$ A$ can do a piece of work in $ 9$ days. $ B$ is $ 50\%$ more efficient than $ A$. The number of days it takes $ B$ to do the same piece of work is: $ \textbf{(A)}\ 13\frac {1}{2} \qquad\textbf{(B)}\ 4\frac {1}{2} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{none of these answers}$

Given two circles $C_1,C_2$ with common center, the radius of $C_2$ is twice the radius of $C_1$. Quadrilateral $A_1A_2A_3A_4$ is inscribed in $C_1$. The extension of $A_4A_1$ meets $C_2$ at $B_1$; the extension of $A_1A_2$ meets $C_2$ at $B_2$; the extension of $A_2A_3$ meets $C_2$ at $B_3$; the extension of $A_3A_4$ meets $C_2$ at $B_4$. Prove that $P(B_1B_2B_3B_4)\ge 2P(A_1A_2A_3A_4)$, and in what case the equality holds? ($P(X)$ denotes the perimeter of quadrilateral $X$)

Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.

Find the sum of all $x$ from $2$ to $1000$ inclusive such that $$\prod_{n=2}^x \log_{n^n}(n+1)^{n+2}$$ is an integer. [i]Proposed by Deyuan Li and Andrew Milas[/i]

In a quadrilateral $ABCD$ sides $AB$ and $CD$ are equal, $\angle A=150^\circ,$ $\angle B=44^\circ,$ $\angle C=72^\circ.$ Perpendicular bisector of the segment $AD$ meets the side $BC$ at point $P.$ Find $\angle APD.$ [i]Proposed by F. Bakharev[/i]

Show that a path on a rectangular grid which starts at the northwest corner, goes through each point on the grid exactly once, and ends at the southeast corner divides the grid into two equal halves: (a) those regions opening north or east; and (b) those regions opening south or west. [img]https://cdn.artofproblemsolving.com/attachments/b/e/aa20c9f9bc44bd1e5a9b9e86d49debf0f821b7.png[/img] (The figure above shows a path meeting the conditions of the problem on a $5 \times 8$ grid. The shaded regions are those opening north or east while the rest open south or west.)

Let $ P,Q $ be the midpoints of the diagonals $ BD, $ respectively, $ AC, $ of the quadrilateral $ ABCD, $ and points $ M,N,R,S $ on the segments $ BC,CD,PQ, $ respectively $ AC, $ except their extremities, such that $$ \frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} . $$ Show that the center of mass of the triangle $ AMN $ is situated on the segment $ RS. $

Csenge and Eszter ate a whole basket of cherries. Csenge ate a quarter of all cherries while Eszter ate four-sevenths of all cherries and forty more. How many cherries were in the basket in total?

A sequence $ \left( x_n\right)_{n\ge 0} $ of real numbers satisfies $ x_0>1=x_{n+1}\left( x_n-\left\lfloor x_n\right\rfloor\right) , $ for each $ n\ge 1. $ Prove that if $ \left( x_n\right)_{n\ge 0} $ is periodic, then $ x_0 $ is a root of a quadratic equation. Study the converse.

Consider ten concentric circles and ten rays as in the following figure. At the points where the inner circle is intersected by the rays write successively, in direction clockwise, the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. In the next circle we write the numbers $11, 12, 13, 14, 15, 16, 17, 18, 19,20$ successively, and so on successively until the last round were we write the numbers $91, 92, 93, 94, 95, 96, 97, 98, 99, 100$ successively. In this orde, the numbers $1, 11, 21, 31, 41, 51, 61, 71, 81, 91$ are in the same ray, and similarly for the other rays. In front of $50$ of those $100$ numbers, we use the sign ''$-$'' such as: a) in each of the ten rays, exist exactly $5$ signs ''$-$'' , and also b) in each of the ten concentric circles, to be exactly $5$ signs ''$-$''. Prove that the sum of the $100$ signed numbers that occur, equals zero. [img]https://cdn.artofproblemsolving.com/attachments/9/d/ffee6518fcd1b996c31cf06d0ce484a821b4ae.gif[/img]

The sides $BC$ and $AD$ of a quadrilateral $ABCD$ are parallel and the diagonals intersect in $O$. For this quadrilateral $|CD| =|AO|$ and $|BC| = |OD|$ hold. Furthermore $CA$ is the angular bisector of angle $BCD$. Determine the size of angle $ABC$. [asy] unitsize(1 cm); pair A, B, C, D, O; D = (0,0); B = 3*dir(180 + 72); C = 3*dir(180 + 72 + 36); A = extension(D, D + (1,0), C, C + dir(180 - 36)); O = extension(A, C, B, D); draw(A--B--C--D--cycle); draw(B--D); draw(A--C); dot("$A$", A, N); dot("$B$", B, SW); dot("$C$", C, SE); dot("$D$", D, N); dot("$O$", O, E); [/asy] Attention: the figure is not drawn to scale.

