Numbers $a, b, c$ are such that $3a + 4b = 3c$ and $4a - 3b = 4c$. Show that $a^2 + b^2 = c^2$.

$\sqrt{\frac{1}{9} + \frac{1}{16}} =$ $\textbf{(A)}\ \frac15 \qquad \textbf{(B)}\ \frac14 \qquad \textbf{(C)}\ \frac27 \qquad \textbf{(D)}\ \frac{5}{12} \qquad \textbf{(E)}\ \frac{7}{12}$

Let $ p$ be the greatest prime factor of 9991. Then, the sum of the digits of $ p$ is $ \text{(A)}\ 4 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 28$

Prove that there exists a positive integer $n$ such that in the decimal representation of each of the numbers $\sqrt{n}$, $\sqrt[3]{n},..., \sqrt[10]{n}$ digits $2015$ stand immediately after the decimal point. [i]A.Golovanov [/i]

Hello all. Post your solutions below. [b]Also, I think it is beneficial to everyone if you all attempt to comment on each other's solutions.[/b] 4/1/31. A group of $100$ friends stands in a circle. Initially, one person has $2019$ mangos, and no one else has mangos. The friends split the mangos according to the following rules: • sharing: to share, a friend passes two mangos to the left and one mango to the right. • eating: the mangos must also be eaten and enjoyed. However, no friend wants to be selfish and eat too many mangos. Every time a person eats a mango, they must also pass another mango to the right. A person may only share if they have at least three mangos, and they may only eat if they have at least two mangos. The friends continue sharing and eating, until so many mangos have been eaten that no one is able to share or eat anymore. Show that there are exactly eight people stuck with mangos, which can no longer be shared or eaten.

Initially on a blackboard, the equation $a_1x^2+b_1x+c=0$ is written where $a_1, b_1, c_1$ are integers and $(a_1+c_1)b_1 > 0$. At each move, if the equation $ax^2+bx+c=0$ is written on the board and there is a $x \in \mathbb{R}$ satisfying the equation, Alice turns this equation into $(b+c)x^2+(c+a)x+(a+b)=0$. Prove that Alice will stop after a finite number of moves.

Zoey read $15$ books, one at a time. The first book took her $1$ day to read, the second book took her $2$ days to read, the third book took her $3$ days to read, and so on, with each book taking her $1$ more day to read than the previous book. Zoey finished the first book on a monday, and the second on a Wednesday. On what day the week did she finish her $15$th book? $\textbf{(A)}\ \text{Sunday}\qquad\textbf{(B)}\ \text{Monday}\qquad\textbf{(C)}\ \text{Wednesday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$

Let $a_n$ be the area of the part enclosed by the curve $y=x^n\ (n\geq 1)$, the line $x=\frac 12$ and the $x$ axis. Prove that : \[0\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}\]

Let $H$ be the orthocenter of an acute-angled triangle $ABC$ with $AC\ne BC$. The line through the midpoints of the segments $AB$ and $HC$ intersects the bisector of $\angle ACB$ at $D$. Suppose that the line $HD$ contains the circumcenter of $\triangle ABC$. Determine $\angle ACB$.

[b]U[/b]ndecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years. (The sizes of the space stations and the base are negligible.) Determine the volume, in cubic light years, of the set of all possible locations for the Admiral’s base.

[u]Round 1[/u] [b]p1.[/b] A $\$100$ TV has its price increased by $10\%$. The new price is then decreased by $10\%$. What is the current price of the TV? [b]p2.[/b] If $9w + 8x + 7y = 42$ and $w + 2x + 3y = 8$, then what is the value of $100w + 101x + 102y$? [b]p3.[/b] Find the number of positive factors of $37^3 \cdot 41^3$. [u]Round 2[/u] [b]p4.[/b] Three hoses work together to fill up a pool, and each hose expels water at a constant rate. If it takes the first, second, and third hoses 4, 6, and 12 hours, respectively, to fill up the pool alone, then how long will it take to fill up the pool if all three hoses work together? [b]p5.[/b] A semicircle has radius $1$. A smaller semicircle is inscribed in the larger one such that the two bases are parallel and the arc of the smaller is tangent to the base of the larger. An even smaller semicircle is inscribed in the same manner inside the smaller of the two semicircles, and this procedure continues indefinitely. What is the sum of all of the areas of the semicircles? [b]p6.[/b] Given that $P(x)$ is a quadratic polynomial with $P(1) = 0$, $P(2) = 0$, and $P(0) = 2012$, find $P(-1)$. [u]Round 3[/u] [b]p7.[/b] Darwin has a paper circle. He labels one point on the circumference as $A$. He folds $A$ to every point on the circumference on the circle and undoes it. When he folds $A$ to any point $P$, he makes a blue mark on the point where $\overline{AP}$ and the made crease intersect. If the area of Darwin paper circle is 80, then what is the area of the region surrounded by blue? [b]p8.[/b] Α rectangular wheel of dimensions $6$ feet by $8$ feet rolls for $28$ feet without sliding. What is the total distance traveled by any corner on the rectangle during this roll? [b]p9[/b]. How many times in a $24$-hour period do the minute hand and hour hand of a $12$-hour clock form a right angle? [u]Round 4[/u] The answers in this section all depend on each other. Find smallest possible solution set. [b]p10.[/b] Let B be the answer to problem $11$. Right triangle $ACD$ has a right angle at $C$. Squares $ACEF$ and $ADGH$ are drawn such that points $D$ and $E$ do not coincide and points $E$ and $H$ do not coincide. The midpoints of the sides of $ADGH$ are connected to form a smaller square with area $B.$ If the area of $ACEF$ is also $B$, then find the length $CD$ rounded up to the nearest integer. [b]p11.[/b] Let $C$ be the answer to problem $12$. Find the sum of the digits of $C$. [b]p12.[/b] Let $A$ be the answer to problem $10$. Given that $a_0 = 1$, $a_1 = 2$, and that $a_n = 3a_{n-1 }-a_{n-2}$ for $n \ge 2$, find $a_A$. PS. You should use hide for answers.Rounds 5-8 are [url=https://artofproblemsolving.com/community/c3h3134466p28406321]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134489p28406583]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A circle is inscribed in a regular octagon with area $2024$. A second regular octagon is inscribed in the circle, and its area can be expressed as $a + b\sqrt{c}$, where $a, b, c$ are integers and $c$ is square-free. Compute $a + b + c$.

