Show that there is precisely one sequence $ a_1,a_2,...$ of integers which satisfies $ a_1\equal{}1, a_2>1,$ and $ a_{n\plus{}1}^3\plus{}1\equal{}a_n a_{n\plus{}2}$ for $ n \ge 1$.

Suppose that the function $f:\mathbb{R}\to\mathbb{R}$ satisfies \[f(x^3 + y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2)\] for all $x,y\in\mathbb{R}$. Prove that $f(1996x)=1996f(x)$ for all $x\in\mathbb{R}$.

show that thee is no function f definedonthe positive real numbes such that : $f(y) > (y-x)f(x)^2$

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

The faces of a cubical die are marked with the numbers $ 1$, $ 2$, $ 2$, $ 3$, $ 3$, and $ 4$. The faces of a second cubical die are marked with the numbers $ 1$, $ 3$, $ 4$, $ 5$, $ 6$, and $ 8$. Both dice are thrown. What is the probability that the sum of the two top numbers will be $ 5$, $ 7$, or $ 9$ ? $ \textbf{(A)}\ \frac {5}{18} \qquad \textbf{(B)}\ \frac {7}{18} \qquad \textbf{(C)}\ \frac {11}{18} \qquad \textbf{(D)}\ \frac {3}{4} \qquad \textbf{(E)}\ \frac {8}{9}$

Let the complex function $F(z)$ be regular on the punctuated disk $\{ 0<|z| < R\}$. By a [i]level curve[/i] we mean a component of the level set of $\mathrm{Re}F(z)$, that is, a maximal connected set on which $\mathrm{Re}F(z)$ is constant. Denote by $A(r)$ the union of those level curves that are entirely contained in the punctuated disk $\{ 0<|z|<r\}$. Prove that if the number of components of $A(r)$ has an upper bound independent of $r$ then $F(z)$ can only have a pole type singularity at $0$.

Find the number of functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ for which $f(h+k)+f(hk)=f(h)f(k)+1$, for all integers $h$ and $k$.

Let $a\# b$ be defined as $ab-a-3$. For example, $4\#5=20-4-3=13$ Compute $(2\#0)\#(1\#4)$.

Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1$. Iteratively, for $n=0,1,2,\dots$, a point $x_{n+1}$ is chosen uniformly form the interval $[0,x_n]$. Let $Z$ be the smallest value of $n$ for which $x_n<\delta$. Find the expected value of $Z$, as a function of $\delta$.

Give an example of a continuous real-valued function $f$ form $[0,1]$ to $[0,1]$ which takes on every value in $[0,1]$ an infinite number of times.

For every rational number $m>0$ we consider the function $f_m:\mathbb{R}\rightarrow\mathbb{R},f_m(x)=\frac{1}{m}x+m$. Denote by $G_m$ the graph of the function $f_m$. Let $p,q,r$ be positive rational numbers. a) Show that if $p$ and $q$ are distinct then $G_p\cap G_q$ is non-empty. b) Show that if $G_p\cap G_q$ is a point with integer coordinates, then $p$ and $q$ are integer numbers. c) Show that if $p,q,r$ are consecutive natural numbers, then the area of the triangle determined by intersections of $G_p,G_q$ and $G_r$ is equal to $1$.

Determine all functions $f:\mathbb R\rightarrow\mathbb R$ satisfying the condition \[f(y^2+2xf(y)+f(x)^2)=(y+f(x))(x+f(y))\] for all real numbers $x$ and $y$.

Determine all totally multiplicative and non-negative functions $f\colon\mathbb{Z}\rightarrow \mathbb{Z}$ with the property that if $a, b\in \mathbb{Z}$ and $b\neq 0$, then there exist integers $q$ and $r$ such that $a-qb+r$ and $f(r)<f(b)$.

Find all functions $f:\mathbb{R}\to \mathbb R$ satisfying \[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\] For all $x,y\in\mathbb R$.

Let $x$ and $y$ be positive real numbers with $x + y =1 $. Prove that $$\frac{(3x-1)^2}{x}+ \frac{(3y-1)^2}{y} \ge1.$$ For which $x$ and $y$ equality holds? (K. Czakler, GRG 21, Vienna)

The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function $ f(x) \equal{} ax^{2} \plus{} bx \plus{} c$ are equal. Their common value must also be which of the following? $ \textbf{(A)}\ \text{the coefficient of }x^{2}\qquad \textbf{(B)}\ \text{the coefficient of }x$ $ \textbf{(C)}\ \text{the y \minus{} intercept of the graph of }y \equal{} f(x)$ $ \textbf{(D)}\ \text{one of the x \minus{} intercepts of the graph of }y \equal{} f(x)$ $ \textbf{(E)}\ \text{the mean of the x \minus{} intercepts of the graph of }y \equal{} f(x)$

Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that : \[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\] [i]Proposed by Mohammad Ahmadi[/i]

Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial \[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n. \]

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$: \[f(x-y+f(y))=f(x)+f(y).\]

