In a math test, there are easy and hard questions. The easy questions worth 3 points and the hard questions worth D points.\\ If all the questions begin to worth 4 points, the total punctuation of the test increases 16 points.\\ Instead, if we exchange the questions scores, scoring D points for the easy questions and 3 for the hard ones, the total punctuation of the test is multiplied by $\frac{3}{2}$.\\ Knowing that the number of easy questions is 9 times bigger the number of hard questions, find the number of questions in this test.

Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1.$ Prove that $ max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2.$ Where for all integers $ i, a_{i \plus{} 4} \equal{} a_{i}$ holds.

Let $P$ be a point inside a convex quadrilateral $ABCD$. Points $K,L,M,N$ are given on the sides $AB,BC,CD,DA$ respectively such that $PKBL$ and $PMDN$ are parallelograms. Let $S,S_1$, and $S_2$ be the areas of $ABCD, PNAK$, and $PLCM$. Prove that $\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}$.

For every $n$, the decreasing sequence $\{x_k\}$ satisfies a condition $$x_1+x_4/2+x_9/3+...+x_n^2/n \le 1$$ Prove that for every $n$, it also satisfies $$x_1+x_2/2+x_3/3+...+x_n/n\le 3$$

Fernanda decided to decorate a square blanket with a ribbon and buttons, placing a button in the center of each square where the ribbon passes and forming the design indicated in the figure. If Fernanda sews the first button in the shaded square on line $0$, on which line does she sew the $2007$th button? [img]https://cdn.artofproblemsolving.com/attachments/2/9/0c9c85ec6448ee3f6f363c8f4bcdd5209f53f6.png[/img]

We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define \[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \] Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$. [i]Proposed by Evan Chen[/i]

Show that for an arbitrary tetrahedron there are two planes such that the ratio of the areas of the projections of the tetrahedron onto the two planes is not less than $\sqrt2$.

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(2m+2n)=f(m)f(n)$ for all natural numbers $m,n$.

Find all the integers $x$ in $[20, 50]$ such that $6x+5\equiv 19 \mod 10$, that is, $10$ divides $(6x+15)+19$.

On the sides of the triangle $ABC$ externally constructed right triangles $ABC_1$, $BCA_1$, $CAB_1$. Prove that the points of intersection of the medians of the triangles $ABC$ and $A_1B_1C_1$ coincide.

For a positive integer $n$, let \[S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}\] Answer the following questions: (1) Show the following inequality. \[\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}\] (2) Express $T_n-2S_n$ in terms of $n$. (3) Find the limit $\lim_{n\to\infty} T_n.$

When $p = \sum_{k=1}^{6} k \ln{k}$, the number $e^p$ is an integer. What is the largest power of $2$ that is a factor of $e^p$? ${\textbf{(A)}\ 2^{12}\qquad\textbf{(B)}\ 2^{14}\qquad\textbf{(C)}\ 2^{16}\qquad\textbf{(D)}}\ 2^{18}\qquad\textbf{(E)}\ 2^{20} $

The table of dimensions $n\times n$, $n\in\mathbb{N}$, is filled with numbers from $1$ to $n^2$, but the difference any two numbers on adjacent fields is at most $n$, and that for every $k = 1, 2,\dots , n^2$ set of fields whose numbers are $1, 2,\dots , k$ is connected, as well as the set of fields whose numbers are $k, k + 1,\dots , n^2$. Neighboring fields are fields with a common side, while a set of fields is considered connected if from each field to every other field of that set can be reached going only to the neighboring fields within that set. We call a pair of adjacent numbers, ie. numbers on adjacent fields, good, if their absolute difference is exactly $n$ (one number can be found in several good pairs). Prove that the table has at least $2 (n - 1)$ good pairs.

A regular tetrahedron of volume $1$ is filled with water of total volume $\frac{7}{16}$. Is it possible that the center of the tetrahedron lies on the surface of the water? How about in a cube of volume $1$?

Let $p$ be a prime number in the form $p=4k+3$. Prove that if the numbers $x_0,y_0,z_0,t_0$ are solutions of the equation $x^{2p}+y^{2p}+z^{2p}=t^{2p}$, then at least one of them is divisible by $p$. [i](Plamen Koshlukov)[/i]

Let $p$ and $q$ be prime numbers. Prove that if $pq(p+1)(q+1)+1$ is a perfect square, then $pq + 1$ is also a perfect square.

Given a triangle $ABC$. On its sides $BC$ , $CA$ and $AB$ , the points $A_1$ , $B_1$ and $C_1$ are taken, respectively , such that $2 \angle B_1 A_1 C_1 + \angle BAC = 180^o$ , $2 \angle A_1 C_1 B_1 + \angle ACB = 180^o$ , $2 \angle C_1 B_1 A_1 + \angle CBA = 180^o$ . Find the locus of the centers of the circles circumscribed about the triangles $A_1 B_1 C_1$ (all possible such triangles are considered).

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

The points $D,E$ and $F$ are chosen on the sides $BC,AC$ and $AB$ of triangle $ABC$, respectively. Prove that triangles $ABC$ and $DEF$ have the same centroid if and only if \[\frac{BD}{DC} = \frac{CE}{EA}=\frac{AF}{FB}\]

There is $3 \times 4 \times 5$ - box with its faces divided into $1 \times 1$ - squares. Is it possible to place numbers in these squares so that the sum of numbers in every stripe of squares (one square wide) circling the box, equals $120$?

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\] where $n\in \mathbb{N}.$

Let $BE$ and $CF$ be the medians of an acute triangle $ABC.$ On the line $BC,$ points $K \ne B$ and $L \ne C$ are chosen such that $BE = EK$ and $CF = FL.$ Prove that $AK = AL.$ [i]Proposed by Heorhii Zhilinskyi[/i]

Let $a, b, c$ be reals such that $$a^2(c^2-2b-1)+b^2(a^2-2c-1)+c^2(b^2-2a-1)=0.$$ Show that $$3(a^2+b^2+c^2)+4(a+b+c)+3 \geq 6abc.$$

Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.

Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.