Let $f:[0,1]\to(0,\infty)$ be a continous function on $[0,1]$ and let $A=\int_0^1 f(t)\mathrm{d}t.$[list=a] [*]Consider the function $F:[0,1]\to[0,A]$ defined by \[F(x)=\int_0^xf(t)\mathrm{d}t.\]Prove that $F(x)$ has an inverse function, which is differentiable. [*]Prove that there exists a unique function $g:[0,1]\to[0,1]$ for which\[\int_0^xf(t)\mathrm{d}t=\int_{g(x)}^1f(t)\mathrm{d}t\]is satisfied for every $x\in [0,1].$ [*]Prove that there exists $c\in[0,1]$ for which\[\lim_{x\to c}\frac{g(x)-c}{x-c}=-1,\]whre $g$ is the function uniquely determined at b. [/list]

Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $ n < 60$ (both Kelly and Jason know that $ n < 60$, but that they don't know what the value of $ n$ is). The computer tells Kelly the unit digit of $ n$, and it tells Jason the number of divisors of $ n$. Then, Kelly and Jason have the following dialogue: Kelly: I don't know what $ n$ is, and I'm sure that you don't know either. However, I know that $ n$ is divisible by at least two different primes. Jason: Oh, then I know what the value of $ n$ is. Kelly: Now I also know what $ n$ is. Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of $ n$?

In coordinate system, there are six points $P_i(x_i,y_i)(i=1,2,\cdots,6)$, satisfying: (1) $x_i,y_i\in\{-2,-1,0,1,2\}$. (2) For any three points, they are not collinear. Prove that there exists a triangle $\triangle P_iP_jP_k(1\leq i<j<k\leq6)$, its area is not larger than $2$.

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that the following conditions are true for every pair of positive integers $(x, y)$: $(i)$: $x$ and $f(x)$ have the same number of positive divisors. $(ii)$: If $x \nmid y$ and $y \nmid x$, then: $$\gcd(f(x), f(y)) > f(\gcd(x, y))$$

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Suppose $k \geq 2$ is an integer such that for all $x, y \in G$ and $i \in \{k-1, k, k+1\}$ the relation $(xy)^{i}= x^{i}y^{i}$ holds. Show that $G$ is Abelian.

Let $p=4k+3$ be a prime number. Find the number of different residues mod p of $(x^{2}+y^{2})^{2}$ where $(x,p)=(y,p)=1.$

There are $n$ blocks placed on the unit squares of a $n \times n$ chessboard such that there is exactly one block in each row and each column. Find the maximum value $k$, in terms of $n$, such that however the blocks are arranged, we can place $k$ rooks on the board without any two of them threatening each other. (Two rooks are not threatening each other if there is a block lying between them.)

Let $ABC$ be a triangle and let the points $D, E$ be on the rays $AB$, $AC$ such that $BCED$ is cyclic. Prove that the following two statements are equivalent: $\bullet$ There is a point $X$ on the circumcircle of $ABC$ such that $BDX$, $CEX$ are tangent to each other. $\bullet$ $AB \cdot AD \le 4R^2$, where $R$ is the radius of the circumcircle of $ABC$.

Prove that for all $ n\ge 2 $ natural numbers there exist $ a_n\in\mathbb{Q} $ such that $$ X^{2n}+a_nX^n+1\Huge\vdots X^2+\frac{1}{2}X+1, $$ and that there isn´t any $ a_n\in\mathbb{R}\setminus\mathbb{Q} $ with this property.

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Let be a natural number $ n\ge 2 $ and $ n $ positive real numbers $ a_1,a_n,\ldots ,a_n $ that satisfy the inequalities $$ \sum_{j=1}^i a_j\le a_{i+1} ,\quad \forall i\in\{ 1,2,\ldots ,n-1 \} . $$ Prove that $$ \sum_{k=1}^{n-1} \frac{a_k}{a_{k+1}}\le n/2 . $$

Find all values of $a$ for which the inequality $$a^2x^2 + y^2 + z^2 \ge ayz+xy+xz$$ holds for all $x$, $y$ and $z$.