Last year, Master Cheung is famous for multi-rotation. This year, he comes to DAMO to make noodles for sweeping monk. One day, software engineer Xiao Li talks with Master Cheung about his job. Xiao Li mainly researches and designs the algorithm to adjust the paramter of different kinds of products. These paramters can normally be obtainly by minimising loss function $f$ on $\mathbb{R}^n$. In the recent project of Xiao Li, this loss function is obtained by other topics. For safety consideration and technique reasons, this topic makes Xiao Li difficult to find the interal details of the function. They only provide a port to calculate the value of $f(\text x)$ for any $\text x\in\mathbb{R}^n$. Therefore, Xiao Li must only use the value of the function to minimise $f$. Also, every times calculating the value of $f$ will use a lot of calculating resources. It is good to know that the dimension $n$ is not very high (around $10$). Also, colleague who provides the function tells Xiao Li to assume $f$ is smooth first. This problem reminds Master Cheung of his antique radio. If you want to hear a programme from the radio, you need to turn the knob of the radio carefully. At the same time, you need to pay attention to the quality of the radio received, until the quality is the best. In this process, no one knows the relationship between the angle of turning the knob and the quality of the radio received. Master Cheung and Xiao Li realizes that minimising $f$ is same as adjusting the machine with multiple knobs: Assume every weight of $\text x$ is controlled by a knob. $f(\text x)$ is a certain performance of the machine. We only need to adjust every knobs again and again and observes the value of $f$ in the same time. Maybe there is hope to find the best $\text x$. As a result, two people suggest an iteration algorithm (named Automated Forward/Backward Tuning, $\text{AFBT}$, to minimise $f$. In $k$-th iteration, the algorithm adjusts the individual weight of $\text{x}_k$ to $2n$ points $\{\text x_k\pm t_k\text e^i:i=1,...,n\}$, where $t_k$ is the step size; then, make $y_k$ be the smallest one among the value of the function of thosse points. Then check if $\text y_k$ sufficiently makes $f$ decrease; then, take $\text x_{k+1}=\text y_k$, then make the step size doubled. Otherwise, make $\text x_{k+1}=\text x_k$ and makes the step size decrease in half. In the algorithm, $\text e^i$ is the $i$-th coordinate vector in $\mathbb{R}^n$. The weight of $i$-th is $1$. Others are $0$; $\mathbf{1}(\cdot)$ is indicator function. If $f(\text x_k)-f(\text y_k)$ is at least the square of $t_k$, then take the value of $\mathbf{1}(f(\text k)-f(y_k)\ge t^2_k)$ as $1$. Otherwise, take it as $0$. $\text{AFBT}$ algorithm Input $\text{x}_0\in \mathbb{R}^n$, $t_0>0$. For $k=0, 1, 2, ...$, perform the following loop: 1: #Calculate loss function. 2: $s_k:=\mathbb{1}[f(\text{x}_k)-f(\text{y}_k)\ge t^2_k]$ #Is it sufficiently decreasing? Yes: $s_k=1$; No: $s_k=0$. 3: $\text{x}_{k+1}:=(1-s_k)\text{x}_k+s_k\text{y}_k$ #Update the point of iteration. 4: $t_{k+1}:=2^{2S_k-1}t_k$ #Update step size. $s_k=1$: Step size doubles; $s_k=0$: Step size decreases by half. Now, we made assumption to the loss function $f:\mathbb{R}^n\to \mathbb{R}$. Assumption 1. Let $f$ be a convex function. For any $\text{x}, \text{y}\in \mathbb{R}^n$ and $\alpha \in [0, 1]$, we have $f((1-\alpha)\text{x}+\text{y})\le (1-\alpha)f(\text{x})+\alpha f(\text{y})$. Assumption 2. $f$ is differentiable on $\mathbb{R}^n$ and $\nabla f$ is L-Lipschitz continuous on $\mathbb{R}^n$. Assumption 3. The level set of $f$ is bounded. For any $\lambda\in\mathbb{R}$, set $\{\text x\in \mathbb{R}^n:f(\text x)\le \lambda\}$ is all bounded. Based on assumption 1 and 2, we can prove that $\left\langle \nabla f(\text x),\text y-\text x \right\rangle \le f(\text y)-f(\text x)\le \left\langle \nabla f(\text x),\text y-\text x\right\rangle+\frac{L}{2}||\text x-\text y||^2$ You can refer to any convex analysis textbook for more properties of convex function. Prove that under the assumption 1-3, for $AFBT$, $\lim_{k \to \infty}f(\text{x}_k)=f^*$

Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not. [i](Russia) Valery Senderov[/i]

Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for any $x,y\in \mathbb{R}$, there holds \[f(x+f(y))+f(xy)=yf(x)+f(y)+f(f(x)).\]

Let $a,b,c$ be non-negative real numbers such that $a+b\leq c+1, b+c\leq a+1$ and $c+a\leq b+1.$ Show that \[a^2+b^2+c^2\leq 2abc+1.\]

Let $x,y,z$ be complex numbers satisfying \begin{align*} z^2 + 5x &= 10z \\ y^2 + 5z &= 10y \\ x^2 + 5y &= 10x \end{align*} Find the sum of all possible values of $z$. [i]Proposed by Aaron Lin[/i]