Let be a natural number $ n\ge 2. $ Find the real numbers $ a $ that satisfy the equation $$ \lfloor nx \rfloor =\sum_{k=1}^{n} \lfloor x+(k-1)a \rfloor , $$ for any real numbers $ x. $ [i]Marius Perianu[/i]

Write the number $2004_{(5)}$ [ $2004$ base $5$ ] as a number in base $6$.

Estimate the number of positive integers less than or equal to $1,000,000$ that can be expressed as the sum of two nonnegative integer squares. Your estimate must be an integer, or you will receive a zero.

Evaluate $\int_0^{\frac{\pi}{4}} \frac{1}{1-\sin x}\sqrt{\frac{\cos x}{1+\cos x+\sin x}}dx.$ Own

Let $a$ and $b$ be parallel lines with $50$ distinct points marked on $a$ and $50$ distinct points marked on $b$. Find the greatest possible number of acute-angled triangles all of whose vertices are marked.

Determine for which values of $n$ it is possible to tile a square of side $n$ with figures of the type shown in the picture [asy] unitsize(0.4 cm); draw((0,0)--(5,0)); draw((0,1)--(5,1)); draw((1,2)--(4,2)); draw((2,3)--(3,3)); draw((0,0)--(0,1)); draw((1,0)--(1,2)); draw((2,0)--(2,3)); draw((3,0)--(3,3)); draw((4,0)--(4,2)); draw((5,0)--(5,1)); [/asy]

Solve the diophantine equation \[x^{2008}- y^{2008} = 2^{2009}.\]

Lavidópolis is a city with 2024 neighborhoods. Lavi Dopes was elected mayor, and since he saw that there were no roads in the city, he asked Gil Bento, the monster engineer, to design the city's roads according to the following rules: 1. Any two neighborhoods are connected by at most one two-way road; 2. For any two neighborhoods, there is exactly one route from one neighborhood to another, which may pass through some intermediate neighborhoods, but never passes through the same neighborhood more than once. Mayor Lavi Dopes wants to try for re-election, but since he knows nothing about the city and only shows up during campaign times (he spent all this time stealing... I mean, thinking about math problems), he wants to find a pair of neighborhoods such that the number of roads that are part of the route connecting them is maximized among all pairs of neighborhoods. To do this, he starts asking Gil Bento various questions, all in the following manner: he chooses two of the 2024 neighborhoods, say A and B, and asks: "Given neighborhoods A and B, how many roads are part of the route connecting A to B?" Knowing that Gil Bento always answers correctly to each question, determine the minimum number of questions that Lavi Dopes needs to ask to achieve his goal, regardless of how Gil Bento has designed the roads of Lavidópolis.

Let $f^{(n)}(x)$ denote the $n^{\text{th}}$ iterate of function $f$, i.e $f^{(1)}(x)=f(x)$, $f^{(n+1)}(x)=f(f^{(n)}(x))$. Let $p(n)$ be a given polynomial with integer coefficients, which maps the positive integers into the positive integers. Is it possible that the functional equation $f^{(n)}(n)=p(n)$ has exactly one solution $f$ that maps the positive integers into the positive integers? [i]Submitted by Dávid Matolcsi and Kristóf Szabó, Budapest[/i]

There is a $2021 \times 2023$ board that has a white piece in the central square, on which Mich and Moka are going to play in turns. First Mich places a green token on any free space so that it is not in the same row or column as the white token, then Moka places a red token on any free space so that it is not in the same row or column as the white token. white or green. From now on, Mich will place green tokens and Moka will place red tokens alternately according to the following rules: $\bullet$ For the placed piece there must be another piece of the same color in its row or column, such that there is no other piece between both pieces. $\bullet$ If there is at least one box that meets the previous rule, then it is mandatory to place a token. When a token is placed, it changes all the tokens that are on squares adjacent to it to the same color. The game ends when one of the players can no longer place tiles. If when the game ends the board has more green tiles then Mich wins, and if it has more red tiles then Moka wins. Determine if either player has a winning strategy.