Let $G{}$ be an arbitrary finite group, and let $t_n(G)$ be the number of functions of the form \[f:G^n\to G,\quad f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n\quad(a_0,\ldots,a_n\in G).\]Determine the limit of $t_n(G)^{1/n}$ as $n{}$ tends to infinity.

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.) [asy]import three; import math; size(180); defaultpen(linewidth(.8pt)); currentprojection=orthographic(2,0.2,1); triple A=(0,0,1); triple B=(sqrt(2)/2,sqrt(2)/2,0); triple C=(sqrt(2)/2,-sqrt(2)/2,0); triple D=(-sqrt(2)/2,-sqrt(2)/2,0); triple E=(-sqrt(2)/2,sqrt(2)/2,0); triple F=(0,0,-1); draw(A--B--E--cycle); draw(A--C--D--cycle); draw(F--C--B--cycle); draw(F--D--E--cycle,dotted+linewidth(0.7));[/asy]$ \textbf{(A)}\ 210 \qquad \textbf{(B)}\ 560 \qquad \textbf{(C)}\ 840 \qquad \textbf{(D)}\ 1260 \qquad \textbf{(E)}\ 1680$

If $b>1$, $x>0$ and $(2x)^{\log_b 2}-(3x)^{\log_b 3}=0$, then $x$ is $\text{(A)}\ \frac{1}{216} \qquad \text{(B)}\ \frac{1}{6} \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ \text{not uniquely determined}$

Any two members of a club with $3n+1$ people plays ping-pong, tennis or chess with each other. Everyone has exactly $n$ partners who plays ping-pong, $n$ who play tennis and $n$ who play chess. Prove that we can choose three members of the club who play three different games amongst each other.

In acute triangle $ABC$, $\angle A = 20^o$. Prove that the triangle is isosceles if and only if $$\sqrt[3]{a^3 + b^3 + c^3 -3abc} = \min\{b, c\}$$, where $a,b, c$ are the side lengths of triangle $ABC$.

Find a positive integer $N$ and $a_1, a_2, \cdots, a_N$ where $a_k = 1$ or $a_k = -1$, for each $k=1,2,\cdots,N,$ such that $$a_1 \cdot 1^3 + a_2 \cdot 2^3 + a_3 \cdot 3^3 \cdots + a_N \cdot N^3 = 20162016$$ or show that this is impossible.

Prove that a necessary and sufficient for the existence of a set $ S \subset \{1,2,...,n \}$ with the property that the integers $ 0,1,...,n\minus{}1$ all have an odd number of representations in the form $ x\minus{}y, x,y \in S$, is that $ (2n\minus{}1)$ has a multiple of the form $ 2.4^k\minus{}1$ [i]L. Lovasz, J. Pelikan[/i]

Let $p$ be a prime number and $h$ be a natural number smaller than $p$. We set $n = ph + 1$. Prove that if $2^{n-1}-1$, but not $2^h-1$, is divisible by $n$, then $n$ is a prime number.

A sequence of complex numbers $ z_0,z_1,z_2,....$ is defined by the rule \[ z_{n \plus{} 1} \equal{} \frac {i z_n}{\overline{z_n}} \]where $ \overline{z_n}$ is the complex conjugate of $ z_n$ and $ i^2 \equal{} \minus{} 1$. Suppose that $ |z_0| \equal{} 1$ and $ z_{2005} \equal{} 1$. How many possible values are there for $ z_0$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 2005\qquad \textbf{(E)}\ 2^{2005}$

Find all non-constant polynoms $ f\in\mathbb{Q} [X] $ that don't have any real roots in the interval $ [0,1] $ and for which there exists a function $ \xi :[0,1]\longrightarrow\mathbb{Q} [X]\times\mathbb{Q} [X], \xi (x):=\left( g_x,h_x \right) $ such that $ h_x(x)\neq 0 $ and $ \int_0^x \frac{dt}{f(t)} =\frac{g_x(x)}{h_x(x)} , $ for all $ x\in [0,1] . $

Mater is confused and starts going around the track in the wrong direction. He can go around 7 times in an hour. Lightning and Chick start in the same place at Mater and at the same time, both going the correct direction. Lightning can go around 91 times per hour, while Chick can go around 84 times per hour. When Lightning passes Chick for the third time, how many times will he have passed Mater (if Lightning is passing Mater just as he passes Chick for the third time, count this as passing Mater)? [i]Proposed by Matthew Weiss

Find the locus of the intersection points of the medians all triangles inscribed in a given circle.

$ABCD$ is a parallelogram. A point $M$ is found on the side $AB$ or its extension such that $\angle MAD = \angle AMO$ where $O$ is the intersection point of the diagonals of the parallelogram. Prove that $MD = MG$. (M Smurov)

In how many ways can the numbers from $2$ to $2022$ be arranged so that the first number is a multiple of $1$, the second number is a multiple of $2$, the third number is a multiple of $3$, and so on untile the last number is a multiple of $2021$?