Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.

Let $n\geqslant 2$ be a fixed positive integer. Determine the minimum value of the expression $$\frac{a_{a_1}}{a_1}+\frac{a_{a_2}}{a_2}+\dots +\frac{a_{a_n}}{a_n},$$ where $a_1, a_2, \dots, a_n$ are positive integers at most $n$. [i]Proposed by David Anghel[/i]

Three circles $C_1,C_2,C_3$, with radii $1, 2, 3$ respectively, are externally tangent. In the area enclosed by these circles, there is a circle $C_4$ which is externally tangent to all three circles. Find the radius of $C_4$. [asy] unitsize(0.4 cm); pair[] O; real[] r; O[1] = (-12/5,16/5); r[1] = 1; O[2] = (0,5); r[2] = 2; O[3] = (0,0); r[3] = 3; O[4] = (-132/115, 351/115); r[4] = 6/23; draw(Circle(O[1],r[1])); draw(Circle(O[2],r[2])); draw(Circle(O[3],r[3])); draw(Circle(O[4],r[4])); label("$C_1$", O[1]); label("$C_2$", O[2]); label("$C_3$", O[3]); [/asy]

Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .

In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$. Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$. Determine $AB$

If $a(x), b(x), c(x)$ and $d(x)$ are polynomials in $ x$, show that $$ \int_{1}^{x} a(x) c(x)\; dx\; \cdot \int_{1}^{x} b(x) d(x) \; dx - \int_{1}^{x} a(x) d(x)\; dx\; \cdot \int_{1}^{x} b(x) c(x)\; dx$$ is divisible by $(x-1)^4.$

Alice and Brian are playing a game on a $1\times (N + 2)$ board. To start the game, Alice places a checker on any of the $N$ interior squares. In each move, Brian chooses a positive integer $n$. Alice must move the checker to the $n$-th square on the left or the right of its current position. If the checker moves off the board, Alice wins. If it lands on either of the end squares, Brian wins. If it lands on another interior square, the game proceeds to the next move. For which values of $N$ does Brian have a strategy which allows him to win the game in a finite number of moves?

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Snowhite wrote on a piece of paper a whole number greater than $1$ and multiplied it by itself. She obtained a number, all digits of which are $1$: $n^2 = 111...111$ Does she know how to multiply? [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a bishop on an arbitrary square. Then the second player can put another bishop on a free square that is not controlled by the first bishop. Then the first player can put a new bishop on a free square that is not controlled by the bishops on the board. Then the second player can do the same, etc. A player who cannot put a new bishop on the board loses the game. Who has a winning strategy? [b]p4.[/b] Four girls Marry, Jill, Ann and Susan participated in the concert. They sang songs. Every song was performed by three girls. Mary sang $8$ songs, more then anybody. Susan sang $5$ songs less then all other girls. How many songs were performed at the concert? [b]p5.[/b] Pinocchio has a $10\times 10$ table of numbers. He took the sums of the numbers in each row and each such sum was positive. Then he took the sum of the numbers in each columns and each such sum was negative. Can you trust Pinocchio's calculations? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In an office, at various times during the day the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. If there are five letters in all, and the boss delivers them in the order $1\ 2\ 3\ 4\ 5$, which of the following could [b]not[/b] be the order in which the secretary types them? $ \textbf{(A)}\ 1\ 2\ 3\ 4\ 5 \qquad\textbf{(B)}\ 2\ 4\ 3\ 5\ 1 \qquad\textbf{(C)}\ 3\ 2\ 4\ 1\ 5 \qquad\textbf{(D)}\ 4\ 5\ 2\ 3\ 1 \\ \qquad\textbf{(E)}\ 5\ 4\ 3\ 2\ 1 $

Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside triangle $ABC$.Lines $AP,BP$ and $CP$ intersect $\omega$ again at points $A_1$, $B_1$ and $C_1$ (other than $A, B, C$), respectively. The tangent lines to $\omega$ at $A_1$ and $B_1$ intersect at $C_2$.The tangent lines to $\omega$ at $B_1$ and $C_1$ intersect at $A_2$. The tangent lines to $\omega$ at $C_1$ and $A_1$ intersect at $B_2$. Prove that the lines $AA_2,BB_2$ and $CC_2$ are concurrent.

