Prove that if $(a,b,c,d)$ are positive integers such that $(a+2^{\frac13}b+2^{\frac23}c)^2=d$ then $d$ is a perfect square (i.e is the square of a positive integer).

Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\frac{1}{2}$-mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A? $\textbf{(A)}\ 2 \frac{3}{4} \qquad\textbf{(B)}\ 3 \frac{3}{4} \qquad\textbf{(C)}\ 4 \frac{1}{2} \qquad\textbf{(D)}\ 5 \frac{1}{2} \qquad\textbf{(E)}\ 6 \frac{3}{4}$

Let $\triangle ABC$ be an isosceles triangle such that $AB=AC$. Let $\omega$ be a circle centered at $A$ with a radius strictly less than $AB$. Draw a tangent from $B$ to $\omega$ at $P$, and draw a tangent from $C$ to $\omega$ at $Q$. Suppose that the line $PQ$ intersects the line $BC$ at point $M$. Prove that $M$ is the midpoint of $BC$.

There are $n$ boyfriend-girlfriend pairs at a party. Initially all the girls sit at a round table. For the first dance, each boy invites one of the girls to dance with.After each dance, a boy takes the girl he danced with to her seat, and for the next dance he invites the girl next to her in the counterclockwise direction. For which values of $n$ can the girls be selected in such a way that in every dance at least one boy danced with his girlfriend, assuming that there are no less than $n$ dances?

Let H be an infinite dimensional, separable, complex Hilbert space and denote $\cal B (\cal H)$ the $\cal H$-algebra of its bounded linear operators. Consider the algebras $l_{\infty} ({\Bbb N}, \cal B (\cal H) ) = $ $\{ (a_n) | A_n \in\cal B (\cal H)$ $(n \in {\Bbb N}), \sup_n ||A_n|| <\infty \}$ $C(\beta {\Bbb N}, \cal B (\cal H) )$ = $\{ f: \beta {\Bbb N} \to \cal B (\cal H)|$ f is continuous $\}$ with pointwise operations and supremum norm. Show that these C*-algebras are not isometrically isomorphic. (Here, $\beta {\Bbb N}$ denotes the Stone-Cech compactification of the set of natural numbers.)

[i]a)[/i] Let $M$ be the set of the lengths of the edges of an octahedron whose sides are congruent quadrangles. Prove that $M$ has at most three elements. [i]b)[/i] Let an octahedron whose sides are congruent quadrangles be given. Prove that each of these quadrangles has two equal sides meeting at a common vertex.

Solve following system equations: \[\left\{ \begin{array}{c} 3x+4y=26\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \sqrt{x^2+y^2-4x+2y+5}+\sqrt{x^2+y^2-20x-10y+125}=10\ \end{array} \right.\ \ \]

Let $x \leq y \leq z \leq t$ be real numbers such that $xy + xz + xt + yz + yt + zt = 1.$ [b]a)[/b] Prove that $xt < \frac 13,$ b) Find the least constant $C$ for which inequality $xt < C$ holds for all possible values $x$ and $t.$

Nine numbers are arranged in a square table: $a_1 \,\,\, a_2 \,\,\,a_3$ $b_1 \,\,\,b_2 \,\,\,b_3$ $c_1\,\,\, c_2 \,\,\,c_3$ . It is known that the six numbers obtained by summing the rows and columns of the table are equal: $a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = c_1 + c_2 + c_3 = a_1 + b_1 + c_1 = a_2 + b_2 + c_2 = a_3 + b_3 + c_3$ . Prove that the sum of products of numbers in the rows is equal to the sum of products of numbers in the columns: $a_1 b_1 c_1 + a_2 b_2c_2 + a_3b_3c_3 = a_1a_2a_3 + b_1 b_2 b_3 + c_1 c_2c_3$ . (V Proizvolov)

Pepito's mother wants to prepare $n$ packages of $3$ candies to give away at the birthday party, and for this she will buy assorted candies of $3$ different flavors. You can buy any number of candies but you can't choose how many of each taste. She wants to put one candy of each flavor in each package, and if this is not possible she will use only candy of one flavor and all the packages will have $3$ candies of that flavor. Determine the least number of candies that must be purchased in order to assemble the n packages. He explains why if he buys fewer candies, he is not sure that he will be able to assemble the packages the way he wants.

Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$. Titu Andreescu, Mathematics Department, College of Texas, USA

A circle with center $I$ touches sides $AB,BC,CA$ of triangle $ABC$ in points $C_{1},A_{1},B_{1}$. Lines $AI, CI, B_{1}I$ meet $A_{1}C_{1}$ in points $X, Y, Z$ respectively. Prove that $\angle Y B_{1}Z = \angle XB_{1}Z$.

Let $f:[0,1]\rightarrow R$ be a continuous function and n be an integer number,n>0.Prove that $\int_0^1f(x)dx \le (n+1)*\int_0^1 x^n*f(x)dx $

Find the greatest positive integer $A$ with the following property: For every permutation of $\{1001,1002,...,2000\}$ , the sum of some ten consecutive terms is great than or equal to $A$.

For each positive integer $n$ let \[x_n=p_1+\cdots+p_n\] where $p_1,\ldots,p_n$ are the first $n$ primes. Prove that for each positive integer $n$, there is an integer $k_n$ such that $x_n<k_n^2<x_{n+1}$

Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^{\circ}$ and with incenter $I$ and circumcenter $O$. Let $H$ be the point diametrically opposite(antipode) to $O$ in the circumcircle of $\triangle BOC$. Prove that $IH=BI+IC$.

Let $n$ points be given on a circle, and let $nk + 1$ chords between these points be drawn, where $2k+1 < n$. Show that it is possible to select $k+1$ of the chords so that no two of them intersect.

In base ten, the number $100! = 100 \cdot 99 \cdot 98 \cdot 97... 2 \cdot 1$ has $158$ digits, and the last $24$ digits are all zeros. Find the number of zeros there are at the end of this same number when it is written in base $24$.

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

On sides of convex quadrilateral $ABCD$ on external side constructed equilateral triangles $ABK, BCL, CDM, DAN$. Let $P,Q$- midpoints of $BL, AN$ respectively and $X$- circumcenter of $CMD$. Prove, that $PQ$ perpendicular to $KX$

[b]p1.[/b] A right triangle has hypotenuse of length $12$ cm. The height corresponding to the right angle has length $7$ cm. Is this possible? [img]https://cdn.artofproblemsolving.com/attachments/0/e/3a0c82dc59097b814a68e1063a8570358222a6.png[/img] [b]p2.[/b] Prove that from any $5$ integers one can choose $3$ such that their sum is divisible by $3$. [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a knight on an arbitrary square. Then the second player can put another knight on a free square that is not controlled by the first knight. Then the first player can put a new knight on a free square that is not controlled by the knights on the board. Then the second player can do the same, etc. A player who cannot put a new knight on the board loses the game. Who has a winning strategy? [b]p4.[/b] Consider a regular octagon $ABCDEGH$ (i.e., all sides of the octagon are equal and all angles of the octagon are equal). Show that the area of the rectangle $ABEF$ is one half of the area of the octagon. [img]https://cdn.artofproblemsolving.com/attachments/d/1/674034f0b045c0bcde3d03172b01aae337fba7.png[/img] [b]p5.[/b] Can you find a positive whole number such that after deleting the first digit and the zeros following it (if they are) the number becomes $24$ times smaller? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of an acute angled triangle $ABC$. On the side $BC$ point $K$ is taken such that $\angle BB_1K=\angle A$. On the side $AB$ a point $M$ is taken such that $\angle BB_1M\angle C$. Let $L$ be the intersection of $BB_1$ and $A_1C_1$. Prove that the quadrilateral $B_1KLM$ is circumscribed. [I]Proposed by A. Khrabrov, D. Rostovski[/i]

With $76$ tiles, of which some are white, other blue and the remaining red, they form a rectangle of $4 \times 19$. Show that there is a rectangle, inside the largest, that has its vertices of the same color.

Let $x,y,z$ be real numbers such that the numbers $$\frac{1}{|x^2+2yz|},\quad\frac{1}{|y^2+2zx|},\quad\frac{1}{|z^2+2xy|}$$ are lengths of sides of a (non-degenerate) triangle. Determine all possible values of $xy+yz+zx$.

Cozi makes a two-way table on chalkboard describing the right or left hand usage of students and teachers in her school. However, when she returns to the chalkboard from lunch, she is dismayed to find that most of the numbers on her table have been erased, leaving behind: \begin{tabular}{c c c c} 5 & ? & ? & Total \\ ? & ? & 6 & Total \\ ? & 11 & ? & \\ Total & Total & & \\ \end{tabular} Fortunately, Cozi remembers that the difference between two of the missing numbers is equal to $12.$ Which of the following could be the total number of students and teachers on the table? $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$