$f,g:\mathbb{R}\rightarrow\mathbb{R}$ find all $f,g$ satisfying $\forall x,y\in \mathbb{R}$: \[g(f(x)-y)=f(g(y))+x.\]

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that we have $f(x + y) = f(x) + f(y) + f(xy)$ for all $ x,y\in \mathbb{R}$

For each non-negative $a$, consider the equation $$x^3 + ax - a^3 - 29 = 0.$$ Let $x_o$ be the positive root of this equation. Prove that for all $a > 0$ such a root exists. What is the smallest value of $x_o$?

Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.

Let $a{}$ and $b{}$ be positive integers. Prove that for any real number $x{}$ \[\sum_{j=0}^a\binom{a}{j}\big(2\cos((2j-a)x)\big)^b=\sum_{j=0}^b\binom{b}{j}\big(2\cos((2j-b)x)\big)^a.\]

Find all $f: R\rightarrow R$ such that (i) The set $\{\frac{f(x)}{x}| x\in R-\{0\}\}$ is finite (ii) $f(x-1-f(x)) = f(x)-1-x$ for all $x$

Let $(a_i)_{i\in \mathbb N}$ be a strictly increasing sequence of positive real numbers such that $\lim_{i \to \infty} a_i = +\infty$ and $a_{i+1}/a_i \leq 10$ for each $i$. Prove that for every positive integer $k$ there are infinitely many pairs $(i, j)$ with $10^k \leq a_i/a_j \leq 10^{k+1}.$

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f:\mathbb{N}\longrightarrow\mathbb{N}$ such that \[n+f(m)\mid f(n)+nf(m)\] for all $m,n\in \mathbb{N}$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$. Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$, which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,n$.

If $p(x)$ is a polynomial with integer coeffcients, let $q(x) = \frac{p(x)}{x(1-x)}$ . If $q(x) = q\left(\frac{1}{1-x}\right)$ for every $x \ne 0$, and $p(2) = -7, p(3) = -11$, find $p(10)$.

Let the set of all bijective functions taking positive integers to positive integers be $\mathcal B.$ Find all functions $\mathbf F:\mathcal B\to \mathbb R$ such that $$(\mathbf F(p)+\mathbf F(q))^2=\mathbf F(p \circ p)+\mathbf F(p\circ q)+\mathbf F(q\circ p)+\mathbf F(q\circ q)$$ for all $p,q \in \mathcal B.$ [i]Proposed by ckliao914[/i]

For each nonzero integer $ n$ find all functions $ f: \mathbb{R} \minus{} \{\minus{}3,0 \} \rightarrow \mathbb{R}$ satisfying: $ f(x\plus{}3)\plus{}f \left( \minus{}\frac{9}{x} \right)\equal{}\frac{(1\minus{}n)(x^2\plus{}3x\minus{}9)}{9n(x\plus{}3)}\plus{}\frac{2}{n}$ for all $ x \not\equal{} 0,\minus{}3.$ Furthermore, for each fixed $ n$ find all integers $ x$ for which $ f(x)$ is an integer.

[u]Round 5[/u] [b]5.1.[/b] Quadrilateral $ABCD$ is such that $\angle ABC = \angle ADC = 90^o$ , $\angle BAD = 150^o$ , $AD = 3$, and $AB = \sqrt3$. The area of $ABCD$ can be expressed as $p\sqrt{q}$ for positive integers $p, q$ where $q$ is not divisible by the square of any prime. Find $p + q$. [b]5.2.[/b] Neetin wants to gamble, so his friend Akshay describes a game to him. The game will consist of three dice: a $100$-sided one with the numbers $1$ to $100$, a tetrahedral one with the numbers $1$ to $4$, and a normal $6$-sided die. If Neetin rolls numbers with a product that is divisible by $21$, he wins. Otherwise, he pays Akshay $100$ dollars. The number of dollars that Akshay must pay Neetin for a win in order to make this game fair is $a/b$ for relatively prime positive integers $a, b$. Find $a + b$. (Fair means the expected net gain is $0$. ) [b]5.3.[/b] What is the sum of the fourth powers of the roots of the polynomial $P(x) = x^2 + 2x + 3$? [u]Round 6[/u] [b]6.1.[/b] Consider the set $S = \{1, 2, 3, 4,..., 25\}$. How many ordered $n$-tuples $S_1 = (a_1, a_2, a_3,..., a_n)$ of pairwise distinct ai exist such that $a_i \in S$ and $i^2 | a_i$ for all $1 \le i \le n$? [b]6.2.[/b] How many ways are there to place $2$ identical rooks and $ 1$ queen on a $ 4 \times 4$ chessboard such that no piece attacks another piece? (A queen can move diagonally, vertically or horizontally and a rook can move vertically or horizontally) [b]6.3.[/b] Let $L$ be an ordered list $\ell_1$, $\ell_2$, $...$, $\ell_{36}$ of consecutive positive integers who all have the sum of their digits not divisible by $11$. It is given that $\ell_1$ is the least element of $L$. Find the least possible value of $\ell_1$. [u]Round 7[/u] [b]7.1.[/b] Spencer, Candice, and Heather love to play cards, but they especially love the highest cards in the deck - the face cards (jacks, queens, and kings). They also each have a unique favorite suit: Spencer’s favorite suit is spades, Candice’s favorite suit is clubs, and Heather’s favorite suit is hearts. A dealer pulls out the $9$ face cards from every suit except the diamonds and wants to deal them out to the $3$ friends. How many ways can he do this so that none of the $3$ friends will see a single card that is part of their favorite suit? [b]7.2.[/b] Suppose a sequence of integers satisfies the recurrence $a_{n+3} = 7a_{n+2} - 14a_{n+1} + 8a_n$. If $a_0 = 4$, $a_1 = 9$, and $a_2 = 25$, find $a_{16}$. Your answer will be in the form $2^a + 2^b + c$, where $2^a < a_{16} < 2^{a+1}$ and $b$ is as large as possible. Find $a + b + c$. [b]7.3.[/b] Parallel lines $\ell_1$ and $\ell_2$ are $1$ unit apart. Unit square $WXYZ$ lies in the same plane with vertex $W$ on $\ell_1$. Line $\ell_2$ intersects segments $YX$ and $YZ$ at points $U$ and $O$, respectively. Given $UO =\frac{9}{10}$, the inradius of $\vartriangle YOU$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of contestants who participated in at least one of the three rounds of the 2020 ABMC April contest. Let $B$ be the number of times the letter b appears in the Accuracy Round. Let $M$ be the number of people who submitted both the speed and accuracy rounds before 2:00 PM EST. Further, let $C$ be the number of times the letter c appears in the Speed Round. Estimate $$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766239p24226402]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.

