Let $ABCD$ be a unit square. A semicircle with diameter $AB$ is drawn so that it lies outside of the square. If $E$ is the midpoint of arc $AB$ of the semicircle, what is the area of triangle $CDE$

Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$ and are strictly monotone in $(0,+\infty )$

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]

Can the squares of a $1990 \times 1990$ chessboard be colored black or white so that half the squares in each row and column are black and cells symmetric with respect to the center are of opposite color?

Let $a,b,c$ be 3 integers, $b$ odd, and define the sequence $\{x_n\}_{n\geq 0}$ by $x_0=4$, $x_1=0$, $x_2=2c$, $x_3=3b$ and for all positive integers $n$ we have \[ x_{n+3} = ax_{n-1}+bx_n + cx_{n+1} . \] Prove that for all positive integers $m$, and for all primes $p$ the number $x_{p^m}$ is divisible by $p$.

Rosa and Sara play with a triangle $ABC$, right at $B$. Rosa begins by marking two interior points of the hypotenuse $AC$, then Sara marks an interior point of the hypotenuse $AC$ different from those of Rosa. Then, from these three points the perpendiculars to the sides $AB$ and $BC$ are drawn, forming the following figure. [img]https://cdn.artofproblemsolving.com/attachments/9/9/c964bbacc4a5960bee170865cc43902410e504.png[/img] Sara wins if the area of the shaded surface is equal to the area of the unshaded surface, in other case wins Rosa. Determine who of the two has a winning strategy.

Find the units digit in the decimal expansion of \[\left(2008+\sqrt{4032000}\right)^{2000}+\left(2008+\sqrt{4032000}\right)^{2001}+\left(2008+\sqrt{4032000}\right)^{2002}+\]\[\cdots+\left(2008+\sqrt{4032000}\right)^{2007}+\left(2008+\sqrt{4032000}\right)^{2008}.\]

Consider a sequence $1^{0.01},\ 2^{0.02},\ 2^{0.02},\ 3^{0.03},\ 3^{0.03},\ 3^{0.03},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ \cdots$. (1) Find the 36th term. (2) Find $\int x^2\ln x\ dx$. (3) Let $A$ be the product of from the first term to the 36th term. How many digits does $A$ have integer part? If necessary, you may use the fact $2.0<\ln 8<2.1,\ 2.1<\ln 9<2.2,\ 2.30<\ln 10<2.31$. [i]2010 National Defense Medical College Entrance Exam, Problem 4[/i]

$\dfrac{10-9+8-7+6-5+4-3+2-1}{1-2+3-4+5-6+7-8+9}=$ $\text{(A)}\ -1 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 10$

Let $C$ be a finite set of chords in a circle such that each chord passes through the midpoint of some other chord. Prove that any two of these chords intersect inside the circle.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have: $$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$

Let $n>1$ be an integer. Find all $r\in \mathbb{R}$ so that the system of equations in real variables $x_1, x_2, \dots, x_n$: \begin{align*} &(r\cdot x_1-x_2)(r\cdot x_1-x_3)\dots (r\cdot x_1-x_n)=\\ =&(r\cdot x_2-x_1)(r\cdot x_2-x_3)\dots (r\cdot x_2-x_n)=\\ &\qquad \qquad \qquad \qquad \vdots \\ =&(r\cdot x_n-x_1)(r\cdot x_n-x_2)\dots (r\cdot x_n-x_{n-1}) \end{align*} has a solution where the numbers $x_1, x_2, \dots, x_n$ are pairwise distinct.

One hundred billion light years from Earth is planet Glorp. The inhabitants of Glorp are intelligent, uniform, amorphous beings with constant density which can modify their shape in any way, and reproduce by splitting. Suppose a Glorpian has somehow formed itself into a spinning cylinder in a frictionless environment. It then splits itself into two Glorpians of equal mass, which proceed to mold themselves into cylinders of the same height, but not the same radius, as the original Glorpian. If the new Glorpians' angular velocities after this are equal and the angular velocity of the original Glorpian is $\omega$, let the angular velocity of the each of the new Glorpians be $\omega'$. Then, find $ \left( \frac {\omega'}{\omega} \right)^{10} $. [i](B. Dejean, 3 points)[/i]

The inscribed circle $\omega$ of the triangle $ABC$ touches its sides $BC, CA$, and $AB$ at the points $D, E$, and $F$, respectively. Let the points $X, Y$, and $Z$ of the circle $\omega$ be diametrically opposite to the points $D, E$, and $F$, respectively. Line $AX, BY$ and $CZ$ intersect the sides $BC, CA$ and $AB$ at the points $D', E'$ and $F'$, respectively. On the segments $AD', BE'$ and $CF'$ noted the points $X', Y'$ and $Z'$, respectively, so that $D'X'= AX$, $E'Y' = BY$, $F'Z' = CZ$. Prove that the points $X', Y'$ and $Z'$ coincide.