At a chess tourney, each player played with all the other players two matches, one time with the white pieces, and one time with the black pieces. One point was given for a victory, and 0,5 points were given for a tied game. In the end of the tourney all the players had the same number of points. a) Prove that there exist two players with the same number of tied games; b) Prove that there exist two players which have the same number of lost games when playing with the white pieces.

We color each unit square of a $8\times 8$ table into green or blue such that there are $a$ green unit squares in each $3 \times 3$ square and $b$ green unit squares in each $2 \times 4$ rectangle. Find all possible values of $(a, b)$. (Le Anh Vinh)

For $n\geq k\geq 3$, let $X=\{1,2,...,n\}$ and let $F_{k}$ a the family of $k$-element subsets of $X$, any two of which have at most $k-2$ elements in common. Show that there exists a subset $M_{k}$ of $X$ with at least $[\log_{2}{n}]+1$ elements containing no subset in $F_{k}$.

Let $\mathbb Z^{\geq 0}$ be the set of all non-negative integers. Consider a function $f:\mathbb Z^{\geq 0} \to \mathbb Z^{\geq 0}$ such that $f(0)=1$ and $f(1)=1$, and that for any integer $n \geq 1$, we have \[f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.\] Determine the value of $f(2023)/f(2022)$.

In a cube, let $M$ be the midpoint of one of the segments. Choose two vertices of the cube, $A$ and $B$. What is the number of distinct possible triangles $\triangle AMB$ up to congruency? [i]Proposed by Harry Kim[/i]

For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$? $ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $

The real numbers $a, b, x$ satisfy the inequalities $| a + x + b | \le 1, | 4a + 2x + b | \le1, | 9a + 6x + 4b | \le 1$. Prove that $| x | \le15$.

The ratio of the perimeter of an equilateral triangle having an altitude equal to the radius of a circle, to the perimeter of an equilateral triangle inscribed in the circle is: $ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 1: 3 \qquad\textbf{(C)}\ 1: \sqrt {3} \qquad\textbf{(D)}\ \sqrt {3}: 2 \qquad\textbf{(E)}\ 2: 3$

Pablo says: “I add $2$ to my birthday and multiply the result by $2$. I add to the number obtained $4$ and multiply the result by $5$. To the new number obtained I add the number of the month of my birthday (for example, if it's June, I add $6$) and I get $342$. " What is Pablo's birthday date? Give all the possibilities

Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

Let $D$ and $E$ be points on sides $AB$ and $AC$ of a triangle $ABC$ such that $DB = BC = CE$. The segments $BE$ and $CD$ intersect at point $P$. Prove that the incenter of triangle $ABC$ lies on the circles circumscribed around the triangles $BDP$ and $CEP$.

There are $m$ pieces of candy held in $n$ trays($n,m\ge 4$). An [i]operation[/i] is defined as follow: take out one piece of candy from any two trays respectively, then put them in a third tray. Determine, with proof, if we can move all candies to a single tray by finite [i]operations[/i].

Let $T$ be a finite set of squarefree integers. (a) Show that there exists an integer polynomial $P(x)$ such that the set of squarefree numbers in the range of $P(n)$ across all $n \in \mathbb{Z}$ is exactly $T$. (b) Suppose that $T$ is allowed to be infinite. Is it still true that for all choices of $T$, such an integer polynomial $P(x)$ exists? [i]Allen Wang[/i]