Compute the sum of the positive integers $n \le 100$ for which the polynomial $x^n + x + 1$ can be written as the product of at least $2$ polynomials of positive degree with integer coefficients.

Let $ABC$ be an equilateral triangle, and let $D,E$ and $F$ be points on $BC,BA$ and $AB$ respectively. Let $\angle BAD= \alpha, \angle CBE=\beta$ and $\angle ACF =\gamma$. Prove that if $\alpha+\beta+\gamma \geq 120^\circ$, then the union of the triangular regions $BAD,CBE,ACF$ covers the triangle $ABC$.

A box contains 36 pink, 18 blue, 9 green, 6 red, and 3 purple cubes that are identical in size. If a cube is selected at random, what is the probability that it is green? $\textbf{(A)}\ \dfrac{1}{9} \qquad \textbf{(B)}\ \dfrac{1}{8} \qquad \textbf{(C)}\ \dfrac{1}{5} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{9}{70}$

A Princeton slot machine has $100$ pictures, each equally likely to occur. One is a picture of a tiger. Alice and Bob independently use the slot machine, and each repeatedly makes independent plays. Alice keeps playing until she sees a tiger, at which point she stops. Similarly, Bob keeps playing until he sees a tiger. Given that Bob plays twice as much as Alice, let the expected number of plays for Alice be $\tfrac{a}{b}$ with $a, b$ relatively prime positive integers. Find the remainder when $a + b$ is divided by $1000$.

In $\bigtriangleup ABC$, $AB=BC=29$, and $AC=42$. What is the area of $\bigtriangleup ABC$? $\textbf{(A) }100\qquad\textbf{(B) }420\qquad\textbf{(C) }500\qquad\textbf{(D) }609\qquad \textbf{(E) }701$

Let $ABCD$ is a trapezoid with $\angle A = \angle B = 90^o$ and let $E$ is a point lying on side $CD$. Let the circle $\omega$ is inscribed to triangle $ABE$ and tangents sides $AB, AE$ and $BE$ at points $P, F$ and $K$ respectively. Let $KF$ intersects segments $BC$ and $AD$ at points $M$ and $N$ respectively, as well as $PM$ and $PN$ intersect $\omega$ at points $H$ and $T$ respectively. Prove that $PH = PT$.

Given a triangle $ABC$, $A',B',C'$ are the midpoints of $\overline{BC},\overline{AC},\overline{AB}$, respectively. $B^*,C^*$ lie in $\overline{AC},\overline{AB}$, respectively, such that $\overline{BB^*},\overline{CC^*}$ are the altitudes of the triangle $ABC$. Let $B^{\#},C^{\#}$ be the midpoints of $\overline{BB^*},\overline{CC^*}$, respectively. $\overline{B'B^{\#}}$ and $\overline{C'C^{\#}}$ meet at $K$, and $\overline{AK}$ and $\overline{BC}$ meet at $L$. Prove that $\angle{BAL}=\angle{CAA'}$

$4026$ points were noted on the plane, not three of which lie on a straight line. The $2013$ points are the vertices of a convex polygon, and the other $2013$ vertices are inside this polygon. It is allowed to paint each point in one of two colors. Coloring will be good if some pairs of dots can be combined segments with the following conditions: $\bullet$ Each segment connects dots of the same color. $\bullet$ No two drawn segments intersect at their inner points. $\bullet$ For an arbitrary pair of dots of the same color, there is a path along the lines from one point to another. Please note that the sides of the convex $2013$ rectangle are not automatically drawn segments, although some (or all) can be drawn as needed. Prove that the total number of good colors does not depend on the specific locations of the points and find that number.

Let $S$ be a fixed point on a sphere $\Sigma$ with center $\Omega$. Consider all tetrahedra $SABC$ inscribed in $\Sigma$ such that $SA,SB,SC$ are pairwise orthogonal. (a) Prove that all the planes $ABC$ pass through a single point. (b) In one such tetrahedron, $H$ and $O$ are the orthogonal projections of $S$ and $\Omega$ onto the plane $ABC$, respectively. Let $R$ denote the circumradius of $\triangle ABC$. Prove that $R^2=OH^2+2SH^2$.

If $ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0$, find the value of $b\sqrt{a^2+b}+a\sqrt{b^2+a}$

Let $a, b$ be positive integers such that $54^a=a^b$. Prove that $a$ is a power of $54$.

Two players play a game as follows. The first player chooses two non-zero integers A and B. The second player forms a quadratic with A, B and 1998 as coefficients (in any order). The first player wins iff the equation has two distinct rational roots. Show that the first player can always win.