A $4\times 8$ grid is divided into $32$ unit squares. There are square tiles of sizes $1 \times 1$, $2 \times 2$, $3 \times 3$ and $4 \times 4$. The goal is to completely cover the grid using exactly $n$ of these tiles. [list=a] [*]Is it possible to do this if $n = 19$? [*]Is it possible to do this if $n = 14$? [*]Is it possible to do this if $n = 7$? [/list] [b]Note:[/b] The tiles cannot overlap or extend beyond the grid.

For every $x \ge -\frac{1}{e}\,$, there is a unique number $W(x) \ge -1$ such that \[ W(x) e^{W(x)} = x. \] The function $W$ is called Lambert's $W$ function. Let $y$ be the unique positive number such that \[ \frac{y}{\log_{2} y} = - \frac{3}{5} \, . \] The value of $y$ is of the form $e^{-W(z \ln 2)}$ for some rational number $z$. What is the value of $z$?

Let $ABC$ be a right triangle with $m(\widehat {C}) = 90^\circ$, and $D$ be its incenter. Let $N$ be the intersection of the line $AD$ and the side $CB$. If $|CA|+|AD|=|CB|$, and $|CN|=2$, then what is $|NB|$?

Find the smallest positive integer $ u$ such that there exists only one positive integer $ a$ and satisfies the inequality \[ 20u < 19a < 21u \ \text{?} \]

$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$

A square of side length $5$ is inscribed in a square of side length $7$. If we construct a grid of $1\times1$ squares for both squares, as shown to the right, then we find that the two grids have $8$ lattice points in common. If we do the same construction by inscribing a square of side length $1489$ in a square of side length $2009$, and construct a grid of $1\times1$ squares in each large square, then how many lattice points will the two grids of $1\times1$ squares have in common? [asy] import graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=11.88,ymin=-4.69,ymax=8.77; pair H_2=(0,3), I_2=(3,7), J_2=(7,4), K_2=(4,0), L_2=(3.01,1.99), M_2=(2.01,4), N_2=(4.01,5.01), O_2=(5.01,3); draw((0,0)--(0,7)); draw((0,7)--(7,7)); draw((7,7)--(7,0)); draw((7,0)--(0,0)); draw((0,6)--(7,6)); draw((0,5)--(7,5)); draw(J_2--(0,4)); draw(H_2--(7,3)); draw((0,2)--(7,2)); draw((0,1)--(7,1)); draw((1,0)--(1,7)); draw((2,7)--(2,0)); draw((3,0)--I_2); draw(K_2--(4,7)); draw((5,0)--(5,7)); draw((6,7)--(6,0)); draw(H_2--I_2); draw(I_2--J_2); draw(J_2--K_2); draw(K_2--H_2); draw(H_2--I_2); draw(I_2--J_2); draw((2.41,6.21)--(6.4,3.2)); draw((5.8,2.4)--(1.81,5.41)); draw((1.2,4.61)--(5.2,1.6)); draw((4.6,0.8)--(0.6,3.8)); draw((3.8,6.4)--(0.8,2.4)); draw((1.61,1.79)--(4.6,5.8)); draw((5.4,5.2)--(2.41,1.19)); draw((3.21,0.59)--(6.2,4.6)); draw((0,7)--(7,7),linewidth(1.2)); draw((7,7)--(7,0),linewidth(1.2)); draw((0,0)--(7,0),linewidth(1.2)); draw((0,7)--(0,0),linewidth(1.2)); dot(H_2,linewidth(4pt)+ds); dot(I_2,linewidth(4pt)+ds); dot(J_2,linewidth(4pt)+ds); dot(K_2,linewidth(4pt)+ds); dot(L_2,linewidth(4pt)+ds); dot(M_2,linewidth(4pt)+ds); dot(N_2,linewidth(4pt)+ds); dot(O_2,linewidth(4pt)+ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]

Choose a point $A$ inside a circle of radius $R$. Construct a pair of perpendicular lines through $A$. Then rotate these lines through the same angle $V$ about $A$. The figure formed inside the circle, as the lines move from their initial to their final position, is in the form of a cross with its centre at $A$. Find the area of this cross. (Problem from Latvia)

