There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone? $ \text{(A)}\ 30\qquad\text{(B)}\ 40\qquad\text{(C)}\ 41\qquad\text{(D)}\ 60\qquad\text{(E)}\ 119 $

The positive integer $N$ is said[i] amiable [/i]if the set $\{1,2,\ldots,N\}$ can be partitioned into pairs of elements, each pair having the sum of its elements a perfect square. Prove there exist infinitely many amiable numbers which are themselves perfect squares. ([i]Dan Schwarz[/i])

John takes seven positive integers $a_1,a_2,...,a_7$ and writes the numbers $a_i a_j$, $a_i+a_j$ and $|a_i -a_j |$ for all $i \ne j$ on the blackboard. Find the greatest possible number of distinct odd integers on the blackboard.

Let $A$ be an $n\times n$ real matrix with minimal polynomial $x^n + x - 1$. Prove that the trace of $(nA^{n-1} + I)^{-1}A^{n-2}$ is zero.

Prove that two natural numbers whose digits are all ones are relatively prime if and only if the numbers of their digits are relatively prime.

Show that there exists a constant $a > 1$ such that, for any positive integers $m$ and $n$, $\frac{m}{n} < \sqrt7$ implies that $$7-\frac{m^2}{n^2} \ge \frac{a}{n^2} .$$

There exist $ r$ unique nonnegative integers $ n_1 > n_2 > \cdots > n_r$ and $ r$ unique integers $ a_k$ ($ 1\le k\le r$) with each $ a_k$ either $ 1$ or $ \minus{} 1$ such that \[ a_13^{n_1} \plus{} a_23^{n_2} \plus{} \cdots \plus{} a_r3^{n_r} \equal{} 2008. \]Find $ n_1 \plus{} n_2 \plus{} \cdots \plus{} n_r$.

Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

If $a, b, x$ and $y$ are real numbers such that $ax + by = 3,$ $ax^2+by^2=7,$ $ax^3+bx^3=16$, and $ax^4+by^4=42,$ find $ax^5+by^5$.

In triangle $ ABC$, angle $ C$ is a right angle and $ CB > CA$. Point $ D$ is located on $ \overline{BC}$ so that angle $ CAD$ is twice angle $ DAB$. If $ AC/AD \equal{} 2/3$, then $ CD/BD \equal{} m/n$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$. $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 22\qquad \textbf{(E)}\ 26$

Determine all natural numbers $n$ for which there are $2n$ distinct positive integers $x_1,…,x_n,y_1,…,y_n$ such that the product $$(11x^2_1+12y^2_1)(11x^2_2+12y^2_2)…(11x^2_n+12y^2_n)$$ is a perfect square.

Color the positive integers by four colors $c_1,c_2,c_3,c_4$. (1)Prove that there exists a positive integer $n$ and $i,j\in\{1,2,3,4\}$,such that among all the positive divisors of $n$, the number of divisors with color $c_i$ is at least greater than the number of divisors with color $c_j$ by $3$. (2)Prove that for any positive integer $A$,there exists a positive integer $n$ and $i,j\in\{1,2,3,4\}$,such that among all the positive divisors of $n$, the number of divisors with color $c_i$ is at least greater than the number of divisors with color $c_j$ by $A$.

In a basketball tournament any two of the $n$ teams $S_1,S_2,...,S_n$ play one match (no draws). Denote by $v_i$ and $p_i$ the number of victories and defeats of team $S_i$ ($i = 1,2,...,n$), respectively. Prove that $v^2_1 +v^2_2 +...+v^2_n = p^2_1 +p^2_2 +...+p^2_n$

Square $ BDEC$ with center $ F$ is constructed to the out of triangle $ ABC$ such that $ \angle A \equal{} 90{}^\circ$, $ \left|AB\right| \equal{} \sqrt {12}$, $ \left|AC\right| \equal{} 2$. If $ \left[AF\right]\bigcap \left[BC\right] \equal{} \left\{G\right\}$ , then $ \left|BG\right|$ will be $\textbf{(A)}\ 6 \minus{} 2\sqrt {3} \qquad\textbf{(B)}\ 2\sqrt {3} \minus{} 1 \qquad\textbf{(C)}\ 2 \plus{} \sqrt {3} \\ \qquad\textbf{(D)}\ 4 \minus{} \sqrt {3} \qquad\textbf{(E)}\ 5 \minus{} 2\sqrt {2}$

