Find all two-digit numbers $x$ the sum of whose digits is the same as that of $2x$, $3x$, ... , $9x$.

In the plane, there are two circles $\Gamma_1, \Gamma_2$ intersecting each other at two points $A$ and $B$. Tangents of $\Gamma_1$ at $A$ and $B$ meet each other at $K$. Let us consider an arbitrary point $M$ (which is different of $A$ and $B$) on $\Gamma_1$. The line $MA$ meets $\Gamma_2$ again at $P$. The line $MK$ meets $\Gamma_1$ again at $C$. The line $CA$ meets $\Gamma_2 $ again at $Q$. Show that the midpoint of $PQ$ lies on the line $MC$ and the line $PQ$ passes through a fixed point when $M$ moves on $\Gamma_1$. [color=red][Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .][/color]

[u]Round 5[/u] [b]5.1.[/b] Julia baked a pie for herself to celebrate pi day this year. If Julia bakes anyone pie on pi day, the following year on pi day she bakes a pie for herself with $1/3$ probability, she bakes her friend a pie with $1/6$ probability, and she doesn't bake anyone a pie with $1/2$ probability. However, if Julia doesn't make pie on pi day, the following year on pi day she bakes a pie for herself with $1/2$ probability, she bakes her friend a pie with $1/3$ probability, and she doesn't bake anyone a pie with $1/6$ probability. The probability that Julia bakes at least $2$ pies on pi day in the next $5$ years can be expressed as $p/q$, for relatively prime positive integers $p$ and $q$. Compute $p + q$. [b]5.2.[/b] Steven is flipping a coin but doesn't want to appear too lucky. If he ips the coin $8$ times, the probability he only gets sequences of consecutive heads or consecutive tails that are of length $4$ or less can be expressed as $p/q$, for relatively prime positive integers $p$ and $q$. Compute $p + q$. [b]5.3.[/b] Let $ABCD$ be a square with side length $3$. Further, let $E$ be a point on side$ AD$, such that $AE = 2$ and $DE = 1$, and let $F$ be the point on side $AB$ such that triangle $CEF$ is right with hypotenuse $CF$. The value $CF^2$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [u]Round 6[/u] [b]6.1.[/b] Let $P$ be a point outside circle $\omega$ with center $O$. Let $A,B$ be points on circle $\omega$ such that $PB$ is a tangent to $\omega$ and $PA = AB$. Let $M$ be the midpoint of $AB$. Given $OM = 1$, $PB = 3$, the value of $AB^2$ can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [b]6.2.[/b] Let $a_0, a_1, a_2,...$with each term defined as $a_n = 3a_{n-1} + 5a_{n-2}$ and $a_0 = 0$, $a_1 = 1$. Find the remainder when $a_{2020}$ is divided by $360$. [b]6.3.[/b] James and Charles each randomly pick two points on distinct sides of a square, and they each connect their chosen pair of points with a line segment. The probability that the two line segments intersect can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [u]Round 7[/u] [b]7.1.[/b] For some positive integers $x, y$ let $g = gcd (x, y)$ and $\ell = lcm (2x, y)$: Given that the equation $xy+3g+7\ell = 168$ holds, find the largest possible value of $2x + y$. [b]7.2.[/b] Marco writes the polynomials $$f(x) = nx^4 +2x^3 +3x^2 +4x+5$$ and $$g(x) = a(x-1)^4 +b(x-1)^3 +6(x-1)^2 + d(x - 1) + e,$$ where $n, a, b, d, e$ are real numbers. He notices that $g(i) = f(i) - |i|$ for each integer $i$ satisfying $-5 \le i \le -1$. Then $n^2$ can be expressed as $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]7.3. [/b]Equilateral $\vartriangle ABC$ is inscribed in a circle with center $O$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $BC$, respectively. Segment $\overline{CD}$ intersects $\overline{AB}$ and $\overline{AE}$ at $Y$ and $X$, respectively. Given that $\vartriangle DXE$ and $\vartriangle AXC$ have equal area, $\vartriangle AXY$ has area $ 1$, and $\vartriangle ABC$ has area $52$, find the area of $\vartriangle BXC$. [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of total webpage visits our website received last month. Let $B$ be the number photos in our photo collection from ABMC onsite 2017. Let $M$ be the mean speed round score. Further, let $C$ be the number of times the letter c appears in our problem bank. Estimate $$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766251p24226451]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$, $b$, $c$ be positive integers with $a \le 10$. Suppose the parabola $y = ax^2 + bx + c$ meets the $x$-axis at two distinct points $A$ and $B$. Given that the length of $\overline{AB}$ is irrational, determine, with proof, the smallest possible value of this length, across all such choices of $(a, b, c)$.