The set of solutions of the equation $\log_{10}\left( a^2-15a\right)=2$ consists of $ \textbf{(A)}\ \text{two integers } \qquad\textbf{(B)}\ \text{one integer and one fraction}\qquad$ $\textbf{(C)}\ \text{two irrational numbers }\qquad\textbf{(D)}\ \text{two non-real numbers} \qquad\textbf{(E)}\ \text{no numbers, that is, the empty set} $

Let $ABC$ be a triangle and $D$ be the foot of the altitude from $C$ to $AB$. If $|CH|=|HD|$ where $H$ is the orthocenter, what is $\tan \widehat {A} \cdot \tan \widehat{B}$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \sqrt 2 \qquad\textbf{(C)}\ 3/2 \qquad\textbf{(D)}\ \sqrt 3 \qquad\textbf{(E)}\ \text{None of the preceding} $

Find the maximum value of: $E(a,b)=\frac{a+b}{(4a^2+3)(4b^2+3)}$ For $a,b$ real numbers.

Fill in the grid with the numbers 1 to 6 so that each number appears exactly once in each row and column. A horizontal gray line marks any cell when it is the middle cell of the three consecutive cells with the largest sum in that row. Similarly, a vertical gray line marks any cell when it is the middle of the three consecutive cells with the largest sum in that column. If there is a tie, multiple lines are drawn in the row or column. A cell can have both lines drawn, with the appearance of a plus sign. [asy] // Change this to see the solution bool DRAW_SOLUTION = true; int n = 6; real LINE_WIDTH = 0.3; void drawHLine(int x, int y) { fill((x,y+0.5-LINE_WIDTH/2)--(x,y+0.5+LINE_WIDTH/2)--(x+1,y+0.5+LINE_WIDTH/2)--(x+1,y+0.5-LINE_WIDTH/2)--cycle, gray(0.8)); } void drawVLine(int x, int y) { fill((x+0.5-LINE_WIDTH/2,y)--(x+0.5+LINE_WIDTH/2,y)--(x+0.5+LINE_WIDTH/2,y+1)--(x+0.5-LINE_WIDTH/2,y+1)--cycle, gray(0.8)); } void drawNum(int x, int y, int num) { label(scale(1.5)*string(num), (x+0.5,y+0.5)); } void drawSolNum(int x, int y, int num) { if (DRAW_SOLUTION) { drawNum(x, y, num); } } drawHLine(2,0); drawHLine(4,1); drawHLine(1,2); drawHLine(3,2); drawHLine(4,3); drawHLine(2,4); drawHLine(3,5); drawVLine(0,4); drawVLine(1,3); drawVLine(2,1); drawVLine(2,3); drawVLine(3,4); drawVLine(4,1); drawVLine(5,2); drawNum(0, 0, 5); drawNum(4, 0, 3); drawNum(1, 2, 2); drawNum(3, 3, 4); for(int i = 0; i <= 6; i += 1) { draw((i,0)--(i,6)); draw((0,i)--(6,i)); } [/asy]

Let the set $A=${$ 1,2,3, \dots ,48n+24$ } , where $ n \in \mathbb {N^*}$ . Prove that there exist a subset $B $ of $A $ with $24n+12$ elements with the property : the sum of the squares of the elements of the set $B $ is equal to the sum of the squares of the elements of the set $A$ \ $B $ .

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

The number $2019$ has the following nice properties: (a) It is the sum of the fourth powers of fuve distinct positive integers. (b) It is the sum of six consecutive positive integers. In fact, $2019 = 1^4 + 2^4 + 3^4 + 5^4 + 6^4$ (1) $2019 = 334 + 335 + 336 + 337 + 338 + 339$ (2) Prove that $2019$ is the smallest number that satises [b]both [/b] (a) and (b). (You may assume that (1) and (2) are correct!)

Let $ABCD$ be a quadrilateral and $M$ the midpoint of the segment $AB$. Outside of the quadrilateral are constructed the equilateral triangles $BCE$, $CDF$ and $DAG$. Let $P$ and $N$ be the midpoints of the segments $GF$ and $EF$. Prove that the triangle $MNP$ is equilateral.