Let $ABCDEF$ be a regular hexagon and $P$ be a point on the shorter arc $EF$ of its circumcircle. Prove that the value of $$\frac{AP+BP+CP+DP}{EP+FP}$$is constant and find its value.

The vertices of a regular $2005$-gon are colored red, white and blue. Whenever two vertices of different colors stand next to each other, we are allowed to recolor them into the third color. (a) Prove that there exists a finite sequence of allowed recolorings after which all the vertices are of the same color. (b) Is that color uniquely determined by the initial coloring?

The infinity sequence $r_{1},r_{2},...$ of rational numbers it satisfies that: $\prod_{i=1}^ {k}r_{i}=\sum_{i=1}^{k} r_{i}$. For all natural k. Show that $\frac{1}{r_{n}}-\frac{3}{4}$ is a square of rationale number for all natural $n\geq3$

An ant walks on the plane as follows: initially, it walks $1$ cm in any direction. After, at each step, it changes the trajectory direction by $60^o$ left or right and walks $1$ cm in that direction. It is possible that it returns to the point from which it started in (a) $2008$ steps? (b) $2009$ steps? [img]https://cdn.artofproblemsolving.com/attachments/8/b/d4c0d03c67432c4e790b465a74a876b938244c.png[/img]

In the acute-angled triangle $ABC$, on the lines $BC$, $AC$, $AB$ we consider the points $D$, $E$ and, respectively, $F$, such that $AD\perp AC, BE\perp AB, CF\perp AC$. Let the point $A', B', C'$ be such that $\{A'\}=BC\cap EF, \{B'\}=AC\cap DF, \{C'\}=AB\cap DE$. Prove that the following inequality is true $$\frac{A'F}{A'E} \cdot \frac{B'D}{B'F} \cdot \frac{C'E}{C'D}\geq8$$

Let $n > 1$ be an integer. In a circular arrangement of $n$ lamps $L_0, \ldots, L_{n-1},$ each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \ldots .$ If $L_{j-1}$ ($j$ is taken mod $n$) is ON then $Step_j$ changes the state of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If $L_{j-1}$ is OFF then $Step_j$ does not change anything at all. Show that: (i) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again, (ii) If $n$ has the form $2^k$ then all the lamps are ON after $n^2-1$ steps, (iii) If $n$ has the form $2^k + 1$ then all lamps are ON after $n^2 - n + 1$ steps.

Let $n$ be a positive integer. Prove that the numbers $$1^1, 3^3, 5^5, ..., (2n-1)^{2n-1}$$ all give different remainders when divided by $2^n$.

Prove for postitive $a$ and natural $n$ $$a^n+\frac{1}{a^n}-2 \geq n^2(a+\frac{1}{a}-2)$$

A square sheet of paper $ABCD$ is so folded that $B$ falls on the mid point of $M$ of $CD$. Prove that the crease will divide $BC$ in the ration $5 : 3$.

The integers $a$, $b$, $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$. Determine the largest possible value of $d$,

$X$ is a set with $100$ members. What is the smallest number of subsets of $X$ such that every pair of elements belongs to at least one subset and no subset has more than $50$ members? What is the smallest number if we also require that the union of any two subsets has at most $80$ members?

Point $X$ is placed on segment $AB$ of a regular hexagon $ABCDEF$ such that the ratio of the area of $AXEF$ to the area of $XBCDE$ is $\frac12$. If $AB = 2018$, find $AX$.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Is the number $1234567890987654321$ prime?

There are four boats on one of the river banks; their names are Eight, Four, Two and One, because that is the number of hours it takes each of them to cross the river. One boat can be tied to another, but not more than one, and then the time it takes to cross is equal to that of the slower of the two boats. A single sailor must take all the boats to the other shore. What is the least amount of time you need to complete the move?

If $a,b,c,d,e$ are real numbers, prove the inequality $a^2 +b^2 +c^2 +d^2+e^2 \ge a(b+c+d+e)$.

Let $ A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $ AC$ and $ BD$ intersect at $ X$ and $ Y$. The line $ XY$ meets $ BC$ at $ Z$. Let $ P$ be a point on the line $ XY$ other than $ Z$. The line $ CP$ intersects the circle with diameter $ AC$ at $ C$ and $ M$, and the line $ BP$ intersects the circle with diameter $ BD$ at $ B$ and $ N$. Prove that the lines $ AM,DN,XY$ are concurrent.

Let $x, y, z$ be real numbers such that $$x+y+z=2, \;\;\;\; xy+yz+zx=1$$ Find the maximum possible value of $x-y$.

Find all functions $f : R \to R$ such that $$f(x)f(y) = f(xy + 1) + f(x - y) - 2$$ for all $x, y \in R$.

Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.

If $a>0,a\neq1$, $F(x)$ is an odd function. $G(x)=F(x)\cdot(\frac{1}{a^x-1}+\frac{1}{2})$, then $G(x)$ is $\text{(A)}$ odd function $\text{(B)}$ even function $\text{(C)}$ not odd or even function $\text{(D)}$ not sure

Let $\Delta ABC$ be a triangle and $L$ be the foot of the bisector of $\angle A$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle ABL$ and $\triangle ACL$ respectively and let $B_1$ and $C_1$ be the projections of $C$ and $B$ through the bisectors of the angles $\angle B$ and $\angle C$ respectively. The incircle of $\Delta ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively and the bisectors of angles $\angle B$ and $\angle C$ meet the perpendicular bisector of $AL$ at points $Q$ and $P$ respectively. Prove that the five lines $PC_0, QB_0, O_1C_1, O_2B_1$ and $BC$ are all concurrent.

Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.

Let $ABC$ be a triangle and $D,E$ and $F$ the midpoints of sides $BC, AC$ and $BC$. Medians $AD$ and $BE$ are perpendicular, $AD = 12$ and $BE = 9$. What is the value of $CF$?

Through the endpoints $A$ and $B$ of a diameter $AB$ of a given circle, the tangents $\ell$ and $m$ have been drawn. Let $C\ne A$ be a point on $\ell$ and let $q_1,q_2$ be two rays from $C$. Ray $q_i$ cuts the circle in $D_i$ and $E_i$ with $D_i$ between $C$ and $E_i, i = 1,2$. Rays $AD_1,AD_2,AE_1,AE_2$ meet $m$ in the respective points $M_1,M_2,N_1,N_2$. Prove that $M_1M_2 = N_1N_2$.