The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.

At the beginning of a month a shop has $10$ different products for sale, each with equal prices. Every day the price of each product is either doubled or trebled. By the beginning of the following month all the prices have become different. Prove that the ratio (the maximal price) /(the minimal price) is greater than $27$. (D. Fomin and Stanislav Smirnov, St Petersburg)

Let a sequence $(a_i)_{i=10}^{\infty}$ be defined as follows: [list=a] [*] $a_{10}$ is some positive integer, which can of course be written in base 10. [*] For $i \geq 10$ if $a_i > 0$, let $b_i$ be the positive integer whose base-$(i + 1)$ representation is the same as $a_i$'s base-$i$ representation. Then let $a_{i + 1} = b_i - 1$. If $a_i = 0$, $a_{i + 1} = 0$. [/list] For example, if $a_{10} = 11$, then $b_{10} = 11_{11} (= 12_{10})$; $a_{11} = 11_{11} - 1 = 10_{11} (= 11_{10})$; $b_{11} = 10_{12} (= 12_{10})$; $a_{12} = 11$. Does there exist $a_{10}$ such that $a_i$ is strictly positive for all $i \geq 10$?

Determine all real numbers $a, b, c, d$ for which $ab+cd=6$ $ac+bd=3$ $ad+bc=2$ $a+b+c+d=6$

Let $x, y,z$ be positive real numbers satisfying $2x^2+3y^2+6z^2+12(x+y+z) =108$. Find the maximum value of $x^3y^2z$. Alexandru Gırban

Find all ordered triples $ (x,y,z)$ of real numbers which satisfy the following system of equations: \[ \left\{\begin{array}{rcl} xy & \equal{} & z \minus{} x \minus{} y \\ xz & \equal{} & y \minus{} x \minus{} z \\ yz & \equal{} & x \minus{} y \minus{} z \end{array} \right. \]

Let $a$, $b$ and $c$ be complex numbers such that $abc = 1$. Find the value of the cubic root of \begin{tabular}{|ccc|} $b + n^3c$ & $n(c - b)$ & $n^2(b - c)$\\ $n^2(c - a)$ & $c + n^3a$ & $n(a - c)$\\ $n(b - a)$ & $n^2(a - b)$ & $a + n^3b$ \end{tabular}

Find the first two values of $40!(\text{mod }1763)$ [hide=Answer]1311, 3074[\hide]

Prove that $ \lfloor{(4+\sqrt11)^{n}}\rfloor $ is odd for every natural number n.

Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$

Let $f$ and $g$ be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose $\deg(f)$ is odd and the sets $\{f(a)\mid a\in \mathbb{Z}\}$ and $\{g(a)\mid a\in \mathbb{Z}\}$ are the same. Prove that there exists an integer $k$ such that $g(x)=f(x+k)$.

[b]p22.[/b] Find the unique ordered pair $(m, n)$ of positive integers such that $x = \sqrt[3]{m} -\sqrt[3]{n}$ satisfies $x^6 + 4x^3 - 36x^2 + 4 = 0$. [b]p23.[/b] Jenny plays with a die by placing it flat on the ground and rolling it along any edge for each step. Initially the face with $1$ pip is face up. How many ways are there to roll the dice for $6$ steps and end with the $1$ face up again? [b]p24.[/b] There exists a unique positive five-digit integer with all odd digits that is divisible by $5^5$. Find this integer. PS. You should use hide for answers.

$a;b\in\mathbb{R^+}$. Prove the following inequality: $$\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\leq\sqrt[3]{2(a+b)(\frac1{a}+\frac1{b})}$$

Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

Let $ A$ be an integer not equal to $ 0$. Solve the following system of equations in $ \mathbb{Z}^3$. $ x \plus{} y^2 \plus{} z^3 \equal{} A$ $ \frac {1}{x} \plus{} \frac {1}{y^2} \plus{} \frac {1}{z^3} \equal{} \frac {1}{A}$ $ xy^2z^3 \equal{} A^2$

$23$ frat brothers are sitting in a circle. One, call him Alex, starts with a gallon of water. On the first turn, Alex gives each person in the circle some rational fraction of his water. On each subsequent turn, every person with water uses the same scheme as Alex did to distribute his water, but in relation to themselves. For instance, suppose Alex gave $\frac{1}{2}$ and $\frac{1}{6}$ of his water to his left and right neighbors respectively on the first turn and kept $\frac{1}{3}$ for himself. On each subsequent turn everyone gives $\frac{1}{2}$ and $\frac{1}{6}$ of the water they started the turn with to their left and right neighbors, respectively, and keep the final third for themselves. After $23$ turns, Alex again has a gallon of water. What possibilities are there for the scheme he used in the first turn? (Note: you may find it useful to know that $1+x+x^2+\cdot +x^{23}$ has no polynomial factors with rational coefficients)

A function $f$, defined on the set of real numbers $\mathbb{R}$ is said to have a [i]horizontal chord[/i] of length $a>0$ if there is a real number $x$ such that $f(a+x)=f(x)$. Show that the cubic $$f(x)=x^3-x\quad \quad \quad \quad (x\in \mathbb{R})$$ has a horizontal chord of length $a$ if, and only if, $0<a\le 2$.

Given two polynomials $f$ and $g$ satisfying $f(x) \ge g(x)$ for all real $x,$ a [i]separating line[/i] between $f$ and $g$ is a line $h(x) = mx+k$ such that $f(x) \ge h(x) \ge g(x)$ for all real $x.$ Consider the set of all possible separating lines between $f(x) = x^2 - 2x + 5$ and $g(x) = 1 - x^2.$ The set of slopes of these lines is a closed interval $[a,b].$ Determine $a^4 + b^4.$

For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?

Prove that if $$\sqrt{x + a} +\sqrt{y+b}+\sqrt{z + c} =\sqrt{y + a} +\sqrt{z + b} +\sqrt{x + c} =\sqrt{z + a} +\sqrt{x+b}+\sqrt{y+c}$$ for some $a, b, c, x, y, z$, then $x = y = z$ or $a = b = c$.

Let $x<0.1$ be a positive real number. Let the [i]foury series[/i] be $4+4x+4x^2+4x^3+\dots$, and let the [i]fourier series[/i] be $4+44x+444x^2+4444x^3+\dots$. Suppose that the sum of the fourier series is four times the sum of the foury series. Compute $x$.

Determine all integers $n$ for which there exist an integer $k\geq 2$ and positive integers $x_1,x_2,\hdots,x_k$ so that $$x_1x_2+x_2x_3+\hdots+x_{k-1}x_k=n\text{ and } x_1+x_2+\hdots+x_k=2019.$$

Determine all solutions to the system of equations: \[x^2 + y^2 + x + y = 12\]\[xy + x + y = 3\] [This is the exact form of problem that appeared on the paper, but I think it means to solve in $\mathbb R.$]

What is the $7$th term of the sequence $\{-1, 4,-2, 3,-3, 2,...\}$? (A) $ -1$ (B) $ -2$ (C) $-3$ (D) $-4$ (E) None of the above

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$.