Ted's favorite number is equal to \[1\cdot\binom{2007}1+2\cdot\binom{2007}2+3\cdot\binom{2007}3+\cdots+2007\cdot\binom{2007}{2007}.\] Find the remainder when Ted's favorite number is divided by $25$. $\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l} \textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\ \textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\ \textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\ \textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\ \textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\ \textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\ \textbf{(S) }18&\textbf{(T) }19&\textbf{(U) }20\\\\ \textbf{(V) }21&\textbf{(W) }22 & \textbf{(X) }23\\\\ \textbf{(Y) }24 \end{array}$

There is a regular $2021$-gon. We put a coin with heads up on every vertex of it. Every time, you can choose one vertex, and flip the coin on the vertices adjacent to it. Can you make all the coin tails up?

Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.

Every cell of table $4 \times 4$ is colored into white. It is permitted to place the cross (pictured below) on the table such that its center lies on the table (the whole figure does not need to lie on the table) and change colors of every cell which is covered into opposite (white and black). Find all $n$ such that after $n$ steps it is possible to get the table with every cell colored black.

Does there exist a solution $(x,y,z,u,v)$ in integers greater than $1998$ to the equation $x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65$?

Let $n$ a positive integer, $n > 1$. The number $n$ is wonderful if the number is divisible by sum of the your prime factors. For example; $90$ is wondeful, because $90 = 2 \times 3^2\times 5$ and $2 + 3 + 5 = 10, 10$ divides $90$. Show that, exist a number "wonderful" with at least $10^{2002}$ distinct prime numbers.

In a school with 5 different grades there are 250 girls and 250 boys. Each grade has the same number of students. for a competition of knowledge wants to form teams of a female and a male who are of the same grade. If in each grade there are at least $19$ females and $19$ males. Find the greatest amount of teams that can be formed.

Let $a, b, c$ be positive real numbers such that $ab +be + ba \le 3abc$. Prove that $$a^3+b^3+c^3 \ge a+b+c.$$

Let $D\subseteq \mathbb{C}$ be a compact set with at least two elements and consider the space $\Omega=\bigtimes_{i=1}^{\infty} D$ with the product topology. For any sequence $(d_n)_{n=0}^{\infty} \in \Omega$ let $f_{(d_n)}(z)=\sum_{n=0}^{\infty}d_nz^n$, and for each point $\zeta \in \mathbb{C}$ with $|\zeta|=1$ we define $S=S(\zeta,(d_n))$ to be the set of complex numbers $w$ for which there exists a sequence $(z_k)$ such that $|z_k|<1$, $z_k \to \zeta$, and $f_{d_n}(z_k) \to w$. Prove that on a residual set of $\Omega$, the set $S$ does not depend on the choice of $\zeta$.

If $ a,b,c>0$ and $ a\plus{}b\plus{}c\equal{}3$ prove that: $ \frac{{a^3 \left( {a \plus{} b} \right)}}{{a^2 \plus{} ab \plus{} b^2 }} \plus{} \frac{{b^3 \left( {b \plus{} c} \right)}}{{b^2 \plus{} bc \plus{} c^2 }} \plus{} \frac{{c^3 \left( {c \plus{} a} \right)}}{{c^2 \plus{} ac \plus{} a^2 }} \ge 2$ P.s. I dont know if it has been posted before.

On the sides of triangle $ABC$, isosceles right-angled triangles $AUB, CVB$, and $AWC$ are placed. These three triangles have their right angles at vertices $U, V$ , and $W$, respectively. Triangle $AUB$ lies completely inside triangle $ABC$ and triangles $CVB$ and $AWC$ lie completely outside $ABC$. See the figure. Prove that quadrilateral $UVCW$ is a parallelogram. [asy] import markers; unitsize(1.5 cm); pair A, B, C, U, V, W; A = (0,0); B = (2,0); C = (1.7,2.5); U = (B + rotate(90,A)*(B))/2; V = (B + rotate(90,C)*(B))/2; W = (C + rotate(90,A)*(C))/2; draw(A--B--C--cycle); draw(A--W, StickIntervalMarker(1,1,size=2mm)); draw(C--W, StickIntervalMarker(1,1,size=2mm)); draw(B--V, StickIntervalMarker(1,2,size=2mm)); draw(C--V, StickIntervalMarker(1,2,size=2mm)); draw(A--U, StickIntervalMarker(1,3,size=2mm)); draw(B--U, StickIntervalMarker(1,3,size=2mm)); draw(rightanglemark(A,U,B,5)); draw(rightanglemark(B,V,C,5)); draw(rightanglemark(A,W,C,5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, N); dot("$U$", U, NE); dot("$V$", V, NE); dot("$W$", W, NW); [/asy]

Consider three fixed spheres $S_1, S_2, S_3$ with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each $S_i$ along a great circle of $S_i$. [i]Stere Ianuș[/i]

Find the four smallest four-digit numbers that meet the following condition: by dividing by $2$, $3$, $4$, $5$ or $6$ the remainder is $ 1$.

