In the coordinate plane, given a circle $K: x^2+y^2=1,\ C: y=x^2-2$. Let $l$ be the tangent line of $K$ at $P(\cos \theta,\ \sin \theta)\ (\pi<\theta <2\pi).$ Find the minimum area of the part enclosed by $l$ and $C$.

Find all natural numbers $x,y,z$, such that $7^{x}+13^{y}=2^{z}$.

The number $n$ is "good", if there is three divisors of $n$($d_1, d_2, d_3$), such that $d_1^2+d_2^2+d_3^2=n$ a) Prove that all good number is divisible by $3$ b) Determine if there are infinite good numbers.

Determine all prime numbers $ p $ and natural numbers $ x, y $ for which $ p^x-y^3 = 1 $.

Set $S=\{1,2,...,2004\}$. We denote by $d_1$ the number of subset of $S$ such that the sum of elements of the subset has remainder $7$ when divided by $32$. We denote by $d_2$ the number of subset of $S$ such that the sum of elements of the subset has remainder $14$ when divided by $16$. Compute $\frac{d_1}{d_2}$.

Let $G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \}$ and $H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}$. Prove that $G$ and $H$ together with the operation of matrix multiplication are two non-isomorphical groups.

A set of $1985$ points is distributed around the circumference of a circle and each of the points is marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with $-1$ is less than $662$, there must be at least one good point.

The circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent line of the two circles closest to point $D$ touches $\Gamma_1$ in A and $\Gamma_2$ in $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the midpoint of line segment $BC$. Prove that $\angle DPM = \angle BDC$.

Given a $\triangle ABC$ with circumcenter $O$ and orthocenter $H(O\ne H)$. Denote the midpoints of $BC, AC$ as $D, E$ and let $D', E'$ be the reflections of $D, E$ w.r.t. point $H$, respectively. If lines $AD'$ and $BE'$ meet at $K$, compute $\frac{KO}{KH}$.

Sequence {$x_n$} satisfies: $x_1=1$ , ${x_n=\sqrt{x_{n-1}^2+x_{n-1}}+x_{n-1}}$ ( ${n>=2}$ ) Find the general term of {$x_n$}

For every $k= 1,2, \ldots$ let $s_k$ be the number of pairs $(x,y)$ satisfying the equation $kx + (k+1)y = 1001 - k$ with $x$, $y$ non-negative integers. Find $s_1 + s_2 + \cdots + s_{200}$.

Let $a_1,a_2,...,a_{15}$ be positive integers for which the number $a_k^{k+1} - a_k$ is not divisible by $17$ for any $k = 1,...,15$. Show that there are integers $b_1,b_2,...,b_{15}$ such that: (i) $b_m - b_n$ is not divisible by $17$ for $1 \le m < n \le 15$, and (ii) each $b_i$ is a product of one or more terms of $(a_i)$.

Prove that the equation \[ x^z + y^z = z^z \] has no solutions in postive integers.

Let $n$ be a positive integer. A grid of $n\times n$ has some black-colored cells. Drini can color a cell if at least three cells that share a side with it are also colored black. Drini discovers that by repeating this process, all the cells in the grid can be colored. Prove that if there are initially $k$ colored cells, then $$k\geq \frac{n^2+2n}{3}.$$

In acute-angled triangle $ABC$ with $|BC| < |BA|$, point $N$ is the midpoint of $AC$. The circle with diameter $AB$ intersects the bisector of $\angle B$ in two points: $B$ and $X$. Prove that $XN$ is parallel to $BC$. [img]https://cdn.artofproblemsolving.com/attachments/5/1/f0ae8f5df8f2cc1bb80de1ee1807dc845a87b3.png[/img]

Pooh has an unlimited supply of $1\times1$, $2\times2$, $3\times3$, and $4\times4$ squares. What is the minimum number of squares he needs to use in order to fully cover a $5\times5$ with no $2$ squares overlapping?

For which non-negative integers $n$ is \[K=5^{2n+3} + 3^{n+3} \cdot 2^n\] prime?