A point $ (x,y)$ in the plane is called a lattice point if both $ x$ and $ y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to $ \textbf{(A)}\ 4.0\qquad \textbf{(B)}\ 4.2\qquad \textbf{(C)}\ 4.5\qquad \textbf{(D)}\ 5.0\qquad \textbf{(E)}\ 5.6$

For each $n \ge 2$ define the polynomial $$f_n(x)=x^n-x^{n-1}-\dots-x-1.$$ Prove that (a) For each $n \ge 2$, $f_n(x)=0$ has a unique positive real root $\alpha_n$; (b) $(\alpha_n)_n$ is a strictly increasing sequence; (c) $\lim_{n \rightarrow \infty} \alpha_n=2.$

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

If $ \theta$ is a constant such that $ 0 < \theta < \pi$ and $ x \plus{} \frac{1}{x} \equal{} 2\cos{\theta}$. then for each positive integer $ n$, $ x^n \plus{} \frac{1}{x^n}$ equals $ \textbf{(A)}\ 2\cos{\theta}\qquad \textbf{(B)}\ 2^n\cos{\theta}\qquad \textbf{(C)}\ 2\cos^n{\theta}\qquad \textbf{(D)}\ 2\cos{n\theta}\qquad \textbf{(E)}\ 2^n\cos^n{\theta}$

There are $ 1993$ towns in a country, and at least $ 93$ roads going out of each town. It's known that every town can be reached from any other town. Prove that this can always be done with no more than $ 62$ transfers.

Give the answer about the following propositions $(p),\ (q)$ whether they are true or not. If the answer is true, then give the proof and if the answer is false, then give the proof by giving the counter example. $(p)$ If we can form a triangle such that one of inner angles of the triangle is $60^\circ$ by choosing 3 points from the vertices of a regular $n$-polygon, then $n$ is a multiple of 3. $(q)$ In $\triangle{ABC},\ \triangle{A'B'C'}$, if $AB=A'B',\ BC=B'C',\ \angle{A}=\angle{A'}$, then these triangles are congruent. 30 points

On the plane circles $k$ and $\ell$ are intersected at points $C$ and $D$, where circle $k$ passes through the center $L$ of circle $\ell$. The straight line passing through point $D$ intersects circles $k$ and $\ell$ for the second time at points $A$ and $B$ respectively in such a way that $D$ is the interior point of segment $AB$. Show that $AB = AC$.

[b]Randy:[/b] "Hi Rachel, that's an interesting quadratic equation you have written down. What are its roots?'' [b]Rachel:[/b] "The roots are two positive integers. One of the roots is my age, and the other root is the age of my younger brother, Jimmy.'' [b]Randy:[/b] "That is very neat! Let me see if I can figure out how old you and Jimmy are. That shouldn't be too difficult since all of your coefficients are integers. By the way, I notice that the sum of the three coefficients is a prime number.'' [b]Rachel:[/b] "Interesting. Now figure out how old I am.'' [b]Randy:[/b] "Instead, I will guess your age and substitute it for $x$ in your quadratic equation $\dots$ darn, that gives me $-55$, and not $0$.'' [b]Rachel:[/b] "Oh, leave me alone!'' (1) Prove that Jimmy is two years old. (2) Determine Rachel's age.

Let $p$ be a prime number. How many solutions does the congruence $x^2+y^2+z^2+1\equiv 0\pmod{p}$ have among the modulo $p$ remainder classes? [i]Proposed by: Zoltán Gyenes, Budapest[/i]

Find all quadruplets of positive integers $(m,n,k,l)$ such that $3^m = 2^k + 7^n$ and $m^k = 1 + k + k^2 + k^3 + \ldots + k^l$.

How many pairs of integers $x, y$ are there between $1$ and $1000$ such that $x^2 + y^2$ is divisible by $7$?