$ A$ and $ B$ together can do a job in $ 2$ days; $ B$ and $ C$ can do it in four days; and $ A$ and $ C$ in $ 2\frac{2}{5}$ days. The number of days required for $ A$ to do the job alone is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 2.8$

A large number of rocks are placed on a table. On each turn, one may remove some rocks from the table following these rules: on the first turn, only one rock may be removed, and on every subsequent turn, one may remove either twice as many rocks or the same number of rocks as they have discarded on the previous turn. Determine the minimum number of turns required to remove exactly $2012$ rocks from the table.

As a gift, Dilhan was given the number $n=1^1\cdot2^2\cdots2021^{2021}$, and each day, he has been dividing $n$ by $2021!$ exactly once. One day, when he did this, he discovered that, for the first time, $n$ was no longer an integer, but instead a reduced fraction of the form $\frac{a}b$. What is the sum of all distinct prime factors of $b$? [i]Proposed by Adam Bertelli[/i]

There are $ n$ students; each student knows exactly $d $ girl students and $d $ boy students ("knowing" is a symmetric relation). Find all pairs $ (n,d) $ of integers .

Let $P(x)$ be a polynomial with integer coefficients that has at least one rational root. Let $n$ be a positive integer. Alan and Allan are playing a game. First, Alan writes down $n$ integers at $n$ different locations on a board. Then Allan may make moves of the following kind: choose a position that has integer $a$ written, then choose a different position that has integer $b$ written, then at the first position erase $a$ and in its place write $a+P(b)$. After any nonnegative number of moves, Allan may choose to end the game. Once Allan ends the game, his score is the number of times the mode (most common element) of the integers on the board appears. Find, in terms of $P(x)$ and $n$, the maximum score Allan can guarantee. [i]Henrick Rabinovitz[/i]

An $n \times m$ matrix is nice if it contains every integer from $1$ to $mn$ exactly once and $1$ is the only entry which is the smallest both in its row and in its column. Prove that the number of $n \times m$ nice matrices is $(nm)!n!m!/(n+m-1)!$.

Let a point $P$ lie inside a triangle $ABC$. The rays starting at $P$ and crossing the sides $BC$, $AC$, $AB$ under the right angle meet the circumcircle of $ABC$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. It is known that lines $AA_{1}$, $BB_{1}$, $CC_{1}$ concur at point $Q$. Prove that all such lines $PQ$ concur.

$A$ and $B$ lie on a circle. $P$ lies on the minor arc $AB$. $Q$ and $R$ (distinct from $P$) also lie on the circle, so that $P$ and $Q$ are equidistant from $A$, and $P$ and $R$ are equidistant from $B$. Show that the intersection of $AR$ and $BQ$ is the reflection of $P$ in $AB$.

In a convex quadrilateral $ABCD$, the lines $AB$ and $CD$ meet at point $E$ and the lines $AD$ and $BC$ meet at point $F$. Let $P$ be the intersection point of diagonals $AC$ and $BD$. Suppose that $\omega_1$ is a circle passing through $D$ and tangent to $AC$ at $P$. Also suppose that $\omega_2$ is a circle passing through $C$ and tangent to $BD$ at $P$. Let $X$ be the intersection point of $\omega_1$ and $AD$, and $Y$ be the intersection point of $\omega_2$ and $BC$. Suppose that the circles $\omega_1$ and $\omega_2$ intersect each other in $Q$ for the second time. Prove that the perpendicular from $P$ to the line $EF$ passes through the circumcenter of triangle $XQY$ . Proposed by Iman Maghsoudi

In a $100\times25$ rectangle table, fill in a positive real number in each blank. Let the number in the $i$th line, the $j$th column be $x_{i,j}(i=1,2,\cdots,100,j=1,2,\cdots,25)$ (shown in Fig.1 ). Then, we rearrange the numbers in each column: $x'_{1,j}\geq x'_{2,j}\geq\cdots\geq x'_{100,j}(j=1,2,\cdots,25)$ (shown in Fig.2 ). Find the minumum value of $k$, satisfying: As long as $\sum_{j=1}^{25}x_{i,j}\leq1$ for numbers in Fig.1 ($i=1,2,\cdots,100$), then $\sum_{j=1}^{25}x'_{i,j}\leq1$ for $i\geq k$ in Fig.2. $$\textbf{Fig.1}\\ \begin{tabular}{|c|c|c|c|} \hline $x_{1,1}$&$x_{1,2}$&$\cdots$&$x_{1,25}$\\ \hline $x_{2,1}$&$x_{2,2}$&$\cdots$&$x_{2,25}$\\ \hline $\cdots$&$\cdots$&$\cdots$&$\cdots$\\ \hline $x_{100,1}$&$x_{100,2}$&$\cdots$&$x_{100,25}$\\ \hline \end{tabular} \qquad\textbf{Fig.2}\\ \begin{tabular}{|c|c|c|c|} \hline $x'_{1,1}$&$x'_{1,2}$&$\cdots$&$x'_{1,25}$\\ \hline $x'_{2,1}$&$x'_{2,2}$&$\cdots$&$x'_{2,25}$\\ \hline $\cdots$&$\cdots$&$\cdots$&$\cdots$\\ \hline $x'_{100,1}$&$x'_{100,2}$&$\cdots$&$x'_{100,25}$\\ \hline \end{tabular}$$

Solve the equation $x^3 - [x] = 3$.

