD =20$, and $CD = 18$, find the area of triangle $\vartriangle DOG$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive numbers $b_1, b_2,..., b_n$ are given so that $b_1 + b_2 + ...+ b_n \le 10$. Further, $a_1 = b_1$ and $a_m = sa_{m-1} + b_m$ for $m > 1$, where $0 \le s < 1$. Show that $a^2_1 + a^2_2 + ... + a^2_n \le \frac{100}{1 - s^2} $

Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]

Let us call a real number $r$ [i]interesting[/i], if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.