There are five distinct real positive numbers. It is known that the total sum of their squares and the total sum of their pairwise products are equal. (a) Prove that we can choose three numbers such that it would not be possible to make a triangle with sides' lengths equal to these numbers. (b) Prove that the number of such triples is at least six (triples which consist of the same numbers in different order are considered the same).

Let $S$ be the set of all positive integers $n$ such that each of the numbers $n + 1$, $n + 3$, $n + 4$, $n + 5$, $n + 6$, and $n + 8$ is composite. Determine the largest integer $k$ with the following property: For each $n \in S$ there exist at least $k$ consecutive composite integers in the set {$n, n +1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$}.

An infinite, strictly increasing sequence $\{a_n\}$ of positive integers satisfies the condition $a_{a_n}\le a_n + a_{n + 3}$ for all $n\ge 1$. Prove that there are infinitely many triples $(k, l, m)$ of positive integers such that $k <l <m$ and $a_k + a_m = 2a_l$.

The lengths of the sides of a triangle are 6, 8 and 10 units. Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.

$a_0, a_1, \ldots, a_{100}$ and $b_1, b_2,\ldots, b_{100}$ are sequences of real numbers, for which the property holds: for all $n=0, 1, \ldots, 99$, either $$a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n,$$ or $$a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n.$$ Given $a_{100}\leq a_0$, what is the maximal value of $b_1+b_2+\cdots+b_{100}$?

Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.

Find the least positive integer $ N$ such that the sum of its digits is 100 and the sum of the digits of $ 2N$ is 110.

Find the largest integer $N$ satisfying the following two conditions: [b](i)[/b] $\left[ \frac N3 \right]$ consists of three equal digits; [b](ii)[/b] $\left[ \frac N3 \right] = 1 + 2 + 3 +\cdots + n$ for some positive integer $n.$

The real numbers $x,y,z$ satisfy $x^2+y^2+z^2=3.$ Prove that the inequality $x^3-(y^2+yz+z^2)x+yz(y+z)\le 3\sqrt{3}.$ and find all triples $(x,y,z)$ for which equality holds.

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 $f(2023)/f(2022)$.