Solve the equation, where $x$ and $y$ are positive integers: $$ x^3-y^3=999$$

The expression $1\oplus2\oplus3\oplus4\oplus5\oplus6\oplus7\oplus8\oplus9$ is written on a blackboard. Bill and Peter play the following game. They replace $\oplus$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning, Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win. [hide=Original Wording]The expression $1*2*3*4*5*6*7*8*9$ is written on a blackboard. Bill and Peter play the following game. They replace $*$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win.[/hide]

A circle $K_1$ of radius $r_1 = 1\slash 2$ is inscribed in a semi-circle $H$ with diameter $AB$ and radius $1.$ A sequence of different circles $K_2, K_3, \ldots$ with radii $r_2, r_3, \ldots$ respectively are drawn so that for each $n\geq 1$, the circle $K_{n+1}$ is tangent to $H$, $K_n$ and $AB.$ Prove that $a_n = 1\slash r_n$ is an integer for each $n$, and that it is a perfect square for $n$ even and twice a perfect square for $n$ odd.

Let $x_1,x_2,\ldots, x_{10}$ be non-negative real numbers such that $\frac{x_1}{1}+ \frac{x_2}{2} +\cdots+ \frac{x_{10}}{10}$ $\leq9$. Find the maximum possible value of $\frac{{x_1}^2}{1}+\frac{{x_2}^2}{2}+\cdots+\frac{{x_{10}}^2}{10}$.

Sam writes three $3$-digit positive integers (that don't end in $0$) on the board and adds them together. Jessica reverses each of the integers, and adds the reversals together. (For example, $\overline{XYZ}$ becomes $\overline{ZYX}$.) What is the smallest possible positive three-digit difference between Sam's sum and Jessica's sum?

Let $ABC$ be an acute triangle and $\Gamma$ be its circumcircle. Line $\ell$ is tangent to $\Gamma$ at $A$ and let $D$ and $E$ be distinct points on $\ell$ such that $AD = AE$. Suppose that $B$ and $D$ lie on the same side of line $AC$. The circumcircle $\Omega_1$ of $\vartriangle ABD$ meets $AC$ again at $F$. The circumcircle $\Omega_2$ of $\vartriangle ACE$ meets $AB$ again at $G$. The common chord of $\Omega_1$ and $\Omega_2$ meets $\Gamma$ again at $H$. Let $K$ be the reflection of $H$ across line $BC$ and let $L$ be the intersection of $BF$ and $CG$. Prove that $A, K$ and $L$ are collinear.

Let $ABCD$ be a trapezoid with bases $AD$ and $BC$, and let $M$ be the midpoint of $CD$. The circumcircle of triangle $BCM$ meets $AC$ and $BD$ again at $E$ and $F$, with $E$ and $F$ distinct, and line $EF$ meets the circumcircle of triangle $AEM$ again at $P$. Prove that $CP$ is parallel to $BD$. [i]Proposed by Ariel García[/i]

Find positive integers $a$, $b$, and $c$ such that \[\sqrt{a}+\sqrt{b}+\sqrt{c}=\sqrt{219+\sqrt{10080}+\sqrt{12600}+\sqrt{35280}}.\] Prove that your solution is correct. (Warning: numerical approximations of the values do not constitute a proof.)

find largest constant C st $\sum_{i=1}^{4} (x_i+1/x_i)^3 \geq C$ for all positive real numbers $x_1,..,x_4$ st $x_1^3+x_3^3+3x_1x_3=x_2+x_4=1$

A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a 76 game schedule. How many games does a team play within its own division? $ \textbf{(A) } 36 \qquad \textbf{(B) } 48 \qquad \textbf{(C) } 54 \qquad \textbf{(D) } 60 \qquad \textbf{(E) } 72 $

Find all triples $(a, b, c)$ of natural numbers such that the sets $$\{ gcd (a, b), gcd(b, c), gcd(c, a), lcm (a, b), lcm (b, c), lcm (c, a)\}$$ and $$\{2, 3, 5, 30, 60\}$$ are the same. Remark: For example, the sets $\{1, 2013\}$ and $\{1, 1, 2013\}$ are equal.

Express $\sqrt{2 + \sqrt{3}}$ in the form $\frac{a + \sqrt{b}}{\sqrt{c}}$, where $a$ is a positive integer and $b$ and $c$ are square-free positive integers.