Let there be a figure with $9$ disks and $11$ edges, as shown below. We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find the minimum sum of all real numbers written in $11$ edges.

Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$

What is the sum of the solutions to the equation $\sqrt[4]x =\displaystyle \frac{12}{7-\sqrt[4]x}$?

Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.

Let $S$ be the set of all real polynomials $f(x) = ax^3 + bx^2 + cx + d$ such that $|f(x)| \le 1$ for all $ -1 \le x \le 1$. Show that the set of possible $|a|$ for $f$ in $S$ is bounded above and find the smallest upper bound.

Let $x, y, z$ be positive real numbers such that $x^2 + xy + y^2 = \frac{25}{4}$, $y^2 + yz + z^2 = 36$, and $z^2 + zx + x^2 = \frac{169}{4}$. Find the value of $xy + yz + zx$.

Find the number of positive integers less than 101 that [i]can not [/i] be written as the difference of two squares of integers.

Let $n$ be a positive integer. Prove that there exist polynomials$f(x)$and $g(x$) with integer coefficients such that \[f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.\]

[b]p1.[/b] A function $f$ defined on the set of positive numbers satisfies the equality $$f(xy) = f(x) + f(y), x, y > 0.$$ Find $f(2007)$ if $f\left( \frac{1}{2007} \right) = 1$. [b]p2.[/b] The plane is painted in two colors. Show that there is an isosceles right triangle with all vertices of the same color. [b]p3.[/b] Show that the number of ways to cut a $2n \times 2n$ square into $1\times 2$ dominoes is divisible by $2$. [b]p4.[/b] Two mirrors form an angle. A beam of light falls on one mirror. Prove that the beam is reflected only finitely many times (even if the angle between mirrors is very small). [b]p5.[/b] A sequence is given by the recurrence relation $a_{n+1} = (s(a_n))^2 +1$, where $s(x)$ is the sum of the digits of the positive integer $x$. Prove that starting from some moment the sequence is periodic. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

Rational numbers $a$, $b$, $c$ satisfy the equation \[(a+b+c)(a+b-c)=c^2\,.\] Show that $a+b=c=0$.

Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.

Suppose that $ x_1, x_2, x_3, \ldots$ are positive real numbers for which \[ x^n_n \equal{} \sum^{n\minus{}1}_{j\equal{}0} x^j_n\] for $ n \equal{} 1, 2, 3, \ldots$ Prove that $ \forall n,$ \[ 2 \minus{} \frac{1}{2^{n\minus{}1}} \leq x_n < 2 \minus{} \frac{1}{2^n}.\]

Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant. [i]Victor Wang.[/i]

Let $R^+$ be the set of positive real numbers. Find all function $f : R^+ \to R$ such that, for all positive real number $x$ and $y$, the following conditions are satisfied: i) $2f (x) + 2f (y) \le f (x + y)$ ii) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$

