In a building whose floors are numbered $1$ to $8$, the builder wants to place elevators so that, for every choice of two floors, there are always at least three elevators that stop on those floors. Furthermore, each elevator can only stop at a maximum of $5$ floors. What is the minimum number of elevators that need to be placed?

Determine all positive integers $d$ such that whenever $d$ divides a positive integer $n$, $d$ will also divide any integer obtained by rearranging the digits of $n$.

In the triangle$ABC$ ($AC > AB$), point $N$ is the midpoint of $BC$, and $I$ is the intersection point of the angle bisectors. Ray $AI$ intersects the circumscribed circle of triangle $ABC$ at point $W$, a perpendicular $WF$ is drawn from it on side $AC$. Find the length of the segment $CF$ , if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$ and $\angle INB = 45^o$. (Gryhoriy Filippovskyi)

Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon.

Each of six fruit baskets contains pears, plums and apples. The number of plums in each basket equals the total number of apples in all other baskets combined while the number of apples in each basket equals the total number of pears in all other baskets combined. Prove that the total number of fruits is a multiple of $31$.

$x\geq y\geq z\geq \frac{\pi}{12},x+y+z=\frac{\pi}{2}$, find the maximum and minumum value of $\cos x\sin y\cos z$.

A regular polyhedron is a polyhedron that is convex and all of its faces are regular polygons. We call a regular polhedron a "[i]Choombam[/i]" iff none of its faces are triangles. a) prove that each choombam can be inscribed in a sphere. b) Prove that faces of each choombam are polygons of at most 3 kinds. (i.e. there is a set $\{m,n,q\}$ that each face of a choombam is $n$-gon or $m$-gon or $q$-gon.) c) Prove that there is only one choombam that its faces are pentagon and hexagon. (Soccer ball) [img]http://aycu08.webshots.com/image/5367/2001362702285797426_rs.jpg[/img] d) For $n>3$, a prism that its faces are 2 regular $n$-gons and $n$ squares, is a choombam. Prove that except these choombams there are finitely many choombams.

Suppose $a$, $b$, $c$ are three positive real numbers with $a + b + c = 3$. Prove that $$\frac{a}{b^2 + c}+\frac{b}{c^2 + a}+\frac{c}{a^2 + b}\geq \frac{3}{2}$$

[b]p1.[/b] Is it possible to put $9$ numbers $1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9$ in a circle in a way such that the sum of any three circularly consecutive numbers is divisible by $3$ and is, moreover: a) greater than $9$ ? b) greater than $15$? [b]p2.[/b] You can cut the figure below along the sides of the small squares into several (at least two) identical pieces. What is the minimal number of such equal pieces? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] [b]p3.[/b] There are $100$ colored marbles in a box. It is known that among any set of ten marbles there are at least two marbles of the same color. Show that the box contains $12$ marbles of the same color. [b]p4.[/b] Is it possible to color squares of a $ 8\times 8$ board in white and black color in such a way that every square has exactly one black neighbor square separated by a side? [b]p5.[/b] In a basket, there are more than $80$ but no more than $200$ white, yellow, black, and red balls. Exactly $12\%$ are yellow, $20\%$ are black. Is it possible that exactly $2/3$ of the balls are white? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

China Mathematical Olympiad 2018 Q6 Given the positive integer $n ,k$ $(n>k)$ and $ a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq \{1,2,\cdots,n\}$ ,$|\mathbb{I} |=k$,have $\sum_{i\in \mathbb{I} }x_i\le \sum_{i\in \mathbb{I} }a_i$ , find the maximum value of $x_1x_2\cdots x_n.$

Determine the number of pairs $(a, b)$, where $1 \le a \le b \le 100$ are positive integers, so that $\frac{a^3+b^3}{a^2+b^2}$ is an integer.

How many triangles are in this figure? [asy] import olympiad; pair A = (0,0); pair B = (0,1); pair C = (0,2); pair D = (0,3); pair E = (0,4); pair F = (1,0); pair G = (2,0); pair H = (3,0); pair I = (4,0); pair J = (1,4); pair K = (2,4); pair L = (3,4); pair M = (4,4); pair N = (4,3); pair O = (4,2); pair P = (4,1); draw(A--E--I--A); draw(M--E--I--M); draw(B--F); draw(C--G); draw(D--H); draw(L--N); draw(O--K); draw(P--J); draw(B--F); draw(B--F); draw(H--P); draw(G--O); draw(F--N); draw(B--L); draw(C--K); draw(D--J); draw(A--M); [/asy] $(A) 56$ $(B) 60$ $(C) 64$ $(D) 68$

Suppose $p$ is an odd prime number. We call the polynomial $f(x)=\sum_{j=0}^n a_jx^j$ with integer coefficients $i$-remainder if $ \sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}$. Prove that the set $\{f(0),f(1),...,f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), (f(x))^2,...,(f(x))^{p-2}$ are $0$-remainder and the polynomial $(f(x))^{p-1}$ is $1$-remainder. [i]Proposed by Yahya Motevassel[/i]

Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores. [asy] //diagram by pog . give me 1,000,000,000 dollars for this diagram size(5cm); defaultpen(0.7); dot((0.5,1)); dot((0.5,1.5)); dot((1.5,1)); dot((1.5,1.5)); dot((2.5,1)); dot((2.5,1.5)); dot((2.5,2)); dot((2.5,2.5)); dot((3.5,1)); dot((3.5,1.5)); dot((3.5,2)); dot((3.5,2.5)); dot((3.5,3)); dot((4.5,1)); dot((4.5,1.5)); dot((5.5,1)); dot((5.5,1.5)); dot((5.5,2)); dot((6.5,1)); dot((7.5,1)); draw((0,0.5)--(8,0.5),linewidth(0.7)); defaultpen(fontsize(10.5pt)); label("$65$", (0.5,-0.1)); label("$70$", (1.5,-0.1)); label("$75$", (2.5,-0.1)); label("$80$", (3.5,-0.1)); label("$85$", (4.5,-0.1)); label("$90$", (5.5,-0.1)); label("$95$", (6.5,-0.1)); label("$100$", (7.5,-0.1)); [/asy] Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students $5$ extra points, which increased the median test score to $85$. What is the minimum number of students who received extra points? (Note that the [i]median[/i] test score equals the average of the $2$ scores in the middle if the $20$ test scores are arranged in increasing order.) $\textbf{(A)} ~2\qquad\textbf{(B)} ~3\qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~5\qquad\textbf{(E)} ~6\qquad$

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.

Let $m$ be a positive integer. Define the sequence $a_0, a_1, a_2, \cdots$ by $a_0 = 0,\; a_1 = m,$ and $a_{n+1} = m^2a_n - a_{n-1}$ for $n = 1,2,3,\cdots$. Prove that an ordered pair $(a,b)$ of non-negative integers, with $a \leq b$, gives a solution to the equation \[ {\displaystyle \frac{a^2 + b^2}{ab + 1} = m^2} \] if and only if $(a,b)$ is of the form $(a_n,a_{n+1})$ for some $n \geq 0$.

Let $\alpha , \beta , \gamma , \delta$ be positive numbers such that for all $x$, $\sin{\alpha x}+\sin {\beta x}=\sin {\gamma x}+\sin {\delta x}$. Prove that $\alpha =\gamma$ or $\alpha=\delta$.

Find all polynomials $P(x), Q(x)$ with real coefficients such that for all real numbers $a$, $P(a)$ is a root of the equation $x^{2023}+Q(a)x^2+(a^{2024}+a)x+a^3+2025a=0.$

Solve the equation: $$\frac{x^2}{3}+\frac{48}{x^3}=10 \left(\frac{x}{3}-\frac{4 }{x} \right)$$

Consider six arbitrary points in space. Every two points are joined by a segment. Prove that there are two triangles that can not be separated. [img]http://i38.tinypic.com/35n615y.png[/img]

For each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides $\sum^{\left\lfloor \frac{n - 1}{2}\right\rfloor}_{i=0} \binom{n}{2 \cdot i + 1} 3^i$. Find all $n$ such that $f(n) = 1996.$ [hide="old version"]For each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides $\sum^{n + 1/2}_{i=1} \binom{2 \cdot i + 1}{n}$. Find all $n$ such that $f(n) = 1996.$[/hide]

In a triangle $ABC,$ incircle touches the sides $BC, CA, AB$ at $D, E, F,$ respectively. A circle $\omega$ passing through $A$ and tangent to line $BC$ at $D$ intersects the line segments $BF$ and $CE$ at $K$ and $L,$ respectively. The line passing through $E$ and parallel to $DL$ intersects the line passing through $F$ and parallel to $DK$ at $P.$ If $R_1, R_2, R_3, R_4$ denotes the circumradius of the triangles $AFD, AED, FPD, EPD,$ respectively, prove that $R_1R_4=R_2R_3.$

In a scalene triangle $\Delta ABC$, $AB<AC$, $PB$ and $PC$ are tangents of the circumcircle $(O)$ of $\Delta ABC$. A point $R$ lies on the arc $\widehat{AC}$(not containing $B$), $PR$ intersects $(O)$ again at $Q$. Suppose $I$ is the incenter of $\Delta ABC$, $ID \perp BC$ at $D$, $QD$ intersects $(O)$ again at $G$. A line passing through $I$ and perpendicular to $AI$ intersects $AB,AC$ at $M,N$, respectively. Prove that, if $AR \parallel BC$, then $A,G,M,N$ are concyclic.

Find all polynomials with degree $\leq n$ and nonnegative coefficients, such that $P(x)P(\frac{1}{x}) \leq P(1)^2$ for every positive $x$

Form a square with sides of length $5$, triangular pieces from the four coreners are removed to form a regular octagonn. Find the area [b]removed[/b] to the nearest integer.