A parabola has focus $F$ and vertex $V$ , where $VF = 1$0. Let $AB$ be a chord of length $100$ that passes through $F$. Determine the area of $\vartriangle VAB$.

We will say that a plane is [i]well-colored[/i] in several colors if it is divided into convex polygons with an area of at least $1/1000$ and each polygon is colored in one color. Points lying on the border of several polygons can be colored in any of their colors. Are there convex is a $9$-gon $R$ and a good coloring of the plane in $7$ colors such that in any polygon obtained from $R$ by a translate to any vector, all colors occupy the same area ($1/7$ of the area of $R$)?

Prove that the sum of the squares of the medians of a triangle is at least $ 9/4 $ if the circumradius of the triangle, the area of the triangle and the inradius of the triangle (in this order) are in arithmetic progression. [i]Dumitru Crăciun[/i]

[b]p1.[/b] A large square is cut into four smaller, congruent squares. If each of the smaller squares has perimeter $4$, what was the perimeter of the original square? [b]p2.[/b] Pie loves to bake apples so much that he spends $24$ hours a day baking them. If Pie bakes a dozen apples in one day, how many minutes does it take Pie to bake one apple, on average? [b]p3.[/b] Bames Jond is sent to spy on James Pond. One day, Bames sees James type in his $4$-digit phone password. Bames remembers that James used the digits $0$, $5$, and $9$, and no other digits, but he does not remember the order. How many possible phone passwords satisfy this condition? [b]p4.[/b] What do you get if you square the answer to this question, add $256$ to it, and then divide by $32$? [b]p5.[/b] Chloe the Horse and Flower the Chicken are best friends. When Chloe gets sad for any reason, she calls Flower, so Chloe must remember Flower's $3$ digit phone number, which can consist of any digits $0-5$. Given that the phone number's digits are unique and add to $5$, the number does not start with $0$, and the $3$ digit number is prime, what is the sum of all possible phone numbers? [b]p6.[/b] Anuj has a circular pizza with diameter $A$ inches, which is cut into $B$ congruent slices, where $A$,$B$ are positive integers. If one of Anuj's pizza slices has a perimeter of $3\pi + 30$ inches, find $A + B$. [b]p7.[/b] Bob really likes to study math. Unfortunately, he gets easily distracted by messages sent by friends. At the beginning of every minute, there is an $\frac{6}{10}$ chance that he will get a message from a friend. If Bob does get a message from a friend, there is a $\frac{9}{10}$ chance that he will look at the message, causing him to waste $30$ seconds before resuming his studying. If Bob doesn't get a message from a friend, there is a $\frac{3}{10}$ chance Bob will still check his messages hoping for a message from his friends, wasting $10$ seconds before he resumes his studying. What is the expected number of minutes in $100$ minutes for which Bob will be studying math? [b]p8.[/b] Suppose there is a positive integer $n$ with $225$ distinct positive integer divisors. What is the minimum possible number of divisors of n that are perfect squares? [b]p9.[/b] Let $a, b, c$ be positive integers. $a$ has $12$ divisors, $b$ has $8$ divisors, $c$ has $6$ divisors, and $lcm(a, b, c) = abc$. Let $d$ be the number of divisors of $a^2bc$. Find the sum of all possible values of $d$. [b]p10.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 17$, $BC = 28$, $AC = 25$. Let the altitude from $A$ to $BC$ and the angle bisector of angle $B$ meet at $P$. Given the length of $BP$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$ where $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime, find $a + b + c$. [b]p11.[/b] Let $a$, $b$, and $c$ be the roots of the cubic equation $x^3-5x+3 = 0$. Let $S = a^4b+ab^4+a^4c+ac^4+b^4c+bc^4$. Find $|S|$. [b]p12.[/b] Call a number palindromeish if changing a single digit of the number into a different digit results in a new six-digit palindrome. For example, the number $110012$ is a palindromeish number since you can change the last digit into a $1$, which results in the palindrome $110011$. Find the number of $6$ digit palindromeish numbers. [b]p13.[/b] Let $P(x)$ be a polynomial of degree $3$ with real coecients and leading coecient $1$. Let the roots of $P(x)$ be $a$, $b$, $c$. Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}= 4$ and $a^2 + b^2 + c^2 = 36$, the coefficient of $x^2$ is negative, and $P(1) = 2$, let the $S$ be the sum of possible values of $P(0)$. Then $|S|$ can be expressed as $\frac{a + b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ such that $gcd(a, b, d) = 1$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d$. [b]p14.[/b] Let $ABC$ be a triangle with side lengths $AB = 7$, $BC = 8$, $AC = 9$. Draw a circle tangent to $AB$ at $B$ and passing through $C$. Let the center of the circle be $O$. The length of $AO$ can be expressed as $\frac{a\sqrt{b}}{c\sqrt{d}}$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, c) = gcd(b, d) = 1$ and $b$,$ d$ are not divisible by the square of any prime. Find $a + b + c + d$. [b]p15.[/b] Many students in Mr. Noeth's BC Calculus class missed their first test, and to avoid taking a makeup, have decided to never leave their houses again. As a result, Mr. Noeth decides that he will have to visit their houses to deliver the makeup tests. Conveniently, the $17$ absent students in his class live in consecutive houses on the same street. Mr. Noeth chooses at least three of every four people in consecutive houses to take a makeup. How many ways can Mr. Noeth select students to take makeups? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We are given an equilateral triangle ABC with the length of its side equal to $1$. There are $n-1$ points on each side of the triangle $ABC$ that equally divide the side into $n$ segments. We draw all possible lines that pass through any two of all those $3(n-1)$ points such that they are parallel to one of three sides of triangle $ABC$. All such lines divide triangle $ABC$ into some lesser triangles whose vertices are called [i]nodes[/i]. We assign a real number for each [i]node[/i] such that the following conditions are satisfied: (I) real numbers $a,b,c$ are assigned to $A,B,C$ respectively; (II) for any rhombus that is consisted of two lesser triangles that share a common side, the sum of the numbers of vertices on its one diagonal is equal to that of vertices on the other diagonal. 1) Find the minimum distance between the [i]node[/i] with the maximal number to the [i]node[/i] with the minimal number; 2) Denote by $S$ the sum of the numbers of all [i]nodes[/i], find $S$.

