Let be a real number $ c $ and a differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that $$ f(c)\neq \frac{1}{b-a}\int_a^b f(x)dx, $$ for any real numbers $ a\neq b. $ Prove that $ f'(c)=0. $ [i]Florin Rotaru[/i]

A square with side $1$ is cut from the paper. Construct a segment with length $1/2024$ using at most $20$ folds. No instruments are available. It is allowed only to fold the paper and to mark the common points of folding lines.

Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?

Let $ ABC$ be a triangle with area $ S$ and points $ D,E,F$ on the sides $ BC,CA,AB$. Perpendiculars at points $ D,E,F$ to the $ BC,CA,AB$ cut circumcircle of the triangle $ ABC$ at points $ (D_1,D_2), (E_1,E_2), (F_1,F_2)$. Prove that: $ |D_1B\cdot D_1C \minus{} D_2B\cdot D_2C| \plus{} |E_1A\cdot E_1C \minus{} E_2A\cdot E_2C| \plus{} |F_1B\cdot F_1A \minus{} F_2B\cdot F_2A| > 4S$

Let $A = \{3 + 10k, 6 + 26k, 5 + 29k, k = 1, 2, 3, 4, ...\}$. Determine the smallest positive integer $r$ such that there exists an integer $b$ with the property that the set $B = \{b + rk, k = 1, 2, 3, 4, ...\}$ is disjoint from $A$.

Evaluate $\displaystyle\sum_{n=0}^\infty \dfrac{\cos n\theta}{2^n}$, where $\cos\theta = \dfrac{1}{5}$.

Fatima and Asma are playing the following game. First, Fatima chooses $2013$ pairwise different numbers, called $a_1, a_2, ..., a_{2013}$. Then, Asma tries to know the value of each number $a_1, a_2, ..., a_{2013}$.. At each time, Asma chooses $1 \le i < j \le 2013$ and asks Fatima ''[i]What is the set $\{a_i,a_j\}$?[/i]'' (For example, if Asma asks what is the set $\{a_i,a_j\}$, and $a_1 = 17$ and $a_2 = 13$, Fatima will answer $\{13. 17\}$). Find the least number of questions Asma needs to ask, to know the value of all the numbers $a_1, a_2, ..., a_{2013}$.

Let $AA_1 , BB_1$, and $CC_1$ be the altitudes of the non-isosceles acute-angled triangle $ABC$. The circles circumscibred around the triangles $ABC$ and $A_1 B_1 C$ intersect again at the point $P , Z$ is the intersection point of the tangents to the circumscribed circle of the triangle $ABC$ conducted at points $A$ and $B$ . Prove that lines $AP , BC$ and $ZC_1$ are concurrent.

Prove that we can give a color to each of the numbers $1,2,3,...,2013$ with seven distinct colors (all colors are necessarily used) such that for any distinct numbers $a,b,c$ of the same color, then $2014\nmid abc$ and the remainder when $abc$ is divided by $2014$ is of the same color as $a,b,c$.

What is the $288$th term of the sequence $a,b,b,c,c,c,d,d,d,d,e,e,e,e,e,f,f,f,f,f,f,...?$

Suppose $A\subseteq \mathbb R^m$ is closed and non-empty. Let $f:A\to A$ is a lipchitz function with constant less than 1. (ie there exist $c<1$ that $|f(x)-f(y)|<c|x-y|,\ \forall x,y \in A)$. Prove that there exists a unique point $x\in A$ such that $f(x)=x$.

A lightbulb is given that emits red, green or blue light and an infinite set $S$ of switches, each with three positions labeled red, green and blue. We know the following: [list=1] [*]For every combination of the switches the lighbulb emits a given color. [*]If all switches are in a position with a given color, the lightbulb emits the same color. [*]If there are two combinations of the switches where each switch is in a different position, the lightbulb emits a different color for the two combinations. [/list] We create the following set $U$ containing some of the subsets of $S$: for each combination of the switches let us observe the color of the lightbulb, and put the set of those switches in $U$ which are in the same position as the color of the lightbulb. Prove that $U$ is an ultrafilter on $S$. In other words, prove that $U$ satisfies the following conditions: [list=1] [*]The empty set is not in $U.$ [*]If two sets are in $U,$ their intersection is also in $U.$ [*]If a set is in $U,$ every subset of $S$ containing it is also in $U.$ [*]Considering a set and its complement in $S,$ exactly one of these sets is contained in $U.$ [/list]

