We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by \[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), \] where $\lfloor x\rfloor$ denotes the integer part of $x$. [b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often. [b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often. [i]Proposed by Johan Meyer, South Africa[/i]

Let $A$ be a square matrix of odd order (at least $3$) whose entries are odd integers. Prove that if $A$ is invertible, then it is not possible for all the minors of the entries of a row to have equal absolute values.

Let $f(x,y,z) = x^2+y^2+z^2+xyz$. Let $p(x,y,z), q(x,y,z)$, $r(x,y,z)$ be polynomials with real coefficients satisfying \[ f(p(x,y,z), q(x,y,z), r(x,y,z)) = f(x,y,z). \] Prove or disprove the assertion that the sequence $p,q,r$ consists of some permutation of $\pm x, \pm y, \pm z$, where the number of minus signs is $0$ or $2.$

Ding and Jianing are playing a game. In this game, they use pieces of paper with $2014$ positions, in which some permutation of the numbers $1, 2, \dots, 2014$ are to be written. (Each number will be written exactly once). Ding fills in a piece of paper first. How many pieces of paper must Jianing fill in to ensure that at least one of her pieces of paper will have a permutation that has the same number as Ding’s in at least one position?

Given the binary operation $\ast$ defined by $a\ast b=a^b$ for all positive numbers $a$ and $b$. The for all positive $a,b,c,n,$ we have $\textbf{(A) }a\ast b=b\ast a\qquad\textbf{(B) }a\ast (b\ast c)=(a\ast b)\ast c\qquad$ $\textbf{(C) }(a\ast b^n)=(a\ast n)\ast b\qquad\textbf{(D) }(a\ast b)^n=a\ast (bn)\qquad \textbf{(E) }\text{None of these}$

Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that \[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]

Prove that of all the quadrilaterals circuscribed around a given circle, the square has the smallest perimeter.

Convex quadrilateral $ABCD$ has $AB = 18, \angle{A} = 60 \textdegree$, and $\overline{AB} \parallel \overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\overline{AB}$ is a side of maximum length. The length of another side is $a$. What is the sum of all possible values of $a$? $\textbf{(A) } 24 \qquad \textbf{(B) } 42 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 66 \qquad \textbf{(E) } 84$

Let $A,B\in\mathbb{C}^{n\times n}$ with $\rho(AB - BA) = 1$. Show that $(AB - BA)^2 = 0$.

Prove that $$\frac{2 }{2013 +1} +\frac{2^{2}}{2013^{2^{1}}+1} +\frac{2^{3}}{2013^{2^{2}}+1} + ...+ \frac{2^{2014}}{2013^{2^{2013}}+1} < \frac{1}{1006}$$

Let $ABC$ be a triangle. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$; similarly, let $Q_1$ and $Q_2$ be points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_1 = CQ_2$. The segments $P_1Q_2$ and $P_2Q_1$ meet at $R$, and the circles $P_1P_2R$ and $Q_1Q_2R$ meet again at $S$, situated inside triangle $P_1Q_1R$. Finally, let $M$ be the midpoint of the side $AC$. Prove that the angles $P_1RS$ and $Q_1RM$ are equal.

A nonzero polynomial $f(x)$ with real coefficients has the property that $f(x)=f^\prime(x)f^{\prime\prime}(x)$. What is the leading coefficient of $f(x)$?

Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ and two natural numbers $ m,n. $ Prove that: $$ \det\left( (AB)^m-(BA)^m\right)\cdot\det\left( (AB)^n-(BA)^n\right)\ge 0. $$

For any positive integer $n$, define $a_n$ to be the product of the digits of $n$. (a) Prove that $n \geq a(n)$ for all positive integers $n$. (b) Find all $n$ for which $n^2-17n+56 = a(n)$.

For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist? [i](A. Golovanov)[/i]

Given a graph with $n$ ($n\ge 4$) vertices . It is known that for any two vertices $A$ and $B$ there exists a vertex which is connected by edges both with $A$ and $B$. Find the smallest possible numbers of edges in the graph. E. Barabanov

Fibonacci numbers are defined as follows: $F_0 = F_1 = 1, F_{n+2} = F_{n+1}+F_n, n \geq 0$. Let $a_n$ be the number of words that consist of $n$ letters $0$ or $1$ and contain no two letters $1$ at distance two from each other. Express $a_n$ in terms of Fibonacci numbers.

Rectangle $ABCD$ has $AB = 24$ and $BC = 7$. Let $d$ be the distance between the centers of the incircles of $\vartriangle ABC$ and $\vartriangle CDA$. Find $d^2$.

