The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$

Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$.

The real numbers $a_0, a_1, ... , a_{n+1}$ satisfy $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_k + a_{k+1}| \le 1$ for $k = 1, 2, ... , n$. Show that $|a_k| \le \frac{ k(n + 1 - k)}{2}$ for all $k$.

Let $P(x),\ Q(x)$ be polynomials such that : \[\int_0^2 \{P(x)\}^2dx=14,\ \int_0^2 P(x)dx=4,\ \int_0^2 \{Q(x)\}^2dx=26,\ \int_0^2 Q(x)dx=2.\] Find the maximum and the minimum value of $\int_0^2 P(x)Q(x)dx$.

Tiles I, II, III and IV are translated so one tile coincides with each of the rectangles $A, B, C$ and $D$. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C$? [asy] size(400); defaultpen(linewidth(0.8)); path p=origin--(8,0)--(8,6)--(0,6)--cycle; draw(p^^shift(8.5,0)*p^^shift(8.5,10)*p^^shift(0,10)*p); draw(shift(20,2)*p^^shift(28,2)*p^^shift(20,8)*p^^shift(28,8)*p); label("8", (4,6+10), S); label("6", (4+8.5,6+10), S); label("7", (4,6), S); label("2", (4+8.5,6), S); label("I", (4,6+10), N); label("II", (4+8.5,6+10), N); label("III", (4,6), N); label("IV", (4+8.5,6), N); label("3", (0,3+10), E); label("4", (0+8.5,3+10), E); label("1", (0,3), E); label("9", (0+8.5,3), E); label("7", (4,10), N); label("2", (4+8.5,10), N); label("0", (4,0), N); label("6", (4+8.5,0), N); label("9", (8,3+10), W); label("3", (8+8.5,3+10), W); label("5", (8,3), W); label("1", (8+8.5,3), W); label("A", (24,10), N); label("B", (32,10), N); label("C", (24,4), N); label("D", (32,4), N); [/asy] $\mathrm{(A)}\ I \qquad \mathrm{(B)}\ II \qquad \mathrm{(C)}\ III \qquad \mathrm{(D)}\ IV \qquad \mathrm{(E)}\text{ cannot be determined}$

Consider a set $S$ of $200$ points on the plane such that $100$ points are the vertices of a convex polygon $A$ and the other $100$ points are in the interior of the polygon. Moreover, no three of the given points are collinear. A triangulation is a way to partition the interior of the polygon $A$ into triangles by drawing the edges between some two points of S such that any two edges do not intersect in the interior, and each point in $S$ is the vertex of at least one triangle. 1. Prove that the number of edges does not depend on the triangulation. 2. Show that for any triangulation, one can color each triangle by one of three given colors such that any two adjacent triangles have different colors.

If $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, then \[\displaystyle\sum_{N=1}^{1024} \lfloor \log_{2}N\rfloor = \] $ \textbf{(A)}\ 8192\qquad\textbf{(B)}\ 8204\qquad\textbf{(C)}\ 9218\qquad\textbf{(D)}\ \lfloor \log_{2}(1024!)\rfloor\qquad\textbf{(E)}\ \text{none of these} $

Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

Erick stands in the square in the 2nd row and 2nd column of a 5 by 5 chessboard. There are \$1 bills in the top left and bottom right squares, and there are \$5 bills in the top right and bottom left squares, as shown below. \[\begin{tabular}{|p{1em}|p{1em}|p{1em}|p{1em}|p{1em}|} \hline \$1 & & & & \$5 \\ \hline & E & & &\\ \hline & & & &\\ \hline & & & &\\ \hline \$5 & & & & \$1 \\ \hline \end{tabular}\] Every second, Erick randomly chooses a square adjacent to the one he currently stands in (that is, a square sharing an edge with the one he currently stands in) and moves to that square. When Erick reaches a square with money on it, he takes it and quits. The expected value of Erick's winnings in dollars is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Consider a convex pentagon circumscribed about a circle. We name the lines that connect vertices of the pentagon with the opposite points of tangency with the circle [i]gergonnians[/i]. (a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent. (b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.

What the smallest number of circles of radius $\sqrt{2}$ that are needed to cover a rectangle $(a)$ of size $6\times 3$? $(b)$ of size $5\times 3$?

Let $K$ be a finite field, $n \ge 2$ an integer, $f \in K[X]$ an irreducible polynomial of degree $n,$ and $g$ the product of all the nonconstant polynomials in $K[X]$ of degree at most $n-1.$ Prove that $f$ divides $g-1.$

Real number $x,y$ satisfy that $4x^2-5xy+4y^2=5,S=x^2+y^2$, then $\frac{1}{S_\text{max}}+\frac{1}{S_\text{min}}=$________.

In the triangle $ABC$ the relationship $AB+AC = 2BC$ holds. Let $I$ and $M$ be the incenter and intersection point of the medians of triangle $ABC$ respectively, $AL$ its angle bisector, and point $P$ the orthocenter of triangle $BIC$. Prove that the points $L, M, P$ lie on a straight line. (Matvii Kurskyi)

If $P$ is the area of a triangle $ABC$ with sides $a,b,c$, prove that $\frac{ab+bc+ca}{4P} \ge \sqrt3$

On Monday Taye has \$2. Everyday he either gains \$3 or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later? $\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$

Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.

Suppose that $n$ is a positive integer and that $a$ is the integer equal to $\frac{10^{2n}-1}{3\left(10^n+1\right)}.$ If the sum of the digits of $a$ is 567, what is the value of $n$?

For every positive real pair $(x,y)$ satisfying the equation $x^3+y^4 = x^2y$, if the greatest value of $x$ is $A$, and the greatest value of $y$ is $B$, then $A/B = ?$ $ \textbf{(A)}\ \frac{2}{3} \qquad \textbf{(B)}\ \frac{512}{729} \qquad \textbf{(C)}\ \frac{729}{1024} \qquad \textbf{(D)}\ \frac{3}{4} \qquad \textbf{(E)}\ \frac{243}{256}$

A sequence infinity $a_{1}, a_{2},...,$ is $generadora$ if: $a_{1}=1,2$ and $a_{n+1}$ is obtained by placing a digit 1 on the left or a digit 2 on the right for all natural n. Prove that there is an infinite $generadora$ sequence such that it does not contain any multiples of 7.

Let $n$ be a positive integer. Suppose for any $\mathcal{S} \subseteq \{1, 2, \cdots, n\}$, $f(\mathcal{S})$ is the set containing all positive integers at most $n$ that have an odd number of factors in $\mathcal{S}$. How many subsets of $\{1, 2, \cdots, n\}$ can be turned into $\{1\}$ after finitely many (possibly zero) applications of $f$?

Determine all positive integers that are expressible in the form \[a^{2}+b^{2}+c^{2}+c,\] where $a$, $b$, $c$ are integers.

Find an integer $n$, where $100 \leq n \leq 1997$, such that \[\frac{2^{n}+2}{n}\] is also an integer.

Let $n$ be a positive integer. Navinim writes down all positive square numbers that divide $n$ on a blackboard. For each number $k$ on the blackboard, Navagem replaces it with $d(k)$. Show that the sum of all numbers on the blackboard now is a perfect square. (Note: $d(k)$ denotes the number of divisors of $k$.) [i](Proposed by Ivan Chan Guan Yu)[/i]

Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.