[b]Problem 4[/b] Find all continuously differentiable functions $ f : \mathbb{R} \rightarrow \mathbb{R} $, such that for every $a \geq 0$ the following relation holds: $$\iiint \limits_{D(a)} xf \left( \frac{ay}{\sqrt{x^2+y^2}} \right) \ dx \ dy\ dz = \frac{\pi a^3}{8} (f(a) + \sin a -1)\ , $$ where $D(a) = \left\{ (x,y,z)\ :\ x^2+y^2+z^2 \leq a^2\ , \ |y|\leq \frac{x}{\sqrt{3}} \right\}\ .$

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$. (a) Prove that $f_n$ is well defined. (b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective. [i]Bogdan Enescu[/i]

Northside's Drum and Bugle Corps raised money for a trip. The drummers and bugle players kept separate sales records. According to the double bar graph, in what month did one group's sales exceed the other's by the greatest percent? [asy] unitsize(12); fill((2,0)--(2,9)--(3,9)--(3,0)--cycle,lightgray); draw((3,0)--(3,9)--(2,9)--(2,0)); draw((2,7)--(1,7)--(1,0)); draw((2,8)--(3,8)); draw((2,7)--(3,7)); for (int a = 1; a <= 6; ++a) { draw((1,a)--(3,a)); } fill((5,0)--(5,3)--(6,3)--(6,0)--cycle,lightgray); draw((6,0)--(6,3)--(5,3)--(5,0)); draw((5,3)--(5,5)--(4,5)--(4,0)); draw((4,4)--(5,4)); draw((4,3)--(5,3)); draw((4,2)--(6,2)); draw((4,1)--(6,1)); fill((8,0)--(8,6)--(9,6)--(9,0)--cycle,lightgray); draw((9,0)--(9,6)--(8,6)--(8,0)); draw((8,6)--(8,9)--(7,9)--(7,0)); draw((7,8)--(8,8)); draw((7,7)--(8,7)); draw((7,6)--(8,6)); for (int a = 1; a <= 5; ++a) { draw((7,a)--(9,a)); } fill((11,0)--(11,12)--(12,12)--(12,0)--cycle,lightgray); draw((12,0)--(12,12)--(11,12)--(11,0)); draw((11,9)--(10,9)--(10,0)); draw((11,11)--(12,11)); draw((11,10)--(12,10)); draw((11,9)--(12,9)); for (int a = 1; a <= 8; ++a) { draw((10,a)--(12,a)); } fill((14,0)--(14,10)--(15,10)--(15,0)--cycle,lightgray); draw((15,0)--(15,10)--(14,10)--(14,0)); draw((14,8)--(13,8)--(13,0)); draw((14,9)--(15,9)); draw((14,8)--(15,8)); for (int a = 1; a <= 7; ++a) { draw((13,a)--(15,a)); } draw((16,0)--(0,0)--(0,13),black); label("Jan",(2,0),S); label("Feb",(5,0),S); label("Mar",(8,0),S); label("Apr",(11,0),S); label("May",(14,0),S); label("$\textbf{MONTHLY SALES}$",(8,14),N); label("S",(0,8),W); label("A",(0,7),W); label("L",(0,6),W); label("E",(0,5),W); label("S",(0,4),W); draw((4,12.5)--(4,13.5)--(5,13.5)--(5,12.5)--cycle); label("Drums",(4,13),W); fill((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle,lightgray); draw((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle); label("Bugles",(15,13),W);[/asy] $\text{(A)}\ \text{Jan} \qquad \text{(B)}\ \text{Feb} \qquad \text{(C)}\ \text{Mar} \qquad \text{(D)}\ \text{Apr} \qquad \text{(E)}\ \text{May}$

A quartic funtion $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

How many ways can you tile a $2\times5$ rectangle with $2\times1$ dominoes of $4$ different colors if no two dominoes of the same color may be adjacent?

Consider the endomorphism ring of an Abelian torsion-free (resp. torsion) group $ G$. Prove that this ring is Neumann-regular if and only if $ G$ is a discrete direct sum of groups isomorphic to the additive group of the rationals (resp. ,a discrete direct sum of cyclic groups of prime order). (A ring $ R$ is called Neumann-regular if for every $ \alpha \in R$ there exists a $ \beta \in R$ such that $ \alpha \beta \alpha\equal{}\alpha$.) [i]E. Freid[/i]

Let $a_{1}$, $\cdots$, $a_{k}$ and $m_{1}$, $\cdots$, $m_{k}$ be integers with $2 \le m_{1}$ and $2m_{i}\le m_{i+1}$ for $1 \le i \le k-1$. Show that there are infinitely many integers $x$ which do not satisfy any of congruences \[x \equiv a_{1}\; \pmod{m_{1}}, x \equiv a_{2}\; \pmod{m_{2}}, \cdots, x \equiv a_{k}\; \pmod{m_{k}}.\]