$a_1,a_2,\ldots$ is a sequence of [u]nonzero integer[/u] numbers that for all $n\in\mathbb{N}$, if $a_n=2^\alpha k$ such that $k$ is an odd integer and $\alpha$ is a nonnegative integer then: $a_{n+1}=2^\alpha-k$. Prove that if this sequence is periodic, then for all $n\in\mathbb{N}$ we have: $a_{n+2}=a_n$. (The sequence $a_1,a_2,\ldots$ is periodic iff there exists natural number $d$ that for all $n\in\mathbb{N}$ we have: $a_{n+d}=a_n$)

The point $P$ lies on the edge $AB$ of a quadrilateral $ABCD$. The angles $BAD, ABC$ and $CPD$ are right, and $AB = BC + AD$. Show that $BC = BP$ or $AD = BP$.

Let $P$ be a regular hexagon. For a point $A$, let $d_1\le d_2\le ...\le d_6$ the distances from $A$ to the six vertices of $P$, ordered by magnitude. Find the locus of all points $A$ in the interior or on the boundary of $P$ such that: (a) $d_3$ takes the smallest possible value. (b) $d_4$ takes the smallest possible value.

Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is $a+b\sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$

Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

A deck of eight cards has cards numbered $1, 2, 3, 4, 5, 6, 7, 8$, in that order, and a deck of five cards has cards numbered $1, 2, 3, 4, 5$, in that order. The two decks are riffle-shuffled together to form a deck with $13$ cards with the cards from each deck in the same order as they were originally. Thus, numbers on the cards might end up in the order $1122334455678$ or $1234512345678$ but not $1223144553678$. Find the number of possible sequences of the $13$ numbers.

A net for a polyhedron is cut along an edge to give two [b]pieces[/b]. For example, we may cut a cube net along the red edge to form two pieces as shown. [asy] size(5.5cm); draw((1,0)--(1,4)--(2,4)--(2,0)--cycle); draw((1,1)--(2,1)); draw((1,2)--(2,2)); draw((1,3)--(2,3)); draw((0,1)--(3,1)--(3,2)--(0,2)--cycle); draw((2,1)--(2,2),red+linewidth(1.5)); draw((3.5,2)--(5,2)); filldraw((4.25,2.2)--(5,2)--(4.25,1.8)--cycle,black); draw((6,1.5)--(10,1.5)--(10,2.5)--(6,2.5)--cycle); draw((7,1.5)--(7,2.5)); draw((8,1.5)--(8,2.5)); draw((9,1.5)--(9,2.5)); draw((7,2.5)--(7,3.5)--(8,3.5)--(8,2.5)--cycle); draw((11,1.5)--(11,2.5)--(12,2.5)--(12,1.5)--cycle); [/asy] Are there two distinct polyhedra for which this process may result in the same two pairs of pieces? If you think the answer is no, prove that no pair of polyhedra can result in the same two pairs of pieces. If you think the answer is yes, provide an example; a clear example will suffice as a proof.

Find all functions $ f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that: $ f(x\plus{}f(y))\equal{}y\plus{}f(x)$ for all $ x,y \in \mathbb{Q}$.

Let $\pi$ be a permutation of the numbers from $ 2$ through $2012$. Find the largest possible value of $$\log_2 \pi(2) \cdot \log_3 \pi(3) ...\log_{2012} \pi(2012).$$

Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

Maria Ivanovna drew on the blackboard a right triangle $ABC$ with a right angle $B$. Three students looked at her and said: $\bullet$ Yura said: "The hypotenuse of this triangle is $10$ cm." $\bullet$ Roma said: "The altitude drawn from the vertex $B$ on the side $AC$ is $6$ cm." $\bullet$ Seva said: "The area of the triangle $ABC$ is $25$ cm$^2$." Determine which of the students was mistaken if it is known that there is exactly one such person.

