The hall in a castle is a regular hexagon where its sides' length is 6 meters. The floor of the hall is to be tiled with equilateral triangular tiles where its sides' length is 50 centimeters. Each tile is divided into three congruent triangles by their altitudes up to its orthocenter (see below). Each of these small triangles are colored such that each tile has different colors and no two tiles have identical colorings. How many colors at least are required? A tile's pattern is: [asy] draw((0,0.000)--(2,0.000)); draw((2,0.000)--(1,1.732)); draw((1,1.732)--(0,0.000)); draw((1,0.577)--(0,0.000)); draw((1,0.577)--(2,0.000)); draw((1,0.577)--(1,1.732)); [/asy]

Let $O$ be the circumcenter of the acute triangle $ABC$ ($AB < AC$). Let $A_1$ and $P$ be the feet of the perpendicular lines drawn from $A$ and $O$ to $BC$, respectively. The lines $BO$ and $CO$ intersect $AA_1$ in $D$ and $E$, respectively. Let $F$ be the second intersection point of $\odot ABD$ and $\odot ACE$. Prove that the angle bisector od $\angle FAP$ passes through the incenter of $\triangle ABC$.

If $\alpha, \beta, \gamma$ are angles whose measures in radians belong to the interval $\left[0, \frac{\pi}{2}\right]$ such that: $$\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1$$ calculate the minimum possible value of $\cos \alpha + \cos \beta + \cos \gamma$.

Let \(n\) be any positive integer. Prove that \[\sum^n_{i=1} \frac{1}{(i^2+i)^{3/4}} > 2-\frac{2}{\sqrt{n+1}}\].

Let $ CL$ be a bisector of triangle $ ABC$. Points $ A_1$ and $ B_1$ are the reflections of $ A$ and $ B$ in $ CL$, points $ A_2$ and $ B_2$ are the reflections of $ A$ and $ B$ in $ L$. Let $ O_1$ and $ O_2$ be the circumcenters of triangles $ AB_1B_2$ and $ BA_1A_2$ respectively. Prove that angles $ O_1CA$ and $ O_2CB$ are equal.

What is the remainder when $13^{16} + 17^{12}$ is divided by $221$?

Let $O$ be a point of three-dimensional space and let $l_1, l_2, l_3$ be mutually perpendicular straight lines passing through $O$. Let $S$ denote the sphere with center $O$ and radius $R$, and for every point $M$ of $S$, let $S_M$ denote the sphere with center $M$ and radius $R$. We denote by $P_1, P_2, P_3$ the intersection of $S_M$ with the straight lines $l_1, l_2, l_3$, respectively, where we put $P_i \neq O$ if $l_i$ meets $S_M$ at two distinct points and $P_i = O$ otherwise ($i = 1, 2, 3$). What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$?

In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long. [asy] draw(Circle((0,0),10)); draw((0,0)--(10,0)--(8.5,5.3)--(-8.5,5.3)--(-3,9.5)--(3,9.5)); dot((0,0)); dot((10,0)); dot((8.5,5.3)); dot((-8.5,5.3)); dot((-3,9.5)); dot((3,9.5)); label("1", (5,0), S); label("s", (8,2.6)); label("d", (0,4)); label("s", (-5,7)); label("s", (0,8.5)); label("O", (0,0),W); label("R", (10,0), E); label("M", (-8.5,5.3), W); label("N", (8.5,5.3), E); label("P", (-3,9.5), NW); label("Q", (3,9.5), NE); [/asy] Of the three equations \[ \textbf{I.}\ d-s=1, \qquad \textbf{II.}\ ds=1, \qquad \textbf{III.}\ d^2-s^2=\sqrt{5} \]those which are necessarily true are $\textbf{(A)}\ \textbf{I}\ \text{only} \qquad\textbf{(B)}\ \textbf{II}\ \text{only} \qquad\textbf{(C)}\ \textbf{III}\ \text{only} \qquad\textbf{(D)}\ \textbf{I}\ \text{and}\ \textbf{II}\ \text{only} \qquad\textbf{(E)}\ \textbf{I, II}\ \text{and} \textbf{III}$