Let be a natural number $ n $ and $ 2n $ positive real numbers $ v_1,v_2,\ldots ,v_{2n} $ such that the last $ n $ of them are greater than $ 1. $ Prove that: $$ \sum_{i=1}^n v_iv_{n+i}\le \max_{1\le k\le n}\left( \left( -1+\prod_{l=n}^{2n} v_l \right) v_k +\sum_{m=1}^n v_m \right) $$

Let $\triangle ABC$ be a triangle such that $AB=AC=40$ and $BC=79.$ Let $X$ and $Y$ be the points on segments $AB$ and $AC$ such that $AX=5, AY=25.$ Given that $P$ is the intersection of lines $XY$ and $BC,$ compute $PX\cdot PY-PB\cdot PC.$

The degrees of all vertices of a graph are not less than 100 and not more than 200. Prove that its vertices can be divided into connected pairs and triples.

Find all positive integers \( m \) for which there exists an infinite subset \( A \) of the positive integers such that: for any pairwise distinct positive integers \( a_1, a_2, \cdots, a_m \in A \), the sum \( a_1 + a_2 + \cdots + a_m \) and the product \( a_1a_2 \cdots a_m \) are both square-free.

Let be a natural number $ n, $ a positive real number $ \lambda , $ and a complex number $ z. $ Prove the following inequalities. $$ 0\le -\lambda +\frac{1}{n}\sum_{\stackrel{w\in\mathbb{C}}{w^n=1 }} \left| z-\lambda w \right|\le |z| $$ [i]Gheorghe Iurea[/i]

A snail crawls $2\frac12$ centimeters in $4\frac14$ minutes. At this rate, how many centimeters can the snail crawl is 85 minutes?

Let $ABC$ be a triangle with $AB = 13, BC = 14,$ and $CA = 15.$ Pick points $Q$ and $R$ on $AC$ and $AB$ such that $\angle CBQ = \angle BCR = 90^\circ.$ There exist two points $P_1\neq P_2$ in the plane of $ABC$ such that $\triangle P_1 QR, \triangle P_2QR,$ and $\triangle ABC$ are similar (with vertices in order). Compute the sum of the distances from $P_1$ to $BC$ and $P_2$ to $BC.$

Let $P$ be a point inside triangle $ABC$, and suppose lines $AP$, $BP$, $CP$ meet the circumcircle again at $T$, $S$, $R$ (here $T \neq A$, $S \neq B$, $R \neq C$). Let $U$ be any point in the interior of $PT$. A line through $U$ parallel to $AB$ meets $CR$ at $W$, and the line through $U$ parallel to $AC$ meets $BS$ again at $V$. Finally, the line through $B$ parallel to $CP$ and the line through $C$ parallel to $BP$ intersect at point $Q$. Given that $RS$ and $VW$ are parallel, prove that $\angle CAP = \angle BAQ$.

In a room, there are $100$ light switches, labeled with the positive integers ${1,2, . . . ,100}$. They’re all initially turned off. On the $i$ th day for $1 \le i \le 100$, Bob flips every light switch with label number $k$ divisible by $i$ a total of $\frac{k}{i}$ times. Find the sum of the labels of the light switches that are turned on at the end of the $100$th day.

The expression $ \frac{1^{4y\minus{}1}}{5^{\minus{}1}\plus{}3^{\minus{}1}}$ is equal to: $ \textbf{(A)}\ \frac{4y\minus{}1}{8} \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ \frac{15}{2} \qquad \textbf{(D)}\ \frac{15}{8} \qquad \textbf{(E)}\ \frac{1}{8}$

A (not necessarily regular) tetrahedron $A_1A_2A_3A_4$ is given in space. For each pair of indices $1\leq i<j\leq 4$, an ellipsoid with foci $A_i,A_j$ and string length $\ell_{ij}$, for positive numbers $\ell_{ij}$, is given (in all 6 ellipsoids were built). For each $i=1,2$, a pair of points $X_i\neq X'_i$ was chosen so that $X_i, X'_i$ both belong to all three ellipsoids with $A_i$ as one of their foci. Prove that the lines $X_1X'_1, X_2X'_2$ share a point in space if and only if \[\ell_{13}+\ell_{24}=\ell_{14}+\ell_{23}\] [i]Remark: An [u]ellipsoid[/u] with foci $P,Q$ and string length $\ell>|PQ|$ is defined here as the set of points $X$ in space for which $|XQ|+|XP|=\ell$.[/i]

If $z_1$ , $z_2$ are the roots of the equation with real coefficients $z^2+az+b = 0$, prove that $ z^n_1 + z^n_2$ is a real number for any natural value of $n$. If particular of the equation $z^2 - 2z + 2 = 0$, express, as a function of $n$, the said sum.

Let $ABC$ be a triangle, $H$ its orthocenter, and $\Gamma$ its circumcircle. Let $J$ be the point diametrically opposite to $A$ on $\Gamma$. The points $D$, $E$ and $F$ are the feet of the altitudes from $A$, $B$ and $C$, respectively. The line $AD$ intersects $\Gamma$ again at $P$. The circumcircle of $EFP$ intersects $\Gamma$ again at $Q$. Let $K$ be the second point of intersection of $JH$ with $\Gamma$. Prove that $K$, $D$ and $Q$ are collinear.