Show that there exist no integer solutions $(x, y, z)$ to the equation $$x^3+2y^3+4z^3=9$$

The integers $1, 2, \dots, n$ are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers. The averages of the numbers on the five slips are $1234$, $345$, $128$, $19$, and $9.5$ in some order. Compute $n$. [i]Proposed by Evan Chen[/i]

A student recorded the exact percentage frequency distribution for a set of measurements, as shown below. However, the student neglected to indicate $N$, the total number of measurements. What is the smallest possible value of $N$? \[ \begin{tabular}{c c} \text{measured value} & \text{percent frequency} \\ \hline 0 & 12.5 \\ 1 & 0\\ 2 & 50\\ 3 & 25 \\ 4 & 12.5 \\ \hline \ & 100 \\ \end{tabular} \] $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 25 \qquad\textbf{(E)}\ 50 $

Let $a = \sqrt[401]{4} - 1$ and for each $n \ge 2$, let $b_n = \binom{n}{1} + \binom{n}{2} a + \ldots + \binom{n}{n} a^{n-1}$. Find $b_{2006} - b_{2005}$.

Three hoops are arranged concentrically as in the diagram. Each hoop is threaded with $ 20$ beads, $ 10$ of which are black and $ 10$ are white. On each hoop the positions of the beads are labelled $ 1$ through $ 20$ as shown. We say there is a match at position $ i$ if all three beads at position $ i$ have the same color. We are free to slide beads around a hoop, not breaking the hoop. Show that it is always possible to move them into a configuration involving no less than $ 5$ matches.

In triangle $ABC$ with $AB = 10$, let$ D$ be a point on side BC such that $AD$ bisects $\angle BAC$. If $\frac{CD}{BD} = 2$ and the area of $ABC$ is $50$, compute the value of $\angle BAD$ in degrees.

Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.

Restore a triangle $ABC$ by the Nagel point, the vertex $B$ and the foot of the altitude from this vertex.

Let $\triangle ABC$ be a triangle with $AB=85$, $BC=125$, $CA=140$, and incircle $\omega$. Let $D$, $E$, $F$ be the points of tangency of $\omega$ with $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively, and furthermore denote by $X$, $Y$, and $Z$ the incenters of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$, also respectively. Find the circumradius of $\triangle XYZ$. [i] Proposed by David Altizio [/i]

Let $|AB|=3$ and the length of the altitude from $C$ be $2$ in $\triangle ABC$. What is the maximum value of the product of the lengths of the other two altitudes? $ \textbf{(A)}\ \frac{144}{25} \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 3\sqrt 2 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{None of the above} $

In the cabinet 2004 telephones are located; each two of these telephones are connected by a cable, which is colored in one of four colors. From each color there is one cable at least. Can one always select several telephones in such a way that among their pairwise cable connections exactly 3 different colors occur?

(a) Let $\gcd(m, k) = 1$. Prove that there exist integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ such that each product $a_ib_j$ ($i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k$) gives a different residue when divided by $mk.$ (b) Let $\gcd(m, k) > 1$. Prove that for any integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i, j) \neq (s, t)$) that give the same residue when divided by $mk.$ [i]Proposed by Hungary.[/i]

Find all pairs of real numbers $(a, b)$ such that the equality $$|ax+by|+ |bx + ay| = 2|x| + 2|y|$$ holds for all reals $x$ and $y$.

For which natural numbers $n$ does $2^n > 10n^2 -60n + 80$ hold?

Determine whether there exist $1976$ nonsimilar triangles with angles $\alpha, \beta, \gamma,$ each of them satisfying the relations \[\frac{\sin \alpha + \sin\beta + \sin\gamma}{\cos \alpha + \cos \beta + \cos \gamma}=\frac{12}{7}\text{ and }\sin \alpha \sin \beta \sin \gamma =\frac{12}{25}\]

Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$

A city has $4$ horizontal and $n\geq3$ vertical boulevards which intersect at $4n$ crossroads. The crossroads divide every horizontal boulevard into $n-1$ streets and every vertical boulevard into $3$ streets. The mayor of the city decides to close the minimum possible number of crossroads so that the city doesn't have a closed path(this means that starting from any street and going only through open crossroads without turning back you can't return to the same street). $a)$Prove that exactly $n$ crossroads are closed. $b)$Prove that if from any street you can go to any other street and none of the $4$ corner crossroads are closed then exactly $3$ crossroads on the border are closed(A crossroad is on the border if it lies either on the first or fourth horizontal boulevard, or on the first or the n-th vertical boulevard).

Let $ a,b,c,d$ be the positive integers such that $ a > b > c > d$ and $ (a \plus{} b \minus{} c \plus{} d) | (ac \plus{} bd)$ . Prove that if $ m$ is arbitrary positive integer , $ n$ is arbitrary odd positive integer, then $ a^n b^m \plus{} c^m d^n$ is composite number

Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have \[a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.\]

In a triangle $ABC$, $AB>AC$. The foot of the altitude from $A$ to $BC$ is $D$, the intersection of bisector of $B$ and $AD$ is $K$, the foot of the altitude from $B$ to $CK$ is $M$ and let $BM$ and $AK$ intersect at point $N$. The line through $N$ parallel to $DM$ intersects $AC$ at $T$. Prove that $BM$ is the bisector of angle $\widehat{TBC}$.

Find all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that \[ f(x)-f(y)\in \mathbb{Q}\quad \text{ for all }\quad x-y\in\mathbb{Q} \]

A box $3\times5\times7$ is divided into unit cube cells. In each of the cells, there is a c[i][/i]ockchafer. At a signal, every c[i][/i]ockchafer moves through a face of its cell to a neighboring cell. (a) What is the minimum number of empty cells after the signal? (b) The same question, assuming that the c[i][/i]ockchafers move to diagonally adjacent cells (sharing exactly one vertex).

Kevin has a set $S$ of $2014$ points scattered on an infinitely large planar gameboard. Because he is bored, he asks Ashley to evaluate \[ x = 4f_4 + 6f_6 + 8f_8 + 10f_{10} + \cdots \] while he evaluates \[ y = 3f_3 + 5f_5+7f_7+9f_9 + \cdots, \] where $f_k$ denotes the number of convex $k$-gons whose vertices lie in $S$ but none of whose interior points lie in $S$. However, since Kevin wishes to one-up everything that Ashley does, he secretly positions the points so that $y-x$ is as large as possible, but in order to avoid suspicion, he makes sure no three points lie on a single line. Find $\left\lvert y-x \right\rvert$. [i]Proposed by Robin Park[/i]

Determine all the pairs of positive odd integers $(a,b),$ such that $a,b>1$ and $$7\varphi^2(a)-\varphi(ab)+11\varphi^2(b)=2(a^2+b^2),$$ where $\varphi(n)$ is Euler's totient function.

Consider a table with three rows and eleven columns. There are zeroes prefilled in the cell of the first row and the first column and in the cell of the second row and the last column. Determine the least real number $\alpha$ such that the table can be filled with non-negative numbers and the following conditions hold simultaneously: (1) the sum of numbers in every column is one, (2) the sum of every two neighboring numbers in the first row is at most one, (3) the sum of every two neighboring numbers in the second row is at most one, (4) the sum of every two neighboring numbers in the third row is at most $\alpha$.