Find [the decimal form of] the largest prime divisor of $100111011_6$.

There are $1 000 000$ soldiers in a line. The sergeant splits the line into $100$ segments (the length of different segments may be different) and permutes the segments (not changing the order of soldiers in each segment) forming a new line. The sergeant repeats this procedure several times (splits the new line in segments of the same lengths and permutes them in exactly the same way as the first time). Every soldier originally from the first segment recorded the number of performed procedures that took him to return to the first segment for the first time. Prove that at most $100$ of these numbers are different.

Let $ABCD$ be a cyclic quadrilateral, $BL$ and $CN$ be the internal angle bisectors in triangles $ABD$ and $ACD$ respectively. The circumcircles of triangles $ABL$ and $CDN$ meet at points $P$ and $Q$. Prove that the line $PQ$ passes through the midpoint of the arc $AD$ not containing $B$.

Find all values ​​of the parameter $a$ for which the equation $$(a-1)^2x^4 + (a^2-a) x^3 + 3x - 1 = 0$$ has a unique solution and for these $a$ solve the equation.

Three circles lie inside a square. Each of them touches externally two remaining circles. Also each circle touches two sides of the square. Prove that two of these circles are congruent.

A regular $n{}$-gon and a regular $m$-gon with distinct vertices are inscribed in the same circle. The vertices of these polygons divide the circle into $n+m$ arcs. Is it always possible to inscribe a regular $(n+m)$-gon in the same circle so that exactly one of its vertices is on each of these arcs?

Let $\odot O$ be the inscribed circle of $\vartriangle ABC$, with $D$, $E$, $N$ the touchpoints with sides $AB$, $AC$, $BC$ respectively. Extension of $NO$ intersects segment $DE$ at point $K$. Extension of $AK$ intersects segment $BC$ at point $M$. Prove that $M$ is the midpoint of $BC$. [img]https://cdn.artofproblemsolving.com/attachments/a/6/a503c500178551ddf9bdb1df0805ed22bc417d.png[/img]

Let $a_n$ denote an angle from the interval for each $\left( 0, \frac{\pi}{2}\right)$ , the tangent of which is equal to $n$ . Prove that $$\sqrt{1+1^2} \sin(a_1-a_{1000}) + \sqrt{1+2^2} \sin(a_2-a_{1000})+...+\sqrt{1+2000^2} \sin(a_{2000}-a_{1000}) = \sin a_{1000} $$

Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]

Find the minimum value of $\int_0^1 \frac{|t-x|}{t+1}dt$

Let $n \geq 1$ be an integer, and let $S$ be a set of integer pairs $(a,b)$ with $1 \leq a < b \leq 2^n$. Assume $|S| > n \cdot 2^{n+1}$. Prove that there exists four integers $a < b < c < d$ such that $S$ contains all three pairs $(a,c)$, $(b,d)$ and $(a,d)$.

Let $a$, $b$, $c$, $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$. Prove that: \[\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.\] [i]Proposed by Zhang Yanzong and Song Qing[/i]

Determine all real numbers $a, b$ and all integers $n\ge 1$ for which$ (a + b)^n = a^n + b^n$ holds.

A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $ P$ is selected at random inside the circumscribed sphere. The probability that $ P$ lies inside one of the five small spheres is closest to $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 0.1\qquad \textbf{(C)}\ 0.2\qquad \textbf{(D)}\ 0.3\qquad \textbf{(E)}\ 0.4$

Triangle $\triangle ABC$ has side lengths $\overline{AB} = 13, \overline{BC} = 14,$ and $\overline{AC} = 15$. Suppose $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $P$ be a point on $\overline{MN}$, such that if the circumcircles of triangles $\triangle BMP$ and $\triangle CNP$ intersect at a second point $Q$ distinct from $P$, then $PQ$ is parallel to $AB$. The value of $AP^2$ can be expressed as a common fraction of the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest.

For how many real values of $x$ is the equation $(x^2 - 7)^3 = 0$ true? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Show that for each prime $p \geq 7$, there exist a positive integer $n$ and integers $x_{i}$, $y_{i}$ $(i = 1, . . . , n)$, not divisible by $p$, such that $x_{i}^{2}+ y_{i}^{2}\equiv x_{i+1}^{2}\pmod{p}$ where $x_{n+1} = x_{1}$

On all three sides of a right triangle $ABC$ external squares are constructed; their centers denoted by $D$, $E$, $F$. Show that the ratio of the area of triangle $DEF$ to the area of triangle $ABC$ is: a) [i](2 points)[/i] greater than $1$; b) [i](2 points)[/i] at least $2$.

Prove that if $\frac{AC}{BC}=\frac{AB + BC}{AC}$ in a triangle $ABC$ , then $\angle B = 2 \angle A$ .

Let $p > 2$ be a prime number. $\mathbb{F}_p[x]$ is defined as the set of polynomials in $x$ with coefficients in $\mathbb{F}_p$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^k$ are equal in $\mathbb{F}_p$ for each nonnegative integer $k$. For example, $(x+2)(2x+3) = 2x^2 + 2x + 1$ in $\mathbb{F}_5[x]$ because the corresponding coefficients are equal modulo 5. Let $f, g \in \mathbb{F}_p[x]$. The pair $(f, g)$ is called [i]compositional[/i] if \[f(g(x)) \equiv x^{p^2} - x\] in $\mathbb{F}_p[x]$. Find, with proof, the number of compositional pairs.

Let $n \geq 2$ be a natural number, and let $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ be a permutation of $\left(1;\;2;\;...;\;n\right)$. For any integer $k$ with $1 \leq k \leq n$, we place $a_k$ raisins on the position $k$ of the real number axis. [The real number axis is the $x$-axis of a Cartesian coordinate system.] Now, we place three children A, B, C on the positions $x_A$, $x_B$, $x_C$, each of the numbers $x_A$, $x_B$, $x_C$ being an element of $\left\{1;\;2;\;...;\;n\right\}$. [It is not forbidden to place different children on the same place!] For any $k$, the $a_k$ raisins placed on the position $k$ are equally handed out to those children whose positions are next to $k$. [So, if there is only one child lying next to $k$, then he gets the raisin. If there are two children lying next to $k$ (either both on the same position or symmetric with respect to $k$), then each of them gets one half of the raisin. Etc..] After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places). For which $n$ does there exist a configuration $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ and numbers $x_A$, $x_B$, $x_C$, such that all three children are happy?

Find all value of parameter $a$ such that equations $x^2-ax+1=0$ and $x^2-x+a=0$ have at least one same solution. For this value $a$ find same solution of this equations(real or imaginary).

What is the maximum possible value of $5-|6x-80|$ over all integers $x$? $\textbf{(A) }{-}1\qquad\textbf{(B) }0\qquad\textbf{(C) }1\qquad\textbf{(D) }3\qquad\textbf{(E) }5$

Let $ABC$ be an acute-angled triangle. $M ,N$ are any two points on the sides $AB , AC$ respectively. The circles with the diameters $BN$ and $CM$ intersect at points $P$ and $Q$. Show that the points $P, Q$ and the orthocenter of the triangle $ABC$ lie on a straight line.