Found problems: 85335
Given a triangle $ABC$ and points $D, E, F$, which are points of contact of the inscribed circle to the sides of the triangle.
i) Prove that $\frac{2pr}{R} \le DE + EF + DF \le p$
($p$ is the semiperimeter, $r$ and $R$ are respectively the radius of the inscribed and circumscribed circle of $\vartriangle ABC$).
ii). Find out when equality is achieved.
A rectangle $3 \times 5$ is divided into $15$ $1 \times 1$ cells. The middle $3$ cells that have no common points with the border of the rectangle are deleted. Is it possible to put in the remaining $12$ cells numbers $1, 2, \ldots, 12$ in some order, so that the sums of the numbers in the cells along each of the four sides of the rectangle are equal?
[i]Proposed by Mariia Rozhkova[/i]
The [i]traveler ant[/i] is walking over several chess boards. He only walks vertically and horizontally through the squares of the boards and does not pass two or more times over the same square of a board.
a) In a $4$x$4$ board, from which squares can he begin his travel so that he can pass through all the squares of the board?
b) In a $5$x$5$ board, from which squares can he begin his travel so that he can pass through all the squares of the board?
c) In a $n$x$n$ board, from which squares can he begin his travel so that he can pass through all the squares of the board?
In Sikinia, there are $2024$ cities. Between some of them there are flight connections, which can be used in either direction. No city has a direct flight to all $2023$ other cities. It is known, however, that there is a positive integer $n$ with the following property: For any $n$ cities in Sikinia, there is another city which is directly connected to all these cities.
Determine the largest possible value of $n$.
In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m \cdot n$ and $n \cdot m$ as the same product.
Let $ABC$ be a triangle inscribed in circle $C(O,R)$. Let $M$ ba apoint on the arc $BC$ . Let $D,E,Z$ be the feet of the perpendiculars drawn from $M$ on lines $AB,AC,BC$ respectively. Prove that $\frac{(BC)^2}{(MZ)^2} \ge 8\frac{R U_a}{(MD)\cdot(ME)}$ where $U_a$ is the altitude drawn on $BC$.
Let $n$ be a natural number. There are $6n + 4$ mathematicians at the conference. $2n+1$ meetings are held. On every meeting mathematicians are sitting at one table with $4$ chairs and $n$ tables with $6$ chairs.
Distances between each two adjacent chairs are equal. Two mathematicians sit in $special$ position if they sit at the same table and they are adjacent or diametrically opposite. For which natural numbers $n$ is possible that after end of all meetings every 2 mathematicians are sitting at the $special$ position less than 2 times.
Let $CD$ be a diameter of circle $K$. Let $AB$ be a chord that is parallel to $CD$. The line segment $AE$, with $E$ on $K$, is parallel to $CB$; $F$ is the point of intersection of line segments $AB$ and $DE$. The line segment $FG$, with $G$ on $DC$, extended is parallel to $CB$. Is $GA$ tangent to $K$ at point $A \?$
In a triangle $ABC$, let $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ respectively. Let $D'$, $E'$ and $F'$ be the second intersection of lines $AD$, $BE$ and $CF$ with the circumcircle of $ABC$. Show that the triangles $DEF$ and $D'E'F'$ are similar.
If $ a\ge b>1$, what is the largest possible value of $ \log_a(a/b)\plus{}\log_b(b/a)$?
$ \textbf{(A)}\ \minus{}2 \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
$600$ integer numbers from $[1,1000]$ colored in red. Natural segment $[n,k]$ is called yummy if for every natural $t$ from $[1,k-n]$ there are two red numbers $a,b$ from $[n,k]$ and $b-a=t$ .
Prove that there is yummy segment with $[a,b]$ with $b-a \geq 199$
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$
[i]Proposed by Liam Baker, South Africa[/i]
Given a cube in which you can put two massive spheres of radius 1.
What's the smallest possible value of the side - length of the cube?
Prove that your answer is the best possible.
The vertices of a regular hexagon $A_1,A_2,...,A_6$ lie respectively on the sides $B_1B_2$, $B_2B_3$, $B_3B_4$, $B_4B_5$, $B_5B_6$, $B_6B_1$ of a convex hexagon $B_1B_2B_3B_4B_5B_6$. Prove that
$$S_{B_1B_2B_3B_4B_5B_6} \le \frac32 S_{A_1A_2A_3A_4A_5A_6}.$$
Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$.
Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.
Let $Q$ be any point on a circle and let $P_1P_2P_3...P_8$ be a regular inscribed octagon. Prove that the sum of the fourth powers of the distances from $Q$ to the diameters $P_1P_5$, $P_2P_6$, $P_3P_7$, $P_4P_8$ is independent of the position of $Q$.
