Prove that $\frac{a}{(a + 1)(b + 1)} +\frac{ b}{(b + 1)(c + 1)} + \frac{c}{(c + 1)(a + 1)} \ge \frac34$ where $a, b$ and $c$ are positive real numbers satisfying $abc = 1$.

Let real numbers $ a,b,c$. Prove that $ \sqrt{2(a^2\plus{}b^2)}\plus{}\sqrt{2(b^2\plus{}c^2)}\plus{}\sqrt{2(c^2\plus{}a^2)}\ge \sqrt{3(a\plus{}b)^2\plus{}3(b\plus{}c)^2\plus{}3(c\plus{}a)^2}$.

Let $n$ and $t$ be positive integers. What is the number of ways to place $t$ dominoes $(1\times 2$ or $2\times 1$ rectangles) in a $2\times n$ table so that there is no $2\times 2$ square formed by $2$ dominoes and each $2\times 3$ rectangle either does not have a horizontal domino in the middle and last cell in the first row or does not have a horizontal domino in the first and middle cell in the second row (or both)?

Find all $n \in \mathbb{N}$ such that $ \lfloor \sqrt{n}\rfloor$ divides $n$.

Prove that for any integer $k > 2$, there exist infinitely many positive integers $n$ such that the least common multiple of $n$, $n + 1$,$...$, $n + k - 1$ is greater than the least common multiple of $n + 1$,$n + 2$,$...$, $n + k$.

On a chalkboard, Benji draws a square with side length $6.$ He then splits each side into $3$ equal segments using $2$ points for a total of $12$ segments and $8$ points. After trying some shapes, Benji finds that by using a circle, he can connect all $8$ points together. What is the area of this circle? [center] [img] https://cdn.artofproblemsolving.com/attachments/d/f/ead2d0c85f474612d49d3a023a43215e11066b.png [/img] [/center]

Al has the cards $1,2,\dots,10$ in a row in increasing order. He first chooses the cards labeled $1$, $2$, and $3$, and rearranges them among their positions in the row in one of six ways (he can leave the positions unchanged). He then chooses the cards labeled $2$, $3$, and $4$, and rearranges them among their positions in the row in one of six ways. (For example, his first move could have made the sequence $3,2,1,4,5,\dots,$ and his second move could have rearranged that to $2,4,1,3,5,\dots$.) He continues this process until he has rearranged the cards with labels $8$, $9$, $10$. Determine the number of possible orderings of cards he can end up with. [i]Proposed by Ray Li[/i]

What is the hundreds digit of $2011^{2011}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9 $

Find all pairs $(x,y)$ of real numbers such that \[16^{x^{2}+y} + 16^{x+y^{2}} = 1\]

Consider the geometric sequence $1/2, \ 1 / 4, \ 1 / 8,...$ Can one choose a subsequence, finite or infinite, for which the ratio of consecutive terms is not $1$ and whose sum is $1/5?$

The diagonals of a convex quadrilateral $ABCD$ meet at point $L$. The orthocenter $H$ of the triangle $LAB$ and the circumcenters $O_1, O_2$, and $O_3$ of the triangles $LBC, LCD$, and $LDA$ were marked. Then the whole configuration except for points $H, O_1, O_2$, and $O_3$ was erased. Restore it using a compass and a ruler.

For an even positive integer $n$ Kevin has a tape of length $4n$ with marks at $-2n,-2n+1,\ldots,2n-1,2n$. He then randomly picks $n$ points in the set $-n,-n+1,-n+2,\ldots,n-1,n$ and places a stone on each of these points. We call a stone 'stuck' if it is on $2n$ or $-2n$, or either all the points to the right, or all the points to the left, all contain stones. Then, every minute, Kevin shifts the unstruck stones in the following manner: [list] [*]He picks an unstuck stone uniformly at random and then flips a fair coin. [*]If the coin came up heads, he then moves that stone and every stone in the largest contiguous set containing that stone one point to the left. If the coin came up tails, he moves every stone in that set one point right instead. [*]He repeats until all the stones are stuck.[/list] Let $p_n$ be the probability that at the end of the process there are exactly $k$ stones in the right half. Evaluate \[\dfrac{p_{n-1}-p_{n-2}+p_{n-3}+\ldots+p_3-p_2+p_1}{p_{n-1}+p_{n-2}+p_{n-3}+\ldots+p_3+p_2+p_1}\] in terms of $n$.

