In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$

A square is divided into 169 identical small squares and in every small square is written 0 or 1. It isn’t allowed in one row or column to have the following arrangements of adjacent digits in this order: 101, 111 or 1001. What is the the biggest possible number of 1’s in the table?

How many lines pass through exactly two points in the following hexagonal grid? [img]https://cdn.artofproblemsolving.com/attachments/2/e/35741c80d0e0ee0ca56f1297b1e377c8db9e22.png[/img]

Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.

Let $x$ and $y$ be two real numbers such that $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$. Find $\tan^2 x + 2 \tan^2 y$.

The graph below shows a portion of the curve defined by the quartic polynomial $ P(x) \equal{} x^4 \plus{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$. Which of the following is the smallest? $ \textbf{(A)}\ P( \minus{} 1)$ $ \textbf{(B)}\ \text{The product of the zeros of }P$ $ \textbf{(C)}\ \text{The product of the non \minus{} real zeros of }P$ $ \textbf{(D)}\ \text{The sum of the coefficients of }P$ $ \textbf{(E)}\ \text{The sum of the real zeros of }P$ [asy] size(170); defaultpen(linewidth(0.7)+fontsize(7));size(250); real f(real x) { real y=1/4; return 0.2125(x*y)^4-0.625(x*y)^3-1.6125(x*y)^2+0.325(x*y)+5.3; } draw(graph(f,-10.5,19.4)); draw((-13,0)--(22,0)^^(0,-10.5)--(0,15)); int i; filldraw((-13,10.5)--(22,10.5)--(22,20)--(-13,20)--cycle,white, white); for(i=-3; i<6; i=i+1) { if(i!=0) { draw((4*i,0)--(4*i,-0.2)); label(string(i), (4*i,-0.2), S); }} for(i=-5; i<6; i=i+1){ if(i!=0) { draw((0,2*i)--(-0.2,2*i)); label(string(2*i), (-0.2,2*i), W); }} label("0", origin, SE);[/asy]

Let $\omega$ be the circumcircle of a triangle $ABC$ (${AB\ne AC}$), $I$ be the incenter, $P$ be the point on $\omega$ for which $\angle API=90^\circ,$ $S$ be the intersection point of lines $AP$ and $BC,$ $W$ be the intersection point of line $AI$ and $\omega.$ Line which passes through point $W$ orthogonally to $AW$ meets $AP$ and $BC$ at points $D$ and $E$ respectively. Prove that $SD=IE.$ (Ye. Azarov)

A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle? [asy] draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); draw((0,5)--(10,5)); draw((3,0)--(3,7)); label("6", (1.5,6)); label("?", (1.5,2.5)); label("14", (6.5,6)); label("35", (6.5,2.5)); [/asy] $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 25 $

Let $ABC$ be a triangle, and let $D, E$ and $F$ be the feet of the altitudes from $A, B$ and $C,$ respectively. A circle $\omega_A$ through $B$ and $C$ crosses the line $EF$ at $X$ and $X'$. Similarly, a circle $\omega_B$ through $C$ and $A$ crosses the line $FD$ at $Y$ and $Y',$ and a circle $\omega_C$ through $A$ and $B$ crosses the line $DE$ at $Z$ and $Z'$. Prove that $X, Y$ and $Z$ are collinear if and only if $X', Y'$ and $Z'$ are collinear. [i]Vlad Robu[/i]

There were $2n,$ $n\ge2,$ teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ all the teams could end up with the same non-zero number of points?

Show that there exists a $10 \times 10$ table of distinct natural numbers such that if $R_i$ is equal to the multiplication of numbers of row $i$ and $S_i$ is equal to multiplication of numbers of column $i$, then numbers $R_1$, $R_2$, ... , $R_{10}$ make a nontrivial arithmetic sequence and numbers $S_1$, $S_2$, ... , $S_{10}$ also make a nontrivial arithmetic sequence. (A nontrivial arithmetic sequence is an arithmetic sequence with common difference between terms not equal to $0$).

Consider a circular necklace with $\displaystyle{2013}$ beads. Each bead can be paintes either green or white. A painting of the necklace is called [i]good[/i] if among any $\displaystyle{21}$ successive beads there is at least one green bead. Prove that the number of good paintings of the necklace is odd. [b]Note.[/b] Two paintings that differ on some beads, but can be obtained from each other by rotating or flipping the necklace, are counted as different paintings. [i]Proposed by Vsevolod Bykov and Oleksandr Rybak, Kiev.[/i]

Let $n \geq 3$ be a fixed integer. The number $1$ is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$, replacing them with the numbers $1$ and $a+b$, then adding one stone to the first bucket and $\gcd(a, b)$ stones to the second bucket. After some finite number of moves, there are $s$ stones in the first bucket and $t$ stones in the second bucket, where $s$ and $t$ are positive integers. Find all possible values of the ratio $\frac{t}{s}$.