The bisectors of the angles $\sphericalangle CAB$ and $\sphericalangle BCA$ intersect the circumcircle of $ABC$ in $P$ and $Q$ respectively. These bisectors intersect each other in point $I$. Prove that $PQ \perp BI$.
A table has $3 000 000$ rows and $100$ columns, divided into unit squares. Each row contains the numbers from $1$ to $100$, each exactly once, and no two rows are the same. Above each column, the number of distinct entries in that column is written in red. Find the smallest possible sum of the red numbers.
Paint the grid points of $L=\{0,1,\dots,n\}^2$ with red or green in such a way that every unit lattice square in $L$ has exactly two red vertices. How many such colorings are possible?
There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,2,3,\cdots,n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that
(1) for each $k$, the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i+n$ is seat $i$);
(2) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.
Find the number of possible values of $n$ with $1<n<1000$.
Consider the isosceles right triangle $ABC$ ($AB = AC$) and the points $M, P \in [AB]$ so that $AM = BP$. Let $D$ be the midpoint of the side $BC$ and $R, Q$ the intersections of the perpendicular from $A$ on$ CM$ with $CM$ and $BC$ respectively. Prove that
a) $\angle AQC = \angle PQB$
b) $\angle DRQ = 45^o$
[b]p1.[/b] Fluffy the Dog is an extremely fluffy dog. Because of his extreme fluffiness, children always love petting Fluffy anywhere. Given that Fluffy likes being petted $1/4$ of the time, out of $120$ random people who each pet Fluffy once, what is the expected number of times Fluffy will enjoy being petted?
[b]p2.[/b] Andy thinks of four numbers $27$, $81$, $36$, and $41$ and whispers the numbers to his classmate Cynthia. For each number she hears, Cynthia writes down every factor of that number on the whiteboard. What is the sum of all the different numbers that are on the whiteboard? (Don't include the same number in your sum more than once)
[b]p3.[/b] Charles wants to increase the area his square garden in his backyard. He increases the length of his garden by $2$ and increases the width of his garden by $3$. If the new area of his garden is $182$, then what was the original area of his garden?
[b]p4.[/b] Antonio is trying to arrange his flute ensemble into an array. However, when he arranges his players into rows of $6$, there are $2$ flute players left over. When he arranges his players into rows of $13$, there are $10$ flute players left over. What is the smallest possible number of flute players in his ensemble such that this number has three prime factors?
[b]p5.[/b] On the AMC $9$ (Acton Math Competition $9$), $5$ points are given for a correct answer, $2$ points are given for a blank answer and $0$ points are given for an incorrect answer. How many possible scores are there on the AMC $9$, a $15$ problem contest?
[b]p6.[/b] Charlie Puth produced three albums this year in the form of CD's. One CD was circular, the second CD was in the shape of a square, and the final one was in the shape of a regular hexagon. When his producer circumscribed a circle around each shape, he noticed that each time, the circumscribed circle had a radius of $10$. The total area occupied by $1$ of each of the different types of CDs can be expressed in the form $a + b\pi + c\sqrt{d}$ where $d$ is not divisible by the square of any prime. Find $a + b + c + d$.
[b]p7.[/b] You are picking blueberries and strawberries to bring home. Each bushel of blueberries earns you $10$ dollars and each bushel of strawberries earns you $8$ dollars. However your cart can only fit $24$ bushels total and has a weight limit of $100$ lbs. If a bushel of blueberries weighs $8$ lbs and each bushel of strawberries weighs $6$ lbs, what is your maximum profit. (You can only pick an integer number of bushels)
[b]p8.[/b] The number $$\sqrt{2218 + 144\sqrt{35} + 176\sqrt{55} + 198\sqrt{77}}$$ can be expressed in the form $a\sqrt5 + b\sqrt7 + c\sqrt{11}$ for positive integers $a, b, c$. Find $abc$.
[b]p9.[/b] Let $(x, y)$ be a point such that no circle passes through the three points $(9,15)$, $(12, 20)$, $(x, y)$, and no circle passes through the points $(0, 17)$, $(16, 19)$, $(x, y)$. Given that $x - y = -\frac{p}{q}$ for relatively prime positive integers $p$, $q$, Find $p + q$.
[b]p10.[/b] How many ways can Alfred, Betty, Catherine, David, Emily and Fred sit around a $6$ person table if no more than three consecutive people can be in alphabetical order (clockwise)?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
The circumcenter of a triangle lies on its incircle. Prove that the ratio of its greatest and smallest sides is less than two.
[i]Proposed by B.Frenkin[/i]
Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that
\[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]
Let $a,b,c,d$ be positive integers satisfying \[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\] Determine all possible values of $a+b+c+d$.