What is the sum of real roots of the equation \[ \left ( x + 1\right )\left ( x + \dfrac 14\right )\left ( x + \dfrac 12\right )\left ( x + \dfrac 34\right )= \dfrac {45}{32}? \] $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ -\dfrac {3}{2} \qquad\textbf{(D)}\ -\dfrac {5}{4} \qquad\textbf{(E)}\ -\dfrac {7}{12} $

Let $\triangle ABC$ be acute. The feet of altitudes from the corners $A, B, C$ are $ D, E, F$, respectively. If $|DF|=3, |FE|=4,$ and $|DE|=5$, then what is the radius of the circle with center $C$ and tangent to $DE$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3$

Show that an integer $p > 3$ is a prime if and only if for every two nonzero integers $a,b$ exactly one of the numbers $N_1 = a+b-6ab+\frac{p-1}{6}$ , $N_2 = a+b+6ab+\frac{p-1}{6}$ is a nonzero integer.

Given: point $ A $, line $ p $, and circle $ k $. Construct a triangle $ ABC $ with angles $ A = 60^\circ $, $ B = 90^\circ $, whose vertex $ B $ lies on line $ p $, and vertex $ C $ - on circle $ k $.

Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.

There are $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of $N$.

Define $ n_a!$ for $ n$ and $ a$ positive to be \[ n_a ! \equal{} n (n\minus{}a)(n\minus{}2a)(n\minus{}3a)...(n\minus{}ka)\] where $ k$ is the greatest integer for which $ n>ka$. Then the quotient $ 72_8!/18_2!$ is equal to $ \textbf{(A)}\ 4^5 \qquad \textbf{(B)}\ 4^6 \qquad \textbf{(C)}\ 4^8 \qquad \textbf{(D)}\ 4^9 \qquad \textbf{(E)}\ 4^{12}$

The circle $C_1$ of center $O$ and the circle $C_2$ intersect at points $A$ and $B$, so that point $O$ lies on circle $C_2$. The lines $d$ and $e$ are tangent at point $A$ to the circles $C_1$ and $C_2$ respectively. If the line $e$ intersects the circle $C_1$ at point $D$, prove that the lines $BD$ and $d$ are parallel.

An equilateral triangle has inside it a point with distances 5,12,13 from the vertices . Find its side.

In triangle $ABC$ medians of triangle $BE$ and $AD$ are perpendicular to each other. Find the length of $\overline{AB}$, if $\overline{BC}=6$ and $\overline{AC}=8$

Let \(\mathbb{R}\) denote the set of real numbers. A subset \(S\subseteq\mathbb{R}\) is called [i]dense[/i] if any non-empty open interval of \(\mathbb{R}\) contains at least one element in \(S\). For a function \(f:\mathbb{R}\to\mathbb{R}\), let \(\mathcal{O}_f(x)\) denote the set \(\left\{x,f(x),f(f(x)),\ldots\right\}\). (a) Is there a function \(g:\mathbb{R}\to\mathbb{R}\), continuous everywhere in \(\mathbb{R}\) such that \(\mathcal{O}_g(x)\) is dense for all \(x\in\mathbb{R}\) for all \(x\in\mathbb{R}\)? (b) Is there a function \(h:\mathbb{R}\to\mathbb{R}\), continuous at all but a single \(x_0\in\mathbb{R}\), such that \(\mathcal{O}_h(x)\) is dense for all \(x\in\mathbb{R}\)? [i]Proposed by the ICMC Problem Committee[/i]

The product $ 8\times .25\times 2\times .125 = $ $ \text{(A)}\ \frac{1}8\qquad\text{(B)}\ \frac{1}4\qquad\text{(C)}\ \frac{1}2\qquad\text{(D)}\ 1\qquad\text{(E)}\ 2 $

Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane $90^\circ$ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$ to the triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$?