Let $k$ be a positive integer such that $p = 8k + 5$ is a prime number. The integers $r_1, r_2, \dots, r_{2k+1}$ are chosen so that the numbers $0, r_1^4, r_2^4, \dots, r_{2k+1}^4$ give pairwise different remainders modulo $p$. Prove that the product \[\prod_{1 \leqslant i < j \leqslant 2k+1} \big(r_i^4 + r_j^4\big)\] is congruent to $(-1)^{k(k+1)/2}$ modulo $p$. (Two integers are congruent modulo $p$ if $p$ divides their difference.) [i]Fedor Petrov[/i]

Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon. [i]Greece[/i]

Quadratic equation $ x^2\plus{}ax\plus{}b\plus{}1\equal{}0$ have 2 positive integer roots, for integers $ a,b$. Show that $ a^2\plus{}b^2$ is not a prime.

In how many ways can the set $N ={1, 2, \cdots, n}$ be partitioned in the form $p(N) = A_{1}\cup A_{2}\cup \cdots \cup A_{k},$ where $A_{i}$ consists of elements forming arithmetic progressions, all with the same common positive difference $d_{p}$ and of length at least one? At least two? [hide="Solution"] [b]Part 1[/b] Claim: There are $2^{n}-2$ ways of performing the desired partitioning. Let $d(k)$ equal the number of ways $N$ can be partitioned as above with common difference $k.$ We are thus trying to evaluate $\sum_{i=1}^{n-1}d(i)$ [b]Lemma: $d(i) = 2^{n-i}$[/b] We may divide $N$ into various rows where the first term of each row denotes a residue $\bmod{i}.$ The only exception is the last row, as no row starts with $0$; the last row will start with $i.$ We display the rows as such: $1, 1+i, 1+2i, 1+3i, \cdots$ $2, 2+i, 2+2i, 2+3i, \cdots$ $\cdots$ $i, 2i, 3i, \cdots$ Since all numbers have one lowest remainder $\bmod{i}$ and we have covered all possible remainders, all elements of $N$ occur exactly once in these rows. Now, we can take $k$ line segments and partition a given row above; all entries within two segments would belong to the same set. For example, we can have: $1| 1+i, 1+2i, 1+3i | 1+4i | 1+5i, 1+6i, 1+7i, 1+8i,$ which would result in the various subsets: ${1},{1+i, 1+2i, 1+3i},{1+4i},{1+5i, 1+6i, 1+7i, 1+8i}.$ For any given row with $k$ elements, we can have at most $k-1$ segments. Thus, we can arrange any number of segments where the number lies between $0$ and $k-1$, inclusive, in: $\binom{k-1}{0}+\binom{k-1}{1}+\cdots+\binom{k-1}{k-1}= 2^{k-1}$ ways. Applying the same principle to the other rows and multiplying, we see that the result always gives us $2^{n-i},$ as desired. We now proceed to the original proof. Since $d(i) = 2^{n-i}$ by the above lemma, we have: $\sum_{i=1}^{n-1}d(i) = \sum_{i=1}^{n-1}2^{n-i}= 2^{n}-2$ Thus, there are $2^{n}-2$ ways of partitioning the set as desired. [b]Part 2[/b] Everything is the same as above, except the lemma slightly changes to $d(i) = 2^{n-i}-i.$ Evaluating the earlier sum gives us: $\sum_{i=1}^{n-1}d(i) = \sum_{i=1}^{n-1}2^{n-i}-i = 2^{n}-\frac{n(n-1)}{2}-2$ [/hide]

[b]a)[/b] Let $ a,b,c $ be three complex numbers. Prove that the element $ \begin{pmatrix} a & a-b & a-b \\ 0 & b & b-c \\ 0 & 0 & c \end{pmatrix} $ has finite order in the multiplicative group of $ 3\times 3 $ complex matrices if and only if $ a,b,c $ have finite orders in the multiplicative group of complex numbers. [b]b)[/b] Prove that a $ 3\times 3 $ real matrix $ M $ has positive determinant if there exists a real number $ \lambda\in\left( 0,\sqrt[3]{4} \right) $ such that $ A^3=\lambda A+I. $ [i]Cristinel Mortici[/i]

A certain frog that was placed on a vertex of a convex polygon chose to jump to another vertex, either clockwise skipping one vertex, either counterclockwise skipping two vertexes, and repeated the procedure. If the number of jumps that the frog made is equal to the number of sides of the polygon, the frog has passed through all its vertexes and ended up on the initial vertex, what´s the set formed by all the possible values that this number can take? [i]Andrei Eckstein[/i]

Let $\omega=\cos\frac{2\pi}{7}+i\cdot\sin\frac{2\pi}{7}$, where $i=\sqrt{-1}$. Find $$\prod_{k=0}^{6}(\omega^{3k}+\omega^k+1).$$