Does \[\left\{ \begin{array}{l} x^y = z \\ y^z = x \\ z^x = y \\ \end{array} \right. \] have any solutions in positive reals apart from $x = y = z= 1$?

In a narrow alley of width $w$ a ladder of length $a$ is placed with its foot at point $P$ between the walls. Resting against one wall at $Q$, the distance $k$ above the ground makes a $45^\circ$ angle with the ground. Resting against the other wall at $R$, a distance $h$ above the ground, the ladder makes a $75^\circ$ angle with the ground. The width $w$ is equal to $\textbf {(A) } a \qquad \textbf {(B) } RQ \qquad \textbf {(C) } k \qquad \textbf {(D) } \frac{h+k}{2} \qquad \textbf {(E) } h$

You play a game with a biased coin, which has probability $\tfrac{3}{4}$ of landing heads. Each time you toss heads, you score $1$ point, while tossing tails earns no points. After any turn, you can stop playing the game and keep the points you currently have. However, if you are still playing when you toss tails for the second time, you lose all of your points. If you play to maximize your expected score, what is your expected score from playing this game?

Let $n\ge2$ be a positive integer. Given a sequence $\left(s_i\right)$ of $n$ distinct real numbers, define the "class" of the sequence to be the sequence $\left(a_1,a_2,\ldots,a_{n-1}\right)$, where $a_i$ is $1$ if $s_{i+1} > s_i$ and $-1$ otherwise. Find the smallest integer $m$ such that there exists a sequence $\left(w_i\right)$ of length $m$ such that for every possible class of a sequence of length $n$, there is a subsequence of $\left(w_i\right)$ that has that class. [i]David Yang.[/i]

Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$. Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.

In a lottery there are $14$ balls, numbered from $1$ to $14$. Four of these balls are drawn at random. D'Angelo wins the lottery if he can split the four balls into two disjoint pairs, where the two balls in each pair have difference at least $5$. The probability that D'Angelo wins the lottery can be expressed as $\frac{m}{n}$, with $m,n$ relatively prime. Find $m+n$. [i]Proposed by Richard Chen[/i]

[i]20 problems for 20 minutes. [/i] [b]p1.[/b] Evaluate $\frac{\sqrt2 \cdot \sqrt6}{\sqrt3}.$ [b]p2.[/b] If $6\%$ of a number is $1218$, what is $18\%$ of that number? [b]p3.[/b] What is the median of $\{42, 9, 8, 4, 5, 1,13666, 3\}$? [b]p4.[/b] Define the operation $\heartsuit$ so that $i\heartsuit u = 5i - 2u$. What is $3\heartsuit 4$? p5. How many $0.2$-inch by $1$-inch by $1$-inch gold bars can fit in a $15$-inch by $12$-inch by $9$-inch box? [b]p6.[/b] A tetrahedron is a triangular pyramid. What is the sum of the number of edges, faces, and vertices of a tetrahedron? [b]p7.[/b] Ron has three blue socks, four white socks, five green socks, and two black socks in a drawer. Ron takes socks out of his drawer blindly and at random. What is the least number of socks that Ron needs to take out to guarantee he will be able to make a pair of matching socks? [b]p8.[/b] One segment with length $6$ and some segments with lengths $10$, $8$, and $2$ form the three letters in the diagram shown below. Compute the sum of the perimeters of the three figures. [img]https://cdn.artofproblemsolving.com/attachments/1/0/9f7d6d42b1d68cd6554d7d5f8dd9f3181054fa.png[/img] [b]p9.[/b] How many integer solutions are there to the inequality $|x - 6| \le 4$? [b]p10.[/b] In a land for bad children, the flavors of ice cream are grass, dirt, earwax, hair, and dust-bunny. The cones are made out of granite, marble, or pumice, and can be topped by hot lava, chalk, or ink. How many ice cream cones can the evil confectioners in this ice-cream land make? (Every ice cream cone consists of one scoop of ice cream, one cone, and one topping.) [b]p11.[/b] Compute the sum of the prime divisors of $245 + 452 + 524$. [b]p12.[/b] In quadrilateral $SEAT$, $SE = 2$, $EA = 3$, $AT = 4$, $\angle EAT = \angle SET = 90^o$. What is the area of the quadrilateral? [b]p13.[/b] What is the angle, in degrees, formed by the hour and minute hands on a clock at $10:30$ AM? [b]p14.[/b] Three numbers are randomly chosen without replacement from the set $\{101, 102, 103,..., 200\}$. What is the probability that these three numbers are the side lengths of a triangle? [b]p15.[/b] John takes a $30$-mile bike ride over hilly terrain, where the road always either goes uphill or downhill, and is never flat. If he bikes a total of $20$ miles uphill, and he bikes at $6$ mph when he goes uphill, and $24$ mph when he goes downhill, what is his average speed, in mph, for the ride? [b]p16.[/b] How many distinct six-letter words (not necessarily in any language known to man) can be formed by rearranging the letters in $EXETER$? (You should include the word EXETER in your count.) [b]p17.[/b] A pie has been cut into eight slices of different sizes. Snow White steals a slice. Then, the seven dwarfs (Sneezy, Sleepy, Dopey, Doc, Happy, Bashful, Grumpy) take slices one by one according to the alphabetical order of their names, but each dwarf can only take a slice next to one that has already been taken. In how many ways can this pie be eaten by these eight persons? [b]p18.[/b] Assume that $n$ is a positive integer such that the remainder of $n$ is $1$ when divided by $3$, is $2$ when divided by $4$, is $3$ when divided by $5$, $...$ , and is $8$ when divided by $10$. What is the smallest possible value of $n$? [b]p19.[/b] Find the sum of all positive four-digit numbers that are perfect squares and that have remainder $1$ when divided by $100$. [b]p20.[/b] A coin of radius $1$ cm is tossed onto a plane surface that has been tiled by equilateral triangles with side length $20\sqrt3$ cm. What is the probability that the coin lands within one of the triangles? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Petya digs the garden bed alone for $a$ minutes longer than he does with Vasya. Vasya digs up the same bed for $b$ minutes longer than he would have done with Petya. How many minutes does it take Vasya and Petya to dig up the same bed together? orthogonal).

Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\] \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]

King Albrecht founded a family. In the family everyone has exactly $ 8$ children. The only, but really important rule is that among the grandchildren of any person at most $x$ can be named Bela. (None of Albrecht’s children is called Bela.) For which $x$ is it possible that after a certain time each newborn in the family has at least one direct ancestor in the Royal family called Bela. No two of Albrecht’s descendants (including himself) have a common child.

Let $x_1, x_2, ... , x_n$ and $y_1, y_2, ..., y_n$ be positive real numbers so that $$x_1 + x_2 + ...+ x_n \ge x_1y_1 + x_2y_2 + ... + x_ny_n.$$ Show that for any non-negative integer $p$ the following inequality holds $$\frac{x_1}{y_1^p} +\frac{ x_2}{y_2^p} + ...+ \frac{x_n}{y_n^p} \ge x_1 + x_2 + ...+ x_n.$$

Find the exact value of $\sin 36^o$.

Determine graphically the number of roots of the equation $\sin x = \lg x$.

Let $a$, $b$, $c$ be the sides of an acute triangle $\triangle ABC$ , then for any $x, y, z \geq 0$, such that $xy+yz+zx=1$ holds inequality:$$a^2x + b^2y + c^2z \geq 4F$$ where $F$ is the area of the triangle $\triangle ABC$

A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of $280$ pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad? $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 96 $

Show that there is no integer-valued function on the integers such that $f(m+f(n))=f(m)-n$ for all $m,n$.

When a fair six-sided dice is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the $5$ faces than can be seen is divisible by $6$? $\textbf{(A)}\ 1/3 \qquad \textbf{(B)}\ 1/2 \qquad \textbf{(C)}\ 2/3 \qquad \textbf{(D)}\ 5/6 \qquad \textbf{(E)}\ 1$

(a) Given $2 + 4 + 6 + \dots + p = 6480$, find $p$. (b) Given $7 + 11 + 15 + \dots + q = 5250$, find $q$. (c) Given $2^2 + 4^2 + 6^2 + \dots + r^2 - 1^2 - 3^2 - 5^2 - \dots - (r-1)^2 = 2485$, compute $r$.

Daniel painted a rectangular painting measuring $2$ meters by $4$ meters with four colors. Knowing that he used more than two colors to paint the four corners of the painting, prove that he painted of the same color two points that are at least $\sqrt5$ meters

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

Let $ABC$ be a right-angled triangle with hypotenuse $AB$. Denote by $D$ the foot of the altitude from $C$. Let $Q, R$, and $P$ be the midpoints of the segments $AD, BD$, and $CD$, respectively. Prove that $\angle AP B + \angle QCR = 180^o$. Czech Republic

On the board is written a number with nine non-zero and distinct digits. Prove that we can delete at most seven digits so that the number formed by the digits left to be a perfect square.

Let $p$ be the product of $n$ real numbers $x_1$, $x_2$,$...$, $x_n$. Prove that if $p - x_k$ is an odd integer for $k = 1, 2,..., n$, then each of the numbers $x_1$, $x_2$,$...$, $x_n$is irrational. (G Galperin)

Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that [i](a)[/i] $ f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n$ for all integers $ m$ and $ n;$ [i](b)[/i] $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \mathbb{Z}.$

We say that a rectangle and a triangle are [i]similar[/i], if they have the same area and the same perimeter. Let $P$ be a rectangle for which the ratio of the longer to the shorter side is at least $\lambda -1+\sqrt{\lambda (\lambda -2)}$ where $\lambda =\frac{3\sqrt{3}}{2}$. Prove that there exists a tringle that is [i]similar[/i] to $P$.