A convex hexagon $ ABCDEF$ is such that each of the diagonals $ AD,BE,CF$ divides the hexagon into two parts of equal area. Prove that these three diagonals are concurrent.

[b]p1.[/b] Vivek has three letters to send out. Unfortunately, he forgets which letter is which after sealing the envelopes and before putting on the addresses. He puts the addresses on at random sends out the letters anyways. What are the chances that none of the three recipients get their intended letter? [b]p2.[/b] David is a horrible bowler. Luckily, Logan and Christy let him use bumpers. The bowling lane is $2$ meters wide, and David's ball travels a total distance of $24$ meters. How many times did David's bowling ball hit the bumpers, if he threw it from the middle of the lane at a $60^o$ degree angle to the horizontal? [b]p3.[/b] Find $\gcd \,(212106, 106212)$. [b]p4.[/b] Michael has two fair dice, one six-sided (with sides marked $1$ through $6$) and one eight-sided (with sides marked $1-8$). Michael play a game with Alex: Alex calls out a number, and then Michael rolls the dice. If the sum of the dice is equal to Alex's number, Michael gives Alex the amount of the sum. Otherwise Alex wins nothing. What number should Alex call to maximize his expected gain of money? [b]p5.[/b] Suppose that $x$ is a real number with $\log_5 \sin x + \log_5 \cos x = -1$. Find $$|\sin^2 x \cos x + \cos^2 x \sin x|.$$ [b]p6.[/b] What is the volume of the largest sphere that FIts inside a regular tetrahedron of side length $6$? [b]p7.[/b] An ant is wandering on the edges of a cube. At every second, the ant randomly chooses one of the three edges incident at one vertex and walks along that edge, arriving at the other vertex at the end of the second. What is the probability that the ant is at its starting vertex after exactly $6$ seconds? [b]p8.[/b] Determine the smallest positive integer $k$ such that there exist $m, n$ non-negative integers with $m > 1$ satisfying $$k = 2^{2m+1} - n^2.$$ [b]p9.[/b] For $A,B \subset Z$ with $A,B \ne \emptyset$, define $A + B = \{a + b|a \in A, b \in B\}$. Determine the least $n$ such that there exist sets $A,B$ with $|A| = |B| = n$ and $A + B = \{0, 1, 2,..., 2012\}$. [b]p10.[/b] For positive integers $n \ge 1$, let $\tau (n)$ and $\sigma (n)$ be, respectively, the number of and sum of the positive integer divisors of $n$ (including $1$ and $n$). For example, $\tau (1) = \sigma (1) = 1$ and $\tau (6) = 4$, $\sigma (6) = 12$. Find the number of positive integers $n \le 100$ such that $$\sigma (n) \le (\sqrt{n} - 1)^2 +\tau (n)\sqrt{n}.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sides of the parallelogram serve as the diagonals of the four squares. The vertices of the squares lying in the part of the plane external to the parallelogram (the sides of the squares emerging from these vertices do not have common points with the parallelogram) serve as the vertices of a quadrilateral of area $a$, the four vertices opposite to them form a quadrilateral of area $b$. Find the area of the parallelogram.