For a sequence $ a_i \in \mathbb{R}, i \in \{0, 1, 2, \ldots\}$ we have $ a_0 \equal{} 1$ and \[ a_{n\plus{}1} \equal{} a_n \plus{} \sqrt{a_{n\plus{}1} \plus{} a_n} \quad \forall n \in \mathbb{N}.\] Prove that this sequence is unique and find an explicit formula for this recursively defined sequence.

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy conditions: $f(x) + f(y) \ge xy$ for all real $x, y$ and for each real $x$ exists real $y$, such that $f(x) + f(y) = xy$.

It is known that the proportion of people with fair hair among people with blue eyes is more than the proportion of people with fair hair among all people. Which is greater , the proportion of people with blue eyes among people with fair hair, or the proportion of people with blue eyes among all people? (Folklore)

Find all real numbers $x_1, \dots, x_{2016}$ that satisfy the following equation for each $1 \le i \le 2016$. (Here $x_{2017} = x_1$.) \[ x_i^2 + x_i - 1 = x_{i+1} \]

Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$. Can number $7$ on the right hand side be replaced with a smaller constant?

Consider two positive real numbers $ a,b $ and the function $ f:(0,\infty )\longrightarrow\left( \sqrt{ab} ,\frac{a+b}{2} \right) $ defined as $ f(x)=-x+\sqrt{x^2+(a+b)x+ab}. $ Prove that it's bijective. [i]D.M. Bătineți-Giurgiu[/i] and [i]Neculai Stanciu[/i]

Given a polynomial of real coefficients P(x) , can it be affirmed that for any real value of x is true of one of the following inequalities: $$P(x) \le P(x)^2; \,\,\, P(x) < 1 + P(x)^2; \,\,\,P(x) \le \frac12 +\frac12 P(x)^2.$$ Find a simple general procedure (among the many existing ones) that allows, provided we are given two polynomials $P(x)$ and $Q(x)$ , find another $M(x)$ such that for every value of $x$, at the same time $-M(x) < P(x)<M(x)$ and $-M(x)< Q(x)<M(x)$.

The set $S$ of positive real numbers $x$ such that \[ \left\lfloor\frac{2x}{5}\right\rfloor + \left\lfloor\frac{3x}{5}\right\rfloor + 1 = \left\lfloor x\right\rfloor \] can be written as $S = \bigcup_{j = 1}^{\infty} I_{j}$, where the $I_{i}$ are disjoint intervals of the form $[a_{i}, b_{i}) = \{x \, | \, a_i \leq x < b_i\}$ and $b_{i} \leq a_{i+1}$ for all $i \geq 1$. Find $\sum_{i=1}^{2017} (b_{i} - a_{i})$.

How many polynomials of degree exactly $5$ with real coefficients send the set $\{1, 2, 3, 4, 5, 6\}$ to a permutation of itself?

$a_1,a_2,...,a_{81}$ are nonzero, $a_i+a_{i+1}>0$ for $i=1,...,80$ and $a_1+a_2+...+a_{81}<0$. What is sign of $a_1*a_2*...*a_{81}$?

Find all solutions to \[ (abcde)^2 = a^2+b^2+c^2+d^2+e^2+f^2. \] in integers. Proposed by [i]Seongjin Shim[/i]