Find all triples $(a, b, c)$ of positive integers such that $a^2 + b^2 = n\cdot lcm(a, b) + n^2$ $b^2 + c^2 = n \cdot lcm(b, c) + n^2$ $c^2 + a^2 = n \cdot lcm(c, a) + n^2$ for some positive integer $n$.

Evaluate: $\sum^{\infty}_{k=1} \tfrac{k}{a^{k-1}}$ for all $|a|<1$.

Circles with radii $R$ and $r$ ($R > r$) are externally tangent. Another common tangent of the circles in drawn. This tangent and the circles bound a region inside which a circle as large as possible is drawn. What is the radius of this circle?

Positive integers $a, b$ and $c$ are all less than $2020$. We know that $a$ divides $b + c$, $b$ divides $a + c$ and $c$ divides $a + b$. How many such ordered triples $(a, b, c)$ are there? Note: In an ordered triple, the order of the numbers matters, so the ordered triple $(0, 1, 2)$ is not the same as the ordered triple $(2, 0, 1)$.

The sides of an equilateral triangle $ABC$ are divided into $n$ equal parts $(n \geq 2) .$ For each point on a side, we draw the lines parallel to other sides of the triangle $ABC,$ e.g. for $n=3$ we have the following diagram: [asy] unitsize(150); defaultpen(linewidth(0.7)); int n = 3; /* # of vertical lines, including AB */ pair A = (0,0), B = dir(-30), C = dir(30); draw(A--B--C--cycle,linewidth(2)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0)); label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE); for(int i = 1; i < n; ++i) { draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n); draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n); draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n); } [/asy] For each $n \geq 2,$ find the number of existing parallelograms.

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a\plus{}b\plus{}c\equal{}3$ . Prove that : $ \frac{1}{2\plus{}a^{2}\plus{}b^{2}}\plus{}\frac{1}{2\plus{}b^{2}\plus{}c^{2}}\plus{}\frac{1}{2\plus{}c^{2}\plus{}a^{2}} \leq \frac{3}{4}$

$ABCD$ is a convex quadrilateral. $K$, $L$, $M$, $N$ are the midpoints of the sides $AB$, $BC$, $CD$, $DA$. $BD$ bisects $KM$ at $Q$. $QA = QB = QC = QD$ , and$\frac{LK}{LM} = \frac{CD}{CB}$. Prove that $ABCD$ is a square

Determine all positive integers $n$ for which $2^{n+1}-n^2$ is a prime number.

Show that the expression $$\frac{n^5 -5n^3 + 4n}{n + 2}$$ where n is any integer, it is always divisible by $24$.

Let $ABC$ be a triangle with circumcircle $(O)$. Let $M,N$ be the midpoint of arc $AB,AC$ which does not contain $C,B$ and let $M',N'$ be the point of tangency of incircle of $\triangle ABC$ with $AB,AC$. Suppose that $X,Y$ are foot of perpendicular of $A$ to $MM',NN'$. If $I$ is the incenter of $\triangle ABC$ then prove that quadrilateral $AXIY$ is cyclic if and only if $b+c=2a$.

Suppose that necklace $\, A \,$ has 14 beads and necklace $\, B \,$ has 19. Prove that for any odd integer $n \geq 1$, there is a way to number each of the 33 beads with an integer from the sequence \[ \{ n, n+1, n+2, \dots, n+32 \} \] so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a ``necklace'' is viewed as a circle in which each bead is adjacent to two other beads.)

Prove the equality $$\prod_{k=1}^{2m}\cos\frac{k\pi}{2m+1}=\frac{(-1)^m}{4m}.$$