Triangle $JHU$ has side lengths $JH=13$, $HU=14$, and $JU=15$. Point $X$ lies on $\overline{HU}$ such that $\triangle{JHX}$ and $\triangle{JUX}$ have equal perimeters. Compute $JX^2$.

Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

The bisector of angle $A$ of triangle $ABC$ ($AB > AC$) meets its circumcircle at point $P$. The perpendicular to $AC$ from $C$ meets the bisector of angle $A$ at point $K$. A cừcle with center $P$ and radius $PK$ meets the minor arc $PA$ of the circumcircle at point $D$. Prove that the quadrilateral $ABDC$ is circumscribed.

Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.

In a rectangle $ABCD, M$ and $N$ are the midpoints of $AD$ and $BC$ respectively and $P$ is a point on line $CD$. The line $PM$ meets $AC$ at $Q$. Prove that MN bisects the angle $\angle QNP$.

Solve the equation $a^3+b^3+c^3=2001$ in positive integers. [i]Mircea Becheanu, Romania[/i]

Let $\mathcal{C}$ be a circle with center $O$, and let $A$ be a point outside the circle. Let the two tangents from the point $A$ to the circle $\mathcal{C}$ meet this circle at the points $S$ and $T$, respectively. Given a point $M$ on the circle $\mathcal{C}$ which is different from the points $S$ and $T$, let the line $MA$ meet the perpendicular from the point $S$ to the line $MO$ at $P$. Prove that the reflection of the point $S$ in the point $P$ lies on the line $MT$.

Altitudes $AA_1, BB_1, CC_1$ of acute triangle $ABC$ intersect at point $H$. On the tangent drawn from point $C$ to the circle $(AB_1C_1)$, the perpendicular $HQ$ is drawn (the point $Q$ lies inside the triangle $ABC$). Prove that the circle passing through the point $B_1$ and touching the line $AB$ at point $A$ is also tangent to line $A_1Q$.

Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.

Find all natural numbers $ n\ge 4 $ that satisfy the property that the affixes of any nonzero pairwise distinct complex numbers $ a,b,c $ that verify the equation $$ (a-b)^n+(b-c)^n+(c-a)^n=0, $$ represent the vertices of an equilateral triangle in the complex plane.

A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n=(x_n,y_n)$, the frog jumps to $P_{n+1}$, which may be any of the points $(x_n+7, y_n+2)$, $(x_n+2,y_n+7)$, $(x_n-5, y_n-10)$, or $(x_n-10,y_n-5)$. There are $M$ points $(x,y)$ with $|x|+|y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000$.

Let $n$ be an integer greater than $3$. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of length $1$. Find the number of convex trapezoids which have vertices among the vertices of the $n^2$ squares of side length $1$, have side lengths less than or equal $3$ and have area equal to $2$ Note: Parallelograms are trapezoids.

In cyclic quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $P$. Let $E$ and $F$ be the respective feet of the perpendiculars from $P$ to lines $AB$ and $CD$. Segments $BF$ and $CE$ meet at $Q$. Prove that lines $PQ$ and $EF$ are perpendicular to each other.

A line $l$ parallel to the side $BC$ of triangle $ABC$ touches its incircle and meets its circumcircle at points $D$ and $E$. Let $I$ be the incenter of $ABC$. Prove that $AI^2 = AD \cdot AE$.

The points $A, B, M$ and $N$ are on a circle with center $O$ such that the radii $OA$ and $OB$ are perpendicular to each other, and $MN$ is parallel to $AB$ and intersects the radius $OA$ at $P$. Find the radius of the circle if $|MP|= 12$ and $|P N| = 2 \sqrt{14}$

Inside triangle \(ABC\), point \(P\) is marked. Point \(Q\) is on segment \(AB\), and point \(R\) is on segment \(AC\) such that the circumcircles of triangles \(BPQ\) and \(CPR\) are tangent to line \(AP\). Lines are drawn through points \(B\) and \(C\) passing through the center of the circumcircle of triangle \(BPC\), and through points \(Q\) and \(R\) passing through the center of the circumcircle of triangle \(PQR\). Prove that there exists a circle tangent to all four drawn lines.

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?