To get to the Stromboli Volcano from the observatory, one has to take a road and a passway, each taking $4$ hours. There are two craters on the top. The first crater erupts for $1$ hour, stays silent for $17$ hours, then repeats the cycle. The second crater erupts for $1$ hour, stays silent for $9$ hours, erupts for $1$ hour, stays silent for $17$ hours, and then repeats the cycle. During the eruption of the first crater, it is dangerous to take both the passway and the road, but the second crater is smaller, so it is still safe to take the road. At noon, scout Vanya saw both craters erupting simultaneously. Will it ever be possible for him to mount the top of the volcano without risking his life?

Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on line $BC$ satisfying $\angle AID=90^{\circ}$. Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to $\overline{BC}$ at $A_1$. Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$, respectively. Prove that if quadrilateral $AB_1A_1C_1$ is cyclic, then $\overline{AD}$ is tangent to the circumcircle of $\triangle DB_1C_1$. [i]Ankan Bhattacharya[/i]

Let $ABC$ be an acute-angle triangle. Suppose that $M$ be the midpoint of $BC$ and $H$ be the orthocenter of $ABC$. Let $F\equiv BH\cap AC$ and $E\equiv CH\cap AB$. Suppose that $X$ be a point on $EF$ such that $\angle XMH=\angle HAM$ and $A,X$ are in the distinct side of $MH$. Prove that $AH$ bisects $MX$.

Evaluate $2010^2-2009\cdot2011.$

Circles $ C_1$ and $ C_2$ with centers at $ O_1$ and $ O_2$ respectively meet at points $ A$ and $ B$. The radii $ O_1B$ and $ O_2B$ meet $ C_1$ and $ C_2$ at $ F$ and$ E$. The line through $ B$ parallel to $ EF$ intersects $ C_1$ again at $ M$ and $ C_2$ again at $ N$. Prove that $ MN \equal{} AE \plus{} AF$.

For each positive integer $n$, put $$p_n =\left(1+\frac{1}{n}\right)^{n},\; P_n =\left(1+\frac{1}{n}\right)^{n+1}, \; h_n = \frac{2 p_n P_{n}}{ p_n + P_n }.$$ Prove that $h_1 < h_2 < h_3 <\ldots$

Let $ABCD$ be a rectangle with area $1$, and let $E$ lie on side $CD$. What is the area of the triangle formed by the centroids of triangles $ABE$, $BCE$, and $ADE$?

We say that a natural number $n$ is interesting if it can be written in the form \[ n = \left\lfloor \frac{1}{a} \right\rfloor + \left\lfloor \frac{1}{b} \right\rfloor + \left\lfloor \frac{1}{c} \right\rfloor, \] where $a,b,c$ are positive real numbers such that $a + b + c = 1.$ Determine all interesting numbers. ( $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.)

\(ABCD\) is a convex quadrilateral, \(AC\) bisects the angle \(\angle BAD\). Points \(E\) and \(F\) are on the sides \(BC\) and \(CD\) respectively such that \(EF \parallel BD\). Extend \(FA\) and \(EA\) to points \(P\) and \(Q\) respectively, such that the circle \(\omega_1\) passing through points \(A\), \(B\), \(P\) and the circle \(\omega_2\) passing through points \(A\), \(D\), \(Q\) are both tangent to line \(AC\). Prove that the points \(B\), \(P\), \(Q\), \(D\) are concyclic.

In a triangle $ ABC$, $ M$ is the midpoint of $ AB$ and $ P$ an arbitrary point on side $ AC$. Using only a straight edge, construct point $ Q$ on $ BC$ such that $ P$ and $ Q$ are at equal distance from $ CM$.

Prove that of any six four-digit numbers, mutual prime in total, you can always choose five numbers that are also relatively prime in total. [hide=original wording]Докажите, что из любых шести четырехзначных чисел, взаимно простых в совокупности, всегда можно выбратьпя ть чисел, также взаимно простых в совокупности.[/hide]