Let $a$ and $b$ be positive real numbers such that $3(a^2+b^2-1) = 4(a+b$). Find the minimum value of the expression $\frac{16}{a}+\frac{1}{b}$ .

A function $f$ with its domain on the positive integers $N =\{1, 2, ...\}$ satisfies the following conditions: (a) $f(1) = 2017$. (b) $\sum_{i=1}^n f(i) = n^2f(n)$, for every positive integer $n > 1$. What is the value of $f(2017)$?

Find the minimum value of $$K(x,y)=16\frac{x^3}{y}+\frac{y^3}{x}-\sqrt{xy}$$ where $x,y$ are the real allowed values

Let $W$ be a regular octahedron and $O$ be its center. In a plane $P$ containing $O$ circles $k_1(O,r_1)$ and $k_2(O,r_2)$ are chosen so that $k_1 \subset P\cap W \subset k_2$. Prove that $\frac{r_1}{r_2}\le \frac{\sqrt3}{2}$

Let $ABC$ be a triangle with circumcircle $\Gamma$ and $AB\neq AC$. Let $D$ and $E$ lie on the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle BAE=\angle DAC$. Let the incenters of $BAE$ and $CAD$ be $X$ and $Y$, respectively, and let the external tangents of the incircles of $BAE$ and $CAD$ intersect at $Z$. Prove that $Z$ lies on the common chord of $\Gamma$ and the circumcircle of $AXY$.

Let $m_n$ be the smallest value of the function ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$ Show that $m_n \to \frac{1}{2}$, as $n \to \infty.$

Let $AP,BQ$ and $CR$ be altitudes of an acute-angled triangle $ABC$. Show that for any point $X$ inside the triangle $PQR$ there exists a tetrahedron $ABCD$ such that $X$ is the point on the face $ABC$ at the greatest distance from $D$ (measured along the surface of the tetrahedron).

Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$? [i]Proposed by Ray Li[/i]

Let $ABC$ be an acute triangle with $|AB| \neq |AC|$ and the midpoints of segments $[AB]$ and $[AC]$ be $D$ resp. $E$. The circumcircles of the triangles $BCD$ and $BCE$ intersect the circumcircle of triangle $ADE$ in $P$ resp. $Q$ with $P \neq D$ and $Q \neq E$. Prove $|AP|=|AQ|$. [i](Notation: $|\cdot|$ denotes the length of a segment and $[\cdot]$ denotes the line segment.)[/i]

Let $ABC$ be a triangle. Circle $\Gamma$ passes through point $A$, meets segments $AB$ and $AC$ again at $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $(BDF)$ at $F$ and the tangent to circle $(CEG)$ at $G$ meet at $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

Take two points $A\ (-1,\ 0),\ B\ (1,\ 0)$ on the $xy$-plane. Let $F$ be the figure by which the whole points $P$ on the plane satisfies $\frac{\pi}{4}\leq \angle{APB}\leq \pi$ and the figure formed by $A,\ B$. Answer the following questions: (1) Illustrate $F$. (2) Find the volume of the solid generated by a rotation of $F$ around the $x$-axis.

[b]a)[/b] Show that the sum of the squares of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [b]b)[/b] Show that the sum of the cubes of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [i]Alexandru Sergiu Alamă[/i]

For each positive integer $ n$, the mean of the first $ n$ terms of a sequence is $ n$. What is the $ 2008$th term of the sequence? $ \textbf{(A)}\ 2008 \qquad \textbf{(B)}\ 4015 \qquad \textbf{(C)}\ 4016 \qquad \textbf{(D)}\ 4,030,056 \qquad \textbf{(E)}\ 4,032,064$

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$.

A truck is initially moving at velocity $v$. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force $F$ to the truck. The truck rolls a distance $x$ before coming to a stop, and the time it takes to stop is $t$. Which of the following expressions is equal the initial momentum of the truck (i.e. the momentum before the driver starts braking)? $\textbf{(A) } Fx\\ \textbf{(B) } Ft/2\\ \textbf{(C) } Fxt\\ \textbf{(D) } 2Ft\\ \textbf{(E) } 2Fx/v$