[u]Round 1[/u] [b]1.1[/b] If the sum of two non-zero integers is $28$, then find the largest possible ratio of these integers. [b]1.2[/b] If Tom rolls a eight-sided die where the numbers $1$ − $8$ are all on a side, let $\frac{m}{n}$ be the probability that the number is a factor of $16$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]1.3[/b] The average score of $35$ second graders on an IQ test was $180$ while the average score of $70$ adults was $90$. What was the total average IQ score of the adults and kids combined? [u]Round 2[/u] [b]2.1[/b] So far this year, Bob has gotten a $95$ and a 98 in Term $1$ and Term $2$. How many different pairs of Term $3$ and Term $4$ grades can Bob get such that he finishes with an average of $97$ for the whole year? Bob can only get integer grades between $0$ and $100$, inclusive. [b]2.2[/b] If a complement of an angle $M$ is one-third the measure of its supplement, then what would be the measure (in degrees) of the third angle of an isosceles triangle in which two of its angles were equal to the measure of angle $M$? [b]2.3[/b] The distinct symbols $\heartsuit, \diamondsuit, \clubsuit$ and $\spadesuit$ each correlate to one of $+, -, \times , \div$, not necessarily in that given order. Given that $$((((72 \,\, \,\, \diamondsuit \,\, \,\,36) \,\, \,\,\spadesuit \,\, \,\,0 ) \,\, \,\, \diamondsuit \,\, \,\, 32) \,\, \,\, \clubsuit \,\, \,\, 3)\,\, \,\, \heartsuit \,\, \,\, 2 = \,\, \,\, 6,$$ what is the value of $$(((((64 \,\, \,\, \spadesuit \,\, \,\, 8) \heartsuit \,\, \,\, 6) \,\, \,\, \spadesuit \,\, \,\, 5) \,\, \,\, \heartsuit \,\, \,\, 1) \,\, \,\, \clubsuit \,\, \,\, 7) \,\, \,\, \diamondsuit \,\, \,\, 1?$$ [u]Round 3[/u] [b]3.1[/b] How many ways can $5$ bunnies be chosen from $7$ male bunnies and $9$ female bunnies if a majority of female bunnies is required? All bunnies are distinct from each other. [b]3.2[/b] If the product of the LCM and GCD of two positive integers is $2021$, what is the product of the two positive integers? [b]3.3[/b] The month of April in ABMC-land is $50$ days long. In this month, on $44\%$ of the days it rained, and on $28\%$ of the days it was sunny. On half of the days it was sunny, it rained as well. The rest of the days were cloudy. How many days were cloudy in April in ABMC-land? [u]Round 4[/u] [b]4.1[/b] In how many ways can $4$ distinct dice be rolled such that a sum of $10$ is produced? [b]4.2[/b] If $p, q, r$ are positive integers such that $p^3\sqrt{q}r^2 = 50$, find the sum of all possible values of $pqr$. [b]4.3[/b] Given that numbers $a, b, c$ satisfy $a + b + c = 0$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}= 10$, and $ab + bc + ac \ne 0$, compute the value of $\frac{-a^2 - b^2 - a^2}{ab + bc + ac}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2826137p24988781]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.

a) Prove that there are no one-to-one (injective) functions $f: \mathbb{N} \to \mathbb{N}\cup \{0\}$ such that \[ f(mn) = f(m)+f(n) , \ \forall \ m,n \in \mathbb{N}. \] b) Prove that for all positive integers $k$ there exist one-to-one functions $f: \{1,2,\ldots,k\}\to\mathbb{N}\cup \{0\}$ such that $f(mn) = f(m)+f(n)$ for all $m,n\in \{1,2,\ldots,k\}$ with $mn\leq k$. [i]Mihai Baluna[/i]

For any given pair of positive integers $m>n$ find all $a\in\mathbb R$ for which the polynomial $x^m-ax^n+1$ can be expressed as a quotient of two nonzero polynomials with real nonnegative coefficients.

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that \[f(xf(y)+f(x)) = 2f(x)+xy\] for every reals $x,y$.

Given the sequence $\{y_n\}_{n=1}^{\infty}$ defined by $y_1=y_2=1$ and \[y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1\] find all integers $k$ such that every term of the sequence is a perfect square.

Let $a,b,c$ be positive real numbers. Prove that $$\frac{a}{\sqrt{(2a+b)(2a+c)}} +\frac{b}{\sqrt{(2b+c)(2b+a)}} +\frac{c}{\sqrt{(2c+a)(2c+b)}} \le 1 $$

Let $ x,y,z $ be numbers distinct from $ -1 $ that verify the equation $$ \frac{1}{1+a} +\frac{1}{1+b} +\frac{1}{1+c} =\frac{3}{2} . $$ Prove that if $ abc=1, $ then $ a $ or $ b $ or $ c $ is equal to $ 1. $

Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$-axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices. Proposed by Mohammad Ahmadi

Let $n$ be an odd positive integer, and let $x_1,x_2,\cdots ,x_n$ be non-negative real numbers. Show that \[ \min_{i=1,\ldots,n} (x_i^2+x_{i+1}^2) \leq \max_{j=1,\ldots,n} (2x_jx_{j+1}) \]where $x_{n+1}=x_1$.