Let $P(x)=x^3-\tfrac{3}{2}x^2+x+\tfrac{1}{4}$. Let $P^{[1]}(x)=P(x)$, and for $n\ge1$, let $P^{n+1}(x)=P^{[n]}(P(x))$. Evaluate: \[ \displaystyle\int_{0}^{1} P^{[2004]} (x) \ \mathrm{d}x. \]

Given that \[ \frac{\cos(x) \plus{} \cos(y) \plus{} \cos(z)}{\cos(x\plus{}y\plus{}z)} \equal{} \frac{\sin(x)\plus{} \sin(y) \plus{} \sin(z)}{\sin(x \plus{} y \plus{} z)} \equal{} a,\] show that \[ \cos(y\plus{}z) \plus{} \cos(z\plus{}x) \plus{} \cos(x\plus{}y) \equal{} a.\]

$300$ couples (one man, one woman) are invited to a party. Everyone at the party either always tells the truth or always lies. Exactly $2/3$ of the men say their partner always tells the truth and the remaining $1/3$ say their partner always lies. Exactly $2/3$ of the women say their partner is the same type as themselves and the remaining $1/3$ say their partner is different. Find $a$, the maximum possible number of people who tell the truth, and $b$, the minimum possible number of people who tell the truth. Express your answer as $(a,b)$.

Quadrilateral $ABCD$ is inscribed inside a circle with $\angle BAC= 70^{\circ}, \angle ADB= 40^{\circ}, AD=4$, and $BC=6$. What is $AC$? $\textbf{(A) }3+\sqrt{5}\qquad\textbf{(B) }6\qquad\textbf{(C) }\frac{9}{2}\sqrt{2}\qquad\textbf{(D) }8-\sqrt{2}\qquad\textbf{(E) }7$

If $ (a,b)$ and $ (c,d)$ are two points on the line whose equation is $ y\equal{}mx\plus{}k$, then the distance between $ (a,b)$ and $ (c,d)$, in terms of $ a$, $ c$, and $ m$, is $ \textbf{(A)}\ |a\minus{}c|\sqrt{1\plus{}m^2} \qquad \textbf{(B)}\ |a\plus{}c|\sqrt{1\plus{}m^2} \qquad \textbf{(C)}\ \frac{|a\minus{}c|}{\sqrt{1\plus{}m^2}} \qquad$ $ \textbf{(D)}\ |a\minus{}c|(1\plus{}m^2) \qquad \textbf{(E)}\ |a\minus{}c|$ $ |m|$

$1$ cow can produce $3$ gallons of milk each day. How many cows would it take to produce $210$ gallons of milk in a week? $\textbf{(A) } 3\qquad\textbf{(B) } 7\qquad\textbf{(C) } 10\qquad\textbf{(D) } 30\qquad\textbf{(E) } 70$

Let $ABC$ be a triangle with $AB=26$, $AC=28$, $BC=30$. Let $X$, $Y$, $Z$ be the midpoints of arcs $BC$, $CA$, $AB$ (not containing the opposite vertices) respectively on the circumcircle of $ABC$. Let $P$ be the midpoint of arc $BC$ containing point $A$. Suppose lines $BP$ and $XZ$ meet at $M$ , while lines $CP$ and $XY$ meet at $N$. Find the square of the distance from $X$ to $MN$. [i]Proposed by Michael Kural[/i]

In maths class Albrecht had to compute $(a+2b-3)^2$ . His result was $a^2 +4b^2-9$ . ‘This is not correct’ said his teacher, ‘try substituting positive integers for $a$ and $b$.’ Albrecht did so, but his result proved to be correct. What numbers could he substitute? a) Show a good substitution. b) Give all the pairs that Albrecht could substitute and prove that there are no more.

The incircle of the triangle $ABC$ is tangent to sides $AC$ and $BC$ at $M$ and $N$, respectively. The bisectors of the angles at $A$ and $B$ intersect $MN$ at points $P$ and $Q$, respectively. Let $O$ be the incentre of $\triangle ABC$. Prove that $MP\cdot OA=BC\cdot OQ$.

No two of $20$ students in a class have the same scores on both written and oral examinations in mathematics. We say that student $A$ is better than $B$ if his two scores are greater than or equal to the corresponding scores of $B$. The scores are integers between $1$ and $10$. (a) Show that there exist three students $A,B,C$ such that $A$ is better than $B$ and $B$ is better than $C$. (b) Would the same be true for a class of $19$ students?