Let $\vartriangle ABC$ be a triangle such that $\angle ABC = 30^o$, $\angle ACB = 15^o$. Let $M$ be midpoint of segment $BC$ and point $N$ lies on segment $MC$, such that the length of $NC$ is equal to length of $AB$. Proce that $AN$ is the bisector of the angle $\angle MAC$. [img]https://cdn.artofproblemsolving.com/attachments/2/7/4c554b53f03288ee69931fdd2c6fbd3e27ab13.png[/img]

Two three-digit numbers are given. The hundreds digit of each of them is equal to the units digit of the other. Find these numbers if their difference is $297$ and the sum of digits of the smaller number is $23$.

[b]p1.[/b] For $25$ bagels they paid as many rubles as the number of bagels you can buy with a ruble. How much does one bagel cost? [b]p2.[/b] Cut the square into the figure into$ 4$ parts of the same shape and size so that each part contains exactly one shaded square. [img]https://cdn.artofproblemsolving.com/attachments/a/2/14f0d435b063bcbc55d3dbdb0a24545af1defb.png[/img] [b]p3.[/b] The numerator and denominator of the fraction are positive numbers. The numerator is increased by $1$, and the denominator is increased by $10$. Can this increase the fraction? [b]p4.[/b] The brother left the house $5$ minutes later than his sister, following her, but walked one and a half times faster than her. How many minutes after leaving will the brother catch up with his sister? [b]p5.[/b] Three apples are worth more than five pears. Can five apples be cheaper than seven pears? Can seven apples be cheaper than thirteen pears? (All apples cost the same, all pears too.) [b]p6.[/b] Give an example of a natural number divisible by $6$ and having exactly $15$ different natural divisors (counting $1$ and the number itself). [b]p7.[/b] In a round dance, $30$ children stand in a circle. Every girl's right neighbor is a boy. Half of the boys have a boy on their right, and all the other boys have a girl on their right. How many boys and girls are there in a round dance? [b]p8.[/b] A sheet of paper was bent in half in a straight line and pierced with a needle in two places, and then unfolded and got $4$ holes. The positions of three of them are marked in figure Where might the fourth hole be? [img]https://cdn.artofproblemsolving.com/attachments/c/8/53b14ddbac4d588827291b27c40e3f59eabc24.png[/img] [b]p9 [/b] The numbers 1$, 2, 3, 4, 5, _, 2000$ are written in a row. First, third, fifth, etc. crossed out in order. Of the remaining $1000 $ numbers, the first, third, fifth, etc. are again crossed out. They do this until one number remains. What is this number? [b]p10.[/b] On the number axis there lives a grasshopper who can jump $1$ and $4$ to the right and left. Can he get from point $1$ to point $2$ of the numerical axis in $1996$ jumps if he must not get to points with coordinates divisible by $4$ (points $0$, $\pm 4$, $\pm 8$ etc.)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by $$\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor$$ where $\left \lfloor \tfrac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\tfrac{x}{y}$. What is the value of $\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )$? $\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}$

Determine the real value of $t$ that minimizes the expression \[ \sqrt{t^2 + (t^2 - 1)^2} + \sqrt{(t-14)^2 + (t^2 - 46)^2}. \]

Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$.

The negation of the statement "No slow learners attend this school" is: $ \textbf{(A)}\ \text{All slow learners attend this school} \\ \textbf{(B)}\ \text{All slow learners do not attend this school} \\ \textbf{(C)}\ \text{Some slow learners attend this school} \\ \textbf{(D)}\ \text{Some slow learners do not attend this school} \\ \textbf{(E)}\ \text{No slow learners do not attend this school}$

For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts: $(1)$ two successive $15\%$ discounts $(2)$ three successive $10\%$ discounts $(3)$ a $25\%$ discount followed by a $5\%$ discount What is the smallest possible positive integer value of $n$? ${ \textbf{(A)}\ \ 27\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}}\ 31\qquad\textbf{(E)}\ 33$

The set $M$ consists of integers, the smallest of which is $1$ and the greatest $100$. Each member of $M$, except $1$, is the sum of two (possibly identical) numbers in $M$. Of all such sets, find one with the smallest possible number of elements.

It is given that $x = -2272$, $y = 10^3+10^2c+10b+a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, c$ are positive integers with $a < b < c$. Find $y.$

The straight line $y=ax+16$ intersects the graph of $y=x^3$ at $2$ distinct points. What is the value of $a$?

The Tower of Hanoi is a puzzle with $n$ disks of different sizes and $3$ vertical rods on it. All of the disks are initially placed on the leftmost rod, sorted by size such that the largest disk is on the bottom. On each turn, one may move the topmost disk of any nonempty rod onto any other rod, provided that it is smaller than the current topmost disk of that rod, if it exists. (For instance, if there were two disks on different rods, the smaller disk could move to either of the other two rods, but the larger disk could only move to the empty rod.) The puzzle is solved when all of the disks are moved to the rightmost rod. The specifications normally include an intelligent monk to move the disks, but instead there is a monkey making random moves (with each valid move having an equal probability of being selected). Given $64$ disks, what is the expected number of moves the monkey will have to make to solve the puzzle?