Found problems: 85335
Convex equiangular hexagon $ABCDEF$ has $AB = CD = EF = \sqrt 3$ and $BC = DE = FA = 2.$ Points $X, Y,$ and $Z$ are situated outside the hexagon such that $AEX, ECY,$ and $CAZ$ are all equilateral triangles. Compute the area of the region bounded by lines $XF, YD, $ and $ZB.$
Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer.
Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers.
[i]Proposed by Matthew Babbitt[/i]
Given that $\sqrt{10} \approx 3.16227766$, find the largest integer $n$ such that $n^2 \leq 10,000,000$.
[i]Proposed by Jacob Stavrianos[/i]
The figure shows a glass prism which is partially filled with liquid. The surface of the prism consists of two isosceles right triangles, two squares with side length $10$ cm and a rectangle. The prism can lie in three different ways. If the prism lies as shown in figure $1$, the height of the liquid is $5$ cm.
[img]https://cdn.artofproblemsolving.com/attachments/4/2/cda98a00f8586132fe519855df123534516b50.png[/img]
a) What is the height of the liquid when it lies as shown in figure $2$?
b) What is the height of the liquid when it lies as shown in figure$ 3$?
Initially, the numbers $1$ and $2$ are written on the board. A move consists of choosing a positive real number $x$ and replacing $(a,b)$ on the board by $\left(a+\frac{x}{b},b+\frac{x}{a}\right)$. Is it possible to create in finitely many moves a situation where the numbers on the board are $2$ and $3$?
Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$
A dealer bought $n$ radios for $d$ dollars, $d$ a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $\$8$ on each radio sold. If the overall profit was $\$72$, then the least possible value of $n$ for the given information is:
$\textbf{(A)}\ 18\qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ 15\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 11$
A set of lines in the plane is in [i]general position[/i] if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its [i]finite regions[/i]. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary.
[i]Note[/i]: Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.
A student is playing computer. Computer shows randomly 2002 positive numbers. Game's rules let do the following operations
- to take 2 numbers from these, to double first one, to add the second one and to save the sum.
- to take another 2 numbers from the remainder numbers, to double the first one, to add the second one, to multiply this sum with previous and to save the result.
- to repeat this procedure, until all the 2002 numbers won't be used.
Student wins the game if final product is maximum possible.
Find the winning strategy and prove it.
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the trangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is
$\text{(A)}\ \frac1{2m+1}\qquad\text{(B)}\ m \qquad\text{(C)}\ 1-m\qquad\text{(D)}\ \frac1{4m} \qquad\text{(E)}\ \frac1{8m^2}$
For each positive real number $r$, define $a_0(r) = 1$ and $a_{n+1}(r) = \lfloor ra_n(r) \rfloor$ for all integers $n \ge 0$.
(a) Prove that for each positive real number $r$, the limit
\[ L(r) = \lim_{n \to \infty} \frac{a_n(r)}{r^n} \]
exists.
(b) Determine all possible values of $L(r)$ as $r$ varies over the set of positive real numbers.
[i]Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.[/i]
Define $\left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2$ for every two points $A = (x_A, y_A)$ and $B = (x_B, y_B)$ in the plane.
Let $S$ be the set of points $(x,y)$ in the plane for which $x,y \in \left\{ 0,1,\dots,100 \right\}$.
Find the number of functions $f : S \to S$ such that $\left\lVert A-B \right\rVert \equiv \left\lVert f(A)-f(B) \right\rVert \pmod{101}$ for any $A, B \in S$.
[i] Proposed by Victor Wang [/i]
Find all positive integers $n$ for which there exists a positive integer $k$ such that the decimal representation of $n^k$ starts and ends with the same digit.
Is it possible to write the numbers from $1$ to $100$ in the cells of a of a $10 \times 10$ square so that:
1. Each cell contains exactly one number;
2. Each number is written exactly once;
3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the
original $10 \times 10$ square, the numbers in them must have the same parity.
The figure below shows examples of such pairs of cells, in which the numbers must have the same parity.
[img]https://i.ibb.co/b3P8t36/Kyiv-MO-2024-7-2.png[/img]
[i]Proposed by Mykhailo Shtandenko[/i]
Prove for all positive reals a,b,c,d:
$ \frac{a\minus{}b}{b\plus{}c}\plus{}\frac{b\minus{}c}{c\plus{}d}\plus{}\frac{c\minus{}d}{d\plus{}a}\plus{}\frac{d\minus{}a}{a\plus{}b} \geq 0$
There are balls that look identical, but their weights all di er by a little. We have a balance that can compare only two balls at a time. What is the minimum number of times, in the worst case, we
have to use to balance to rank all balls by weight?
Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.
[b]p1.[/b] Suppose $5$ bales of hay are weighted two at a time in all possible ways. The weights obtained are $110$, $112$, $113$, $114$, $115$, $116$, $117$, $118$, $120$, $121$. What is the difference between the heaviest and the lightest bale?
[b]p2.[/b] Paul and Paula are playing a game with dice. Each have an $8$-sided die, and they roll at the same time. If the number is the same they continue rolling; otherwise the one who rolled a higher number wins. What is the probability that the game lasts at most $3$ rounds?
[b]p3[/b]. Find the unique positive integer $n$ such that $\frac{n^3+5}{n^2-1}$ is an integer.
[b]p4.[/b] How many numbers have $6$ digits, some four of which are $2, 0, 1, 4$ (not necessarily consecutive or in that order) and have the sum of their digits equal to $9$?
[b]p5.[/b] The Duke School has $N$ students, where $N$ is at most $500$. Every year the school has three sports competitions: one in basketball, one in volleyball, and one in soccer. Students may participate in all three competitions. A basketball team has $5$ spots, a volleyball team has $6$ spots, and a soccer team has $11$ spots on the team. All students are encouraged to play, but $16$ people choose not to play basketball, $9$ choose not to play volleyball and $5$ choose not to play soccer. Miraculously, other than that all of the students who wanted to play could be divided evenly into teams of the appropriate size. How many players are there in the school?
[b]p6.[/b] Let $\{a_n\}_{n\ge 1}$ be a sequence of real numbers such that $a_1 = 0$ and $a_{n+1} =\frac{a_n-\sqrt3}{\sqrt3 a_n+1}$ . Find $a_1 + a_2 +.. + a_{2014}$.
[b]p7.[/b] A soldier is fighting a three-headed dragon. At any minute, the soldier swings her sword, at which point there are three outcomes: either the soldier misses and the dragon grows a new head, the soldier chops off one head that instantaneously regrows, or the soldier chops off two heads and none grow back. If the dragon has at least two heads, the soldier is equally likely to miss or chop off two heads. The dragon dies when it has no heads left, and it overpowers the soldier if it has at least five heads. What is the probability that the soldier wins
[b]p8.[/b] A rook moves alternating horizontally and vertically on an infinite chessboard. The rook moves one square horizontally (in either direction) at the first move, two squares vertically at the second, three horizontally at the third and so on. Let $S$ be the set of integers $n$ with the property that there exists a series of moves such that after the $n$-th move the rock is back where it started. Find the number of elements in the set $S \cap \{1, 2, ..., 2014\}$.
[b]p9.[/b] Find the largest integer $n$ such that the number of positive integer divisors of $n$ (including $1$ and $n$) is at least $\sqrt{n}$.
[b]p10.[/b] Suppose that $x, y$ are irrational numbers such that $xy$, $x^2 + y$, $y^2 + x$ are rational numbers. Find $x + y$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Are there any (not necessarily positive) integers $m$ and $n$ such that
a) $\frac{1}{m}-\frac{1}{n}=\frac{1}{m-n}$ ?
b) $\frac{1}{m}-\frac{1}{n}=\frac{1}{n-m}$
In triangle $ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals:
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ \text{none of these} $
The product of 22 integers is 1. Show that their sum can not be 0.
Let $E$ be an ellipse where the length of the major axis is $26$, the length of the minor axis is $24$, and the foci are at points $R$ and $S$. Let $A$ and $B$ be points on the ellipse such that $RASB$ forms a non-degenerate quadrilateral, lines $RA$ and $SB$ intersect at $P$ with segment $PR$ containing $A$, and lines $RB$ and $AS$ intersect at Q with segment $QR$ containing $B$. Given that $RA = AS$, $AP = 26$, the perimeter of the non-degenerate quadrilateral $RP SQ$ is $m +\sqrt{n}$, where $m$ and $n$ are integers. Compute $m + n$.
Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.
Hexagon $ABCDEF$ has an inscribed circle $\Omega$ that is tangent to each of its sides. If $AB = 12$, $\angle FAB = 120^o$, and $\angle ABC = 150^o$, and if the radius of $\Omega$ can be written as $m +\sqrt{n}$ for positive integers $m, n$, find $m + n$.
Let $\left \{ x_n \right \} _{n\ge 1}$ and $\left \{ y_n \right \} _{n\ge 1}$ be two infinite sequences of integers. Prove that there exists an infinite sequence of integers $\left \{ z_n \right \} _{n\ge 1}$ such that for any positive integer \( n \), the following holds:
\[
\sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right).
\]