Determine the largest real number \(C\) such that $$\frac{1}{x}+\frac{1}{2y}+\frac{1}{3z}\geqslant C$$ for all real numbers \(x,y,z\neq 0\) satisfying the equation $$\frac{x}{yz}+\frac{4y}{xz}+\frac{9z}{xy}=24$$

Suppose $abc\neq 0$. Express in terms of $a,b,$ and $c$ the solutions $x,y,z,u,v,w$ of the equations $$x+y=a,\quad z+u=b,\quad v+w=c,\quad ay=bz,\quad ub=cv,\quad wc=ax.\quad$$

Let $P(x) = x^3 - 6x^2 - 5x + 4$. Suppose that $y$ and $z$ are real numbers such that \[ zP(y) = P(y - n) + P(y + n) \] for all reals $n$. Evaluate $P(y)$.

Find all functions $f:\mathbb R\to\mathbb R$ satisfying the equality $$ f(f(x)+f(y))=(x+y)f(x+y) $$ for all real $x$ and $y$. [i](B. Serankou)[/i]

Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying $$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds. (Karl Czakler)

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

Determine all polynomials $P(x)$ with degree $n\geq 1$ and integer coefficients so that for every real number $x$ the following condition is satisfied $$P(x)=(x-P(0))(x-P(1))(x-P(2))\cdots (x-P(n-1))$$

Let $R[x]$ be the whole set of real coefficient polynomials, and define the mapping $T: R[x] \to R[x]$ as follows: For $$f (x) = a_nx^{n} + a_{n-1}x^{n- 1} +...+ a_1x + a_0,$$ let $$T(f(x))=a_{n}x^{n+1} + a_{n-1}x^{n} + (a_n+a_{n-2})x^{n-1 } + (a_{n-1}+a_{n-3})x^{n-2}+...+(a_2+a_0)x+a_1.$$ Assume $P_0(x)= 1$, $P_n(x) = T(P_{n-1}(x))$ ( $n=1,2,...$), find the constant term of $P_n(x)$.

Determine the smallest odd integer $n \ge 3$, for which there exist $n$ rational numbers $x_1 , x_2 , . . . , x_n$ with the following properties: $(a)$ \[\sum_{i=1}^{n} x_i =0 , \ \sum_{i=1}^{n} x_i^2 = 1.\] $(b)$ \[x_i \cdot x_j \ge - \frac 1n \ \forall \ 1 \le i,j \le n.\]

Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation: \[Q(a+b) = \frac{P(a) - P(b)}{a - b}\] $\forall a, b \in \mathbb{Z}^+$ and $a>b$

show that the polynomial $x^4+3x^3+6x^2+9x+12$ cannot be written as the product of 2 polynomials of degree 2 with integer coefficients.

Prove that the equation $x^{19} + x^{17} = x^{16 }+ x^7 + a$ for any $a \in R$ has at least two imaginary roots

Let $p(x)$ be the monic cubic polynomial with roots $\sin^2(1^{\circ})$, $\sin^2(3^{\circ})$, and $\sin^2(9^{\circ})$. Suppose that $p\left(\frac{1}{4}\right)=\frac{\sin(a^{\circ})}{n\sin(b^{\circ})}$, where $0 <a,b \le 90$ and $a,b,n$ are positive integers. What is $a+b+n$? [i]Proposed by Andrew Yuan[/i]

Set \[A_n=\sum_{k=1}^n \frac{k^6}{2^k}.\] Find $\lim_{n\to\infty} A_n.$

A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by \[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\] for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$

Let $\alpha\neq0$ be a real number. Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f\left(x^2+y^2\right)=f(x-y)f(x+y)+\alpha yf(y)\] holds for all $x, y\in\mathbb R.$

Show that the following implication holds for any two complex numbers $x$ and $y$: if $x+y$, $x^2+y^2$, $x^3+y^3$, $x^4+y^4\in\mathbb Z$, then $x^n+y^n\in\mathbb Z$ for all natural n.

Consider the curve $\Gamma$ defined by the equation $y^2 = x^3 +bx+b^2$, where $b$ is a nonzero rational constant. Inscribe in the curve $\Gamma$ a triangle whose vertices have rational coordinates.

If $a(x), b(x), c(x)$ and $d(x)$ are polynomials in $ x$, show that $$ \int_{1}^{x} a(x) c(x)\; dx\; \cdot \int_{1}^{x} b(x) d(x) \; dx - \int_{1}^{x} a(x) d(x)\; dx\; \cdot \int_{1}^{x} b(x) c(x)\; dx$$ is divisible by $(x-1)^4.$

Show that among the square roots of the first $ 2015 $ natural numbers, we cannot choose an arithmetic sequence composed of $ 45 $ elements.

Initially, a non-constant polynomial $S(x)$ with real coefficients is written down on a board. Whenever the board contains a polynomial $P(x)$, not necessarily alone, one can write down on the board any polynomial of the form $P(C + x)$ or $C + P(x)$ where $C$ is a real constant. Moreover, if the board contains two (not necessarily distinct) polynomials $P(x)$ and $Q(x)$, one can write $P(Q(x))$ and $P(x) + Q(x)$ down on the board. No polynomial is ever erased from the board. Given two sets of real numbers, $A = \{ a_1, a_2, \dots, a_n \}$ and $B = \{ b_1, \dots, b_n \}$, a polynomial $f(x)$ with real coefficients is $(A,B)$-[i]nice[/i] if $f(A) = B$, where $f(A) = \{ f(a_i) : i = 1, 2, \dots, n \}$. Determine all polynomials $S(x)$ that can initially be written down on the board such that, for any two finite sets $A$ and $B$ of real numbers, with $|A| = |B|$, one can produce an $(A,B)$-[i]nice[/i] polynomial in a finite number of steps. [i]Proposed by Navid Safaei, Iran[/i]

A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$ 1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ . 2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$ 3)Describe the possible values of $a_{1435}$ 4)Prove that the values that you got in (3) are correct

Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that $S$ is a positive integer, compute the number of positive divisors of $S$.

Let $A$ and $B$ be given real numbers. Let the sum of real numbers $0\le x_1\le x_2\le\ldots \le x_n$ be $A$, and let the sum of real numbers $0\le y_1\le y_2\le \ldots\le y_n$ be $B$. Find the largest possible value of \[\sum_{i=1}^n (x_i-y_i)^2.\] [i]Proposed by Péter Csikvári, Budapest[/i]

Find all sequences of positive integers $(a_n)_{n\ge 1}$ which satisfy $$a_{n+2}(a_{n+1}-1)=a_n(a_{n+1}+1)$$ for all $n\in \mathbb{Z}_{\ge 1}$. [i](Bogdan Blaga)[/i]