There is a sequence with $a(2)=0$, $a(3)=1$ and $a(n)=a\left(\left\lfloor\dfrac n2\right\rfloor\right)+a\left(\left\lceil\dfrac n2\right\rceil\right)$ for $n\geq 4$. Find $a(2014)$. [Note that $\left\lfloor\dfrac n2\right\rfloor$ and $\left\lceil\dfrac n2\right\rceil$ denote the floor function (largest integer $\leq\tfrac n2$) and the ceiling function (smallest integer $\geq\tfrac n2$), respectively.]

Find the smallest positive number $a$ for which $$\sin a^o = \sin a$$ (on the left ($a^o$) is an angle of $a$ degrees, on the right is an angle in $a$ radians).

Vasya has a numeric expression $$ \Box \cdot \Box +\Box \cdot \Box $$ and 4 cards with numbers that can be put on 4 free places in the expression. Vasya tried to put cards in all possible ways and all the time obtained the same value as a result. Prove that equal numbers are written on three of his cards.

Solve in reals: \begin{eqnarray*}a^2=b^3+c^3 \\ b^2=c^3+d^3 \\ c^2=d^3+e^3 \\ d^2=e^3+a^3 \\ e^2=a^3+b^3 \end{eqnarray*}

Quadratic trinomials $f$ and $g$ with integer coefficients obtain only positive values and the inequality $\dfrac{f(x)}{g(x)}\geq \sqrt{2}$ is true $\forall x\in\mathbb{R}$. Prove that $\dfrac{f(x)}{g(x)}>\sqrt{2}$ is true $\forall x\in\mathbb{R}$ [I]Proposed by A. Khrabrov[/i]

An integer $1$ is written on the blackboard. We are allowed to perform the following operations:to change the number $x$ to $3x+1$ of to $[\frac{x}{2}]$. Prove that we can get all positive integers using this operations.

The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$. Determine the value of $k$.

Call a triple of numbers [b]Nice[/b] if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ [b] Nice[/b] triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios Proposed by [i]Morteza Saghafian[/i]

Find positive real numbers $x,y,z$ that are solutions of the system $x+y+z=xy+yz+zx$ and $xyz=1$ , and have the smallest possible sum.

For the real numbers $a_1,...,a_n, b_1,...,b_m$ , the following relations hold: 1) $|a_i|= |b_j|=1$, $i=1,...,n$ ,$j=1,...,m$ 2) $a_1\sqrt{2+a_2\sqrt{2+...+a_n\sqrt2}}=b_1\sqrt{2+b_2\sqrt{2+...+b_m\sqrt2}}$ Prove that $n = m$ and $a_i=b_i$ , $i=1,...,n$

Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below. \[ \begin{array}{l} 2x_1+x_2+x_3+x_4+x_5=6 \\ x_1+2x_2+x_3+x_4+x_5=12 \\ x_1+x_2+2x_3+x_4+x_5=24 \\ x_1+x_2+x_3+2x_4+x_5=48 \\ x_1+x_2+x_3+x_4+2x_5=96 \\ \end{array} \]

For a positive integer k, let $f_{1}(k)$ be the square of the sum of the digits of k. (For example $f_{1}(123)=(1+2+3)^{2}=36$.) Let $f_{n+1}(k)=f_{1}(f_{n}(k))$. Determine the value of the $f_{2007}(2^{2006})$. Justify your claim.

Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression: (b-a)(b^3+3a^3)

Let $a$ and $b$ be real numbers. It is known that the graph of the parabola $y =ax^2 +b$ cuts the graph of the curve $y = x+1/x$ in exactly three points. Prove that $3ab < 1$.

Let $a_1, a_2, ...,a_n$ be positive real numbers such that $$a_1 + a_2 + ... + a_n = a_1^2 + a_2^2 + ... + a_n^2$$ Prove that $$\sum_{1\le i<j\le n} a_ia_j(1 - a_ia_j) \ge 0$$ Võ Quốc Bá Cẩn.

Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial $f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.

Let $a, b, c, d$ be integers such that $a > b > c > d \ge -2021$ and $$\frac{a + b}{b + c}=\frac{c + d}{d + a}$$ (and $b + c \ne 0 \ne d + a$). What is the maximum possible value of $ac$?

Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy equation: $$ f(x^3+y^3) =f(x^3) + 3x^2f(x)f(y) + 3f(x)f(y)^2 + y^6f(y) $$ for all reals $x,y$

Find all polynomials $P(x) = x^n +a_1x^{n-1} +...+a_n$ whose zeros (with their multiplicities) are exactly $a_1,a_2,...,a_n$.

Let $n\geq 2$ be an integer. For each natural $m$ and each integer sequence $0<k_1<k_2<\cdots <k_m$ for which $k_1+\cdots+k_m=n$, Michael wrote down the number $\frac{1}{k_1\cdot k_2\cdots k_m} $ on the board. Prove that the sum of the numbers on the board is less than $1$.

Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer. [i]Proposed by fattypiggy123[/i]

Solve the system $\begin{cases} x^3 - y^3 = 26 \\ x^2y - xy^2 = 6 \end{cases}$ in $C$ [hide=other version]solved below Solve the system $\begin{cases} x^3 - y^3 = 2b \\ x^2y - xy^2 = b \end{cases}$[/hide]

Find all functions $f$ mapping non-empty finite sets of integers, to integers, such that $$f(A+B)=f(A)+f(B)$$ for all non-empty sets of integers $A$ and $B$. $A+B$ is defined as $\{a+b: a \in A, b \in B\}$.

[b]p1.[/b] A [i]multimagic [/i] square is a $3 \times 3$ array of distinct positive integers with the property that the product of the $3$ numbers in each row, each column, and each of the two diagonals of the array is always the same. (a) Prove that the numbers $1, 2, 3, . . . , 9$ cannot be used to form a multimagic square. (b) Give an example of a multimagic square. [b]p2.[/b] A sequence $a_1, a_2, a_3, ... , a_n$ of real numbers is called an arithmetic progression if $$a_1 - a_2 = a_2 - a_3 = ... = a_{n-1} - a_n.$$ Prove that there exist distinct positive integers $n_1, n_2, n_3, ... , n_{2014}$ such that $$\frac{1}{n_1},\frac{1}{n_2}, ... ,\frac{1}{n_{2014}}$$ is an arithmetic progression. [b]p3.[/b] Let $\lfloor x \rfloor$ be the largest integer that is less than or equal to $x$. For example, $\lfloor 3.9 \rfloor = 3$ and $\lfloor 4\rfloor = 4$. Determine (with proof) all real solutions of the equation $$x^2 - 25 \lfloor x\rfloor + 100 = 0.$$ [b]p4.[/b] An army has $10$ cannons and $8$ carts. Each cart can carry at most one cannon. It takes one day for a cart to cross the desert. What is the least number of days that it takes to get the cannons across the desert? (Cannons can be left part way and picked up later during the procedure.) Prove that the amount of time that your solution requires to move the cannons across the desert is the smallest possible. [b]p5.[/b] Let $C$ be a convex polygon with $4031$ sides. Let $p$ be the length of its perimeter and let $d$ be the sum of the lengths of its diagonals. Show that $$\frac{d}{p}> 2014.$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].