The circuit diagram drawn (see figure ) contains a battery $B$, a lamp $L$ and five switches $S_1$ to $S_5$. The probability that switch $S_3$ is closed (makes contact) is $\frac23$, for the other four switches that probability is $\frac12$ (the probabilities are mutually independent). Calculate the probability that the light is on. [asy] unitsize (2 cm); draw((-1,1)--(-0.5,1)); draw((-0.25,1)--(1,1)--(1,0.25)); draw((1,-0.25)--(1,-1)--(0.05,-1)); draw((-0.05,-1)--(-1,-1)--(-1,0.25)); draw((-1,0.5)--(-1,1)); draw((-1,1)--(-0.5,0.5)); draw((-0.25,0.25)--(0,0)); draw((-1,0)--(-0.75,0)); draw((-0.5,0)--(0,0)); draw((0,1)--(0,0.75)); draw((0,0.5)--(0,0)); draw((-0.25,1)--(-0.5,1.25)); draw((-1,0.25)--(-1.25,0.5)); draw((-0.5,0.5)--(-0.25,0.5)); draw((0,0.75)--(0.25,0.5)); draw((-0.75,0)--(-0.5,-0.25)); draw(Circle((1,0),0.25)); draw(((1,0) + 0.25*dir(45))--((1,0) + 0.25*dir(225))); draw(((1,0) + 0.25*dir(135))--((1,0) + 0.25*dir(315))); draw((0.05,-0.9)--(0.05,-1.1)); draw((-0.05,-0.8)--(-0.05,-1.2)); label("$L$", (1.25,0), E); label("$B$", (-0.1,-1.1), SW); label("$S_1$", (-0.5,1.25), NE); label("$S_2$", (-1.25,0.5), SW); label("$S_3$", (-0.5,0.5), SW); label("$S_4$", (0.25,0.5), NE); label("$S_5$", (-0.5,-0.25), SW); [/asy]

We consider the inscriptible pentagon $ABCDE$ in which $AB=BC=CD$ and the centroid of the pentagon coincides with the circumcenter. Prove that the pentagon $ABCDE$ is regular. [i]The centroid of a pentagon is the point in the plane of the pentagon whose position vector is equal to the average of the position vectors of the vertices.[/i]

The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if $a$ is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than." The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that: $ \textbf{(A)}\ x^2<49 \qquad\textbf{(B)}\ x^2>1 \qquad\textbf{(C)}\ 1<x^2<49\qquad\textbf{(D)}\ -1<x<7\qquad\textbf{(E)}\ -7<x<1 $

The numbers from$ 1$ to $2016$ are divided into three (disjoint) subsets $A, B$ and $C$, each one contains exactly $672$ numbers. Prove that you can find three numbers, each from a different subset, such that the sum of two of them is equal to the third. [hide=original wording]Skaitļi no 1 līdz 2016 ir sadalīti trīs (nešķeļošās) apakškopās A, B un C, katranotām satur tieši 672 skaitļus. Pierādīt, ka var atrast trīs tādus skaitļus, katru no citas apakškopas, ka divu no tiem summa ir vienāda ar trešo. [/hide]

Find all integers that can be written in the form $\frac{(x+y+z)^2}{xyz}$, where $x,y,z$ are positive integers.

Find all functions $f : (0, +\infty) \to (0, +\infty)$ which are increasing on $[1, +\infty)$ and for all positive reals $a, b, c$ they fulfill the following relation $f(ab)f(bc)f(ca)=f(a^2b^2c^2)+f(a^2)+f(b^2)+f(c^2)$.

The twenty-first century began on January $1$, $2001$ and runs through December $31$, $2100$. Note that March $1$, $2014$ fell on Saturday, so there were five Mondays in March $2014$. In how many years of the twenty-first century does March have five Mondays?

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Across all $x \in \mathbb{R}$, find the maximum value of the expression $$\sin x + \sin 3x + \sin 5x.$$

Consider the sequence formed by the first digits of the powers of $5$:$$1,5,2,1,6,...$$ Prove any segment in this sequence, when written in reversed order, will be encountered in the sequence of the first digits of the powers of $2:$ $$1,2,4,8,1,3,6,1...$$

Given the nine-sided regular polygon $ A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $ \{A_1,A_2,...A_9\}$? $ \textbf{(A)} \ 30 \qquad \textbf{(B)} \ 36 \qquad \textbf{(C)} \ 63 \qquad \textbf{(D)} \ 66 \qquad \textbf{(E)} \ 72$

Let $ABC$ be a triangle with $\angle BAC = 75^o$ and $\angle ABC = 45^o$. If $BC =\sqrt3 + 1$, what is the perimeter of $\vartriangle ABC$?

How many different numbers with $6$ digits and multiples of $45$ can be written by adding one digit to the left and one to the right of $2008$?

In trapezoid $ABCD$, $AD$ is parallel to $BC$. Knowing that $AB=AD+BC$, prove that the bisector of $\angle A$ also bisects $CD$.