Find the area of the region of the points such that the total of three tangent lines can be drawn to two parabolas $y=x-x^2,\ y=a(x-x^2)\ (a\geq 2)$ in such a way that there existed the points of tangency in the first quadrant.

In a parallelogram $ABCD$ with $\angle A < 90$, the circle with diameter $AC$ intersects the lines $CB$ and $CD$ again at $E$ and $F$ , and the tangent to this circle at $A$ meets the line $BD$ at $P$ . Prove that the points $P$, $E$, $F$ are collinear.

Let $ABC$ be a triangle and $D,E$ be points on $BC,CA$ such that $AD,BE$ are angle bisectors of $\triangle ABC$. Let $F,G$ be points on the circumcircle of $\triangle ABC$ such that $AF||DE$ and $FG||BC$. Prove that $\frac{AG}{BG}= \frac{AB+AC}{AB+BC}$.

Two persons are playing the following game. They have a pile of stones and take turns removing stones from it, with the first player taking the first turn. At each turn, the first player removes either 1 or 10 stones from the pile, and the second player removes either m or n stones. The player who can not make his move loses. It is known that for any number of stones in the pile, the first player can always win (regardless of the second player's moves). What are the possible values of m and n?

Four semicircles of radius $1$ are placed in a square, as shown below. The diameters of these semicircles lie on the sides of the square and each semicircle touches a vertex of the square. Find the absolute difference between the shaded area and the "hatched" area. [asy] import patterns; add("hatch",hatch(1.2mm)); add("checker",checker(2mm)); real r = 1 + sqrt(3); filldraw((0,0)--(r,0)--(r,r)--(0,r)--cycle,gray(0.4),linewidth(1.5)); fill((1,0)--(r,1)--(r-1,r)--(0,r-1)--cycle,white); fill((1,0)--(r,1)--(r-1,r)--(0,r-1)--cycle,pattern("hatch")); filldraw(arc((1,0),1,0,180)--(0,0)--cycle,white,linewidth(1.5)); filldraw(arc((r,1),1,90,270)--(r,0)--cycle,white,linewidth(1.5)); filldraw(arc((r-1,r),1,180,360)--(r,r)--cycle,white,linewidth(1.5)); filldraw(arc((0,r-1),1,270,450)--(0,r)--cycle,white,linewidth(1.5)); [/asy] [i]Proposed by Connor Gordon[/i]

Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Prove that if $\frac{H_BC}{AC} = \frac{H_CA}{AB}$, then the line symmetric to $BC$ with respect to line $H_BH_C$ is tangent to the circumscribed circle of triangle $H_BH_CA$. [i](Proposed by Mykhailo Bondarenko)[/i]

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Let $ S$ be a set of $ 100$ points in the plane. The distance between every pair of points in $ S$ is different, with the largest distance being $ 30$. Let $ A$ be one of the points in $ S$, let $ B$ be the point in $ S$ farthest from $ A$, and let $ C$ be the point in $ S$ farthest from $ B$. Let $ d$ be the distance between $ B$ and $ C$ rounded to the nearest integer. What is the smallest possible value of $ d$?

For finite sets $A,M$ such that $A \subseteq M \subset \mathbb{Z}^+$, we define $$f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}.$$ Given a positive integer $k$, we call $M$ [i]k-colorable[/i] if it is possible to color the subsets of $M$ with $k$ colors so that for any $A \subseteq M$, if $f_M(A)\neq A$ then $f_M(A)$ and $A$ have different colors. Determine the least positive integer $k$ such that every finite set $M \subset\mathbb{Z}^+$ is k-colorable.

In a kindergarten, a nurse took $n$ congruent cardboard rectangles and gave them to $n$ kids, one per each. Each kid has cut its rectangle into congruent squares(the squares of different kids could be of different sizes). It turned out that the total number of the obtained squares is a prime number. Prove that all the initial squares were in fact squares.