Let $ABCD$ be a parallelogram. Suppose that there exists a point $P$ in the interior of the parallelogram which is on the perpendicular bisector of $AB$ and such that $\angle PBA = \angle ADP$ Show that $\angle CPD = 2 \angle BAP$

A Gaussian prime is a Gaussian integer $ z\equal{}a\plus{}bi$ (where $ a$ and $ b$ are integers) with no Gaussian integer factors of smaller absolute value. Factor $ \minus{}4\plus{}7i$ into Gaussian primes with positive real parts. $ i$ is a symbol with the property that $ i^2\equal{}\minus{}1$.

Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

[b]Problem 4 [/b] Let $m$ be a positive integer and let $p$ be a prime divisor of $m$. Suppose that the complex polynomial $a_0 + a_1x + \ldots + a_nx^n$ with $n < \frac{p}{p-1}\varphi(m)$ and $a_n \neq 0$ is divisible by the cyclotomic polynomial $\phi_m(x)$. Prove that there are at least $p$ nonzero coefficients $a_i\ .$ The cyclotomic polynomial $\phi_m(x)$ is the monic polynomial whose roots are the $m$-th primitive complex roots of unity. Euler’s totient function $\varphi(m)$ denotes the number of positive integers less than or equal to $m$ which are coprime to $m$.

Three circles $\omega_1,\omega_2$ and $\omega_3$ pass through one point $D{}$. Let $A{}$ be the intersection of $\omega_1$ and $\omega_3$, and $E{}$ be the intersections of $\omega_3$ and $\omega_2$ and $F{}$ be the intersection of $\omega_2$ and $\omega_1$. It is known that $\omega_3$ passes through the center $B{}$ of the circle $\omega_2$. The line $EF$ intersects $\omega_1$ a second time at the point $G{}$. Prove that $\angle GAB=90^\circ$. [i]Proposed by K. Knop[/i]

Let $AD$ be the angle bisector of triangle $ABC$, and let the line $\ell$ touch circumcircles of triangles $ADB$ and $ADC$ at points $M$ and $N$ accordingly. Prove that the circle passing through the midpoints of the segments $BD$, $DC$ and $MN$ is tangent to the line $\ell$.

There are $k$ heaps on the table, each containing a different positive number of stones. Juri and Mari make moves alternatingly, Juri starts. On each move, the player making the move has to pick a heap and remove one or more stones in it from the table; in addition, the player is allowed to distribute any number of remaining stones from that heap in any way between other non-empty heaps. The player to remove the last stone from the table wins. For which positive integers $k$ does Juri have a winning strategy for any initial state that satisfies the conditions?

Is it possible to replace stars with plusses or minusses in the following expression $$1 \star 2 \star 3 \star 4 \star 5 \star 6 \star 7 \star 8 \star 9 \star 10 = 0$$ so that to obtain a true equality?

Let $A=\{a_1,a_2,\ldots,a_n\}$ be a set with $a_1<a_2\ldots<a_n$ and $B=\{b_1,b_2,\ldots,b_n\}$ be a set with $b_1<b_2\ldots<b_n$. Show that for every bijective function $f:A\rightarrow B$ the following relation takes place $$\max_{1\leq i\leq n} |a_i-f(a_i)| \geq \max_{1\leq i\leq n} |a_i-b_i|.$$

Let $n \geq 1$ be an integer. Ruben takes a test with $n$ questions. Each question on this test is worth a different number of points. The first question is worth $1$ point, the second question $2$, the third $3$ and so on until the last question which is worth $n$ points. Each question can be answered either correctly or incorrectly. So an answer for a question can either be awarded all, or none of the points the question is worth. Let $f(n)$ be the number of ways he can take the test so that the number of points awarded equals the number of questions he answered incorrectly. Do there exist infinitely many pairs $(a; b)$ with $a < b$ and $f(a) = f(b)$?

