Find $\int_0^x \frac{dt}{1+t^2}+\int_0^{\frac{1}{x}} \frac{dt}{1+t^2}\ (x>0).$

Let $B$ be a subset of $\mathbb{Z}_{3}^{n}$ with the property that for every two distinct members $(a_{1},\ldots,a_{n})$ and $(b_{1},\ldots,b_{n})$ of $B$ there exist $1\leq i\leq n$ such that $a_{i}\equiv{b_{i}+1}\pmod{3}$. Prove that $|B| \leq 2^{n}$.

Determine all pairs of positive real numbers $(a, b)$ for which there exists a function $ f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}} $ satisfying for all positive real numbers $x$ the equation $ f(f(x))=af(x)- bx $

Let $G$ be group with identity $e$ and $\phi :G\to G$ be a function such that : \[ \phi(g_1)\cdot \phi(g_2)\cdot \phi(g_3)=\phi(h_1)\cdot \phi(h_2)\cdot \phi(h_3) \] Whenever $g_1\cdot g_2\cdot g_3=e=h_1\cdot h_2\cdot h_3$ Show there exists $a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism. (that is $\psi(x\cdot y)=\psi (x)\cdot \psi(y)$ for all $x,y\in G$ )

Find all functions $f:\mathbb{N}^* \rightarrow \mathbb{N}$ such that [list] [*] $f(x \cdot y) = f(x) + f(y)$ [*] $f(30) = 0$ [*] $f(x)=0$ always when the units digit of $x$ is $7$ [/list]

Let $k<<n$ denote that $k<n$ and $k\mid n$. Let $f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\}$ be such that, if $n\leq 2013$ and $k<<n$, then $f(k)<<f(n)$. What’s the least possible value of $M$?

[b]4.[/b] Let $f(x)$ be a real- or complex-value integrable function on $(0,1)$ with $\mid f(x) \mid \leq 1 $. Set $ c_k = \int_0^1 f(x) e^{-2 \pi i k x} dx $ and construct the following matrices of order $n$: $ T= (t_{pq})_{p,q=0}^{n-1}, T^{*}= (t_{pq}^{*})_{p,q =0}^{n-1} $ where $t_{pq}= c_{q-p}, t^{*}= \overline {c_{p-q}}$ . Further, consider the following hyper-matrix of order $m$: $ S= \begin{bmatrix} E & T & T^2 & \dots & T^{m-2} & T^{m-1} \\ T^{*} & E & T & \dots & T^{m-3} & T^{m-2} \\ T^{*2} & T^{*} & E & \dots & T^{m-3} & T^{m-2} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ T^{*m-1} & T^{*m-2} & T^{*m-3} & \dots & T^{*} & E \end{bmatrix} $ ($S$ is a matrix of order $mn$ in the ordinary sense; E denotes the unit matrix of order $n$). Show that for any pair $(m , n) $ of positive integers, $S$ has only non-negative real eigenvalues. [b](R. 19)[/b]

Find all functions $ f : \mathbb{R}\to\mathbb{R}$ that satisfy \[ f (x^{3} \plus{} y^{3}) \equal{} x^{2}f (x) \plus{} yf (y^{2}) \] for all $ x, y \in\mathbb R.$

Known $f:\mathbb{N}_0 \to \mathbb{N}_0$ function for $\forall x,y\in \mathbb{N}_0$ the following terms are paid $(a). f(0,y)=y+1$ $(b). f(x+1,0)=f(x,1)$ $(c). f(x+1,y+1)=f(x,f(x+1,y)).$ Find the value if $f(4,1981)$

The function $ f$ satisfies \[f(x) \plus{} f(2x \plus{} y) \plus{} 5xy \equal{} f(3x \minus{} y) \plus{} 2x^2 \plus{} 1\] for all real numbers $ x$, $ y$. Determine the value of $ f(10)$.

Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x) \geq 0$ for all $x$ in $[0,2\pi]$. Show that \[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]

Let be three polynomials of degree two $ p_1,p_2,p_3\in\mathbb{R} [X] $ and the function $$ f:\mathbb{R}\longrightarrow\mathbb{R} ,\quad f(x)=\max\left( p_1(x),p_2(x),p_3(x)\right) . $$ Then, $ f $ is differentiable if and only if any of these three polynomials dominates the other two.

Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$. (a) Prove that $f_n$ is well defined. (b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective. [i]Bogdan Enescu[/i]

On the $x$-$y$ plane, 3 half-lines $y=0,\ (x\geq 0),\ y=x\tan \theta \ (x\geq 0),\ y=-\sqrt{3}x\ (x\leq 0)$ intersect with the circle with the center the origin $O$, radius $r\geq 1$ at $A,\ B,\ C$ respectively. Note that $\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}$. If the area of quadrilateral $OABC$ is one third of the area of the regular hexagon which inscribed in a circle with radius 1, then evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} r^2d\theta .$ [i]2011 Waseda University of Education entrance exam/Science[/i]

Let the definite integral $ I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots )$. (1) Find $ I_0,\ I_{\minus{}1},\ I_2$. (2) Find $ I_1$. (3) Express $ I_{n\plus{}2}$ in terms of $ I_n$. (4) Find $ I_{\minus{}3},\ I_{\minus{}2},\ I_3$. (5) Evaluate the definite integrals $ \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx$ in using the avobe results. You are not allowed to use the formula of integral for $ \sqrt{x^2\plus{}1}$ directively here.

Let $f$, $g$, $h : [a, b] \to \mathbb{R}$, three integrable functions such that:$$\int \limits_a^b fgdx=\int \limits_a^bghdx=\int \limits_a^bhfdx=\int \limits_a^bg^2dx\int \limits_a^bh^2dx=1$$Then$$\int \limits_a^bg^2dx=\int \limits_a^bh^2dx=1$$

Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ and relatively prime to $n$. Find all natural numbers $n$ and primes $p$ such that $\phi(n)=\phi(np)$.

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

Let the $f:\mathbb{N}\rightarrow\mathbb{N}$ be the function such that (i) For all positive integers $n,$ $f(n+f(n))=f(n)$ (ii) $f(n_o)=1$ for some $n_0$ Prove that $f(n)\equiv 1.$

Let $f: N \to N$ be a function satisfying (a) $1\le f(x)-x \le 2019$ $\forall x \in N$ (b) $f(f(x))\equiv x$ (mod $2019$) $\forall x \in N$ Show that $\exists x \in N$ such that $f^k(x)=x+2019 k, \forall k \in N$

Let $ X_1,X_2,\ldots, X_n$ be independent, identically distributed, nonnegative random variables with a common continuous distribution function $ F$. Suppose in addition that the inverse of $ F$, the quantile function $ Q$, is also continuous and $ Q(0)=0$. Let $ 0=X_{0: n} \leq X_{1: n} \leq \ldots \leq X_{n: n}$ be the ordered sample from the above random variables. Prove that if $ EX_1$ is finite, then the random variable \[ \Delta = \sup_{0\leq y \leq 1} \left| \frac 1n \sum_{i=1}^{\lfloor ny \rfloor +1} (n+1-i)(X_{i: n}-X_{i-1: n})- \int_0^y (1-u)dQ(u) \right|\] tends to zero with probability one as $ n \rightarrow \infty$. [i]S. Csorgp, L. Horvath[/i]

Find the number of all real functions $f$ which map the sum of $n$ elements into the sum of their images, such that $f^{n-1}$ is a constant function and $f^{n-2}$ is not. Here $f^0(x) = x$ and $f^k = f \circ f^{k-1}$ for $k \ge 1$.

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

Let $f(x)=Ax^2+Bx+C$, $g(x)=ax^2+bx+c$ be two quadratic polynomial functions with real coefficients that satisfy the relation \[|f(x)| \ge |g(x)|\] for all real $x$. Prove that $|b^2-4ac| \le |B^2-4AC|.$ My solution was nearly complete...