Circle $\odot O$ with diameter $\overline{AB}$ has chord $\overline{CD}$ drawn such that $\overline{AB}$ is perpendicular to $\overline{CD}$ at $P$. Another circle $\odot A$ is drawn, sharing chord $\overline{CD}$. A point $Q$ on minor arc $\overline{CD}$ of $\odot A$ is chosen so that $\text{m} \angle AQP + \text{m} \angle QPB = 60^\circ$. Line $l$ is tangent to $\odot A$ through $Q$ and a point $X$ on $l$ is chosen such that $PX=BX$. If $PQ = 13$ and $BQ = 35$, find $QX$. [i]Proposed by Aaron Lin[/i]

Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \) and \( 0 < f'(t) \leq 1 \) for all \( t \in [0, 1] \). Show that: \[ \left( \int_0^1 f(t) \, dt \right)^2 \geq \int_0^1 f(t)^3 \, dt. \]

Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$. Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.

Given a triangle $ABC$ with integer side lengths, where $BD$ is an angle bisector of $\angle ABC$, $AD=4$, $DC=6$, and $D$ is on $AC$, compute the minimum possible perimeter of $\triangle ABC$.

The natural numbers $p, q$ satisfy the relation $p^p + q^q = p^q + q^p$. Prove that $p = q$.

How many six-letter words formed from the letters of AMC do not contain the substring AMC? (For example, AMAMMC has this property, but AAMCCC does not.)

For all real numbers $x$, $x[x\{x(2-x)-4\}+10]+1=$ $\textbf{(A) }-x^4+2x^3+4x^2+10x+1$ $\textbf{(B) }-x^4-2x^3+4x^2+10x+1$ $\textbf{(C) }-x^4-2x^3-4x^2+10x+1$ $\textbf{(D) }-x^4-2x^3-4x^2-10x+1$ $\textbf{(E) }-x^4+2x^3-4x^2+10x+1$

What is the smallest positive odd integer having the same number of positive divisors as $360$?

Let $x, y, z \in R^+$. Prove that $$\frac{x}{x +\sqrt{(x + y)(x + z)}}+\frac{y}{y +\sqrt{(y + z)(y + x)}}+\frac{z}{z +\sqrt{(x + z)(z + y)}} \le 1$$

Find all integer solutions to the equation $$\frac{xyz}{w}+\frac{yzw}{x}+\frac{zwx}{y}+\frac{wxy}{z}=4$$

Camilla drove $20$ miles in the city at a constant speed and $40$ miles in the country at a constant speed that was $20$ miles per hour greater than her speed in the city. Her entire trip took one hour. Find the number of minutes that Camilla drove in the country rounded to the nearest minute.

Let $m_1< m_2 < \ldots m_{k-1}< m_k$ be $k$ distinct positive integers such that their reciprocals are in arithmetic progression. 1.Show that $k< m_1 + 2$. 2. Give an example of such a sequence of length $k$ for any positive integer $k$.

What is the sum of the last two digits of $7^{42} + 7^{43}$ in base $10$. $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }8\qquad\text{(D) }9\qquad\text{(E) }11$

Vasya chose a certain number $x$ and calculated the following: $a_1=1+x^2+x^3, a_2=1+x^3+x^4, a_3=1+x^4+x^5, ..., a_n=1+x^{n+1}+x^{n+2} ,...$ It turned out that $a_2^2 = a_1a_3$. Prove that for all $n\ge 3$, the equality $a_n^2 = a_{n-1}a_{n+1}$ holds.

The internal bisectors of angles $\angle DAB$ and $\angle BCD$ of a quadrilateral $ABCD$ intersect at the point $X_1$, and the external bisectors of these angles intersect at the point $X_2$. The internal bisectors of angles $\angle ABC$ and $\angle CDA$ intersect at the point $Y_1$, and the external bisectors of these angles intersect at the point $Y_2$. Prove that the angle between the lines $X_1X_2$ and $Y_1Y_2$ equals the angle between the diagonals $AC$ and $BD$. [i](A. Voidelevich)[/i]