Integers $x_1,x_2,\cdots,x_{100}$ satisfy \[ \frac {1}{\sqrt{x_1}} + \frac {1}{\sqrt{x_2}} + \cdots + \frac {1}{\sqrt{x_{100}}} = 20. \]Find $ \displaystyle\prod_{i \ne j} \left( x_i - x_j \right) $.

A grasshopper is jumping along the number line. Initially it is situated at zero. In $k$-th step, the length of his jump is $k$. a) If the jump length is even, then it jumps to the left, otherwise it jumps to the right (for example, firstly it jumps one step to the right, then two steps to the left, then three steps to the right, then four steps to the left...). Will it visit on every integer at least once? b) If the jump length is divisible by three, then it jumps to the left, otherwise it jumps to the right (for example, firstly it jumps one step to the right, then two steps to the right, then three steps to the left, then four steps to the right...). Will it visit every integer at least once? Proposed by Matko Ljulj

Any two of the real numbers $a_1,a_2,a_3,a_4,a_5$ differ by no less than $1$. There exists some real number $k$ satisfying \[a_1+a_2+a_3+a_4+a_5=2k\]\[a_1^2+a_2^2+a_3^2+a_4^2+a_5^2=2k^2\] Prove that $k^2\ge 25/3$.

Let $a,b,c,d$ be distinct integers such that $$(x-a)(x-b)(x-c)(x-d) -4=0$$ has an integer root $r.$ Show that $4r=a+b+c+d.$

If $a = 2b + c$, $b = 2c + d$, $2c = d + a -1$, $d = a - c$, what is $b$?

[b]p1.[/b] What is the last digit of $$1! + 2! + ... + 10!$$ where $n!$ is defined to equal $1 \cdot 2 \cdot ... \cdot n$? [b]p2.[/b] What pair of positive real numbers, $(x, y)$, satisfies $$x^2y^2 = 144$$ $$(x - y)^3 = 64?$$ [b]p3.[/b] Paul rolls a standard $6$-sided die, and records the results. What is the probability that he rolls a $1$ ten times before he rolls a $6$ twice? [b]p4.[/b] A train is approaching a $1$ kilometer long tunnel at a constant $40$ km/hr. It so happens that if Roger, who is inside, runs towards either end of the tunnel at a contant $10$ km/hr, he will reach that end at the exact same time as the train. How far from the center of the tunnel is Roger? [b]p5.[/b] Let $ABC$ be a triangle with $A$ being a right angle. Let $w$ be a circle tangent to $\overline{AB}$ at $A$ and tangent to $\overline{BC}$ at some point $D$. Suppose $w$ intersects $\overline{AC}$ again at $E$ and that $\overline{CE} = 3$, $\overline{CD} = 6$. Compute $\overline{BD}$. [b]p6.[/b] In how many ways can $1000$ be written as a sum of consecutive integers? [b]p7.[/b] Let $ABC$ be an isosceles triangle with $\overline{AB} = \overline{AC} = 10$ and $\overline{BC} = 6$. Let $M$ be the midpoint of $\overline{AB}$, and let $\ell$ be the line through $A$ parallel to $\overline{BC}$. If $\ell$ intersects the circle through $A$, $C$ and $M$ at $D$, then what is the length of $\overline{AD}$? [b]p8.[/b] How many ordered triples of pairwise relatively prime, positive integers, $\{a, b, c\}$, have the property that $a + b$ is a multiple of $c$, $b + c$ is a multiple of $a$, and $a + c$ is a multiple of $b$? [b]p9.[/b] Consider a hexagon inscribed in a circle of radius $r$. If the hexagon has two sides of length $2$, two sides of length $7$, and two sides of length $11$, what is $r$? [b]p10.[/b] Evaluate $$\sum^{\infty}_{i=0} \sum^{\infty}_{j=0} \frac{\left( (-1)^i + (-1)^j\right) \cos (i) \sin (j)}{i!j!} ,$$ where angles are measured in degrees, and $0!$ is defined to equal $1$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a cyclic quadrilateral such that $AB \cdot BC=2 \cdot AD \cdot DC$. Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \le 9AC^2$. [color=#FF0000] Moderator says: Do not double post [url]http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=590175[/url][/color]