If $a$, $b$ and $c$ are the roots of the equation $x^3 - x^2 - x - 1 = 0$, (i) show that $a$, $b$ and $c$ are distinct: (ii) show that \[\frac{a^{1982} - b^{1982}}{a - b} + \frac{b^{1982} - c^{1982}}{b - c} + \frac{c^{1982} - a^{1982}}{c - a}\] is an integer.

When the integer $ {\left(\sqrt{3} \plus{} 5\right)}^{103} \minus{} {\left(\sqrt{3} \minus{} 5\right)}^{103}$ is divided by 9, what is the remainder?

Let $f,g,h:\mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is differentiable, $g$ and $h$ are monotonic, and $f'=f+g+h$. Prove that the set of the points of discontinuity of $g$ coincides with the respective set of $h$.

Compute the number of positive integers $n < 2012$ that share exactly two positive factors with 2012. [i]Proposed by Aaron Lin[/i]

Let $a_1, a_2, a_3, \dots$ be a sequence of integers such that $\text{(i)}$ $a_1=0$ $\text{(ii)}$ for all $i\geq 1$, $a_{i+1}=a_i+1$ or $-a_i-1$. Prove that $\frac{a_1+a_2+\cdots+a_n}{n}\geq-\frac{1}{2}$ for all $n\geq 1$.

Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$

There are two bowls on the table, in one there are $p$, in the other $q$ stones ($p, q \in N*$ ). Two players $A$ and $B$ take turns playing, starting with $A$. Who's turn: $\bullet$ takes a stone from one of the bowls $\bullet$or removes one stone from each bowl $\bullet$ or puts a stone from one of the bowls into the other. Whoever takes the last stone wins. Under what conditions can $A$ and under what conditions can $B$ force the win? The answer must be justified.

Two circles $\omega , \Omega$ intersect in distinct points $A,B$ a line through $B$ intersects $\omega , \Omega$ in $C,D$ respectively such that $B$ lies between $C,D$ another line through $B$ intersects $\omega , \Omega$ in $E,F$ respectively such that $E$ lies between $B,F$ and $FE=CD$. Furthermore $CF$ intersects $\omega , \Omega$ in $P,Q$ respectively and $M,N$ are midpoints of the arcs $PB,QB$. Prove that $CNMF$ is a cyclic quadrilateral.

Let $ \left( a_n \right)_{n\ge 1} $ be an arithmetic progression of positive real numbers, and $ m $ be a natural number. Calculate: [b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^{2m+2}} \sum_{1\le i<j\le n} a_i^ma_j^m $ [b]b)[/b] $ \lim_{n\to\infty } \frac{1}{a_n^{2m+2}} \sum_{1\le i<j\le n} a_i^ma_j^m $ [i]Dumitru Acu[/i]

Find all values of positive integers $a$ and $b$ such that it is possible to put $a$ ones and $b$ zeros in every of vertices in polygon with $a+b$ sides so it is possible to rotate numbers in those vertices with respect to primary position and after rotation one neighboring $0$ and $1$ switch places and in every other vertices other than those two numbers remain the same.

Find, with proof, all functions $f$ mapping integers to integers with the property that for all integers $m,n$, $$f(m)+f(n)= \max\left(f(m+n),f(m-n)\right).$$

[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]

Bernardo randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a $ 3$-digit number. Silvia randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a $ 3$-digit number. What is the probability that Bernardo's number is larger than Silvia's number? $ \textbf{(A)}\ \frac {47}{72}\qquad \textbf{(B)}\ \frac {37}{56}\qquad \textbf{(C)}\ \frac {2}{3}\qquad \textbf{(D)}\ \frac {49}{72}\qquad \textbf{(E)}\ \frac {39}{56}$