Can we arrange $5$ points in space to form a pentagon with equal sides such that the angle between each pair of adjacent edges is $90^o$?

[b]p1.[/b] (a) Brian has a big job to do that will take him two hours to complete. He has six friends who can help him. They all work at the same rate, somewhat slower than Brian. All seven working together can finish the job in $45$ minutes. How long will it take to do the job if Brian worked with only three of his friends? (b) Brian could do his next job in $N$ hours, working alone. This time he has an unlimited list of friends who can help him, but as he moves down the list, each friend works more slowly than those above on the list. The first friend would take $kN$ ($k > 1$) hours to do the job alone, the second friend would take $k^2N$ hours alone, the third friend would take $k^3N$ hours alone, etc. Theoretically, if Brian could get all his infinite number of friends to help him, how long would it take to complete the job? [b]p2.[/b] (a) The centers of two circles of radius $1$ are two opposite vertices of a square of side $1$. Find the area of the intersection of the two circles. (b) The centers of two circles of radius $1$ are two consecutive vertices of a square of side $1$. Find the area of the intersection of the two circles and the square. (c) The centers of four circles of radius $1$ are the vertices of a square of side $1$. Find the area of the intersection of the four circles. [b]p3.[/b] For any real number$ x$, $[x]$ denotes the greatest integer that does not exceed $x$. For example, $[7.3] = 7$, $[10/3] = 3$, $[5] = 5$. Given natural number $N$, denote as $f(N)$ the following sum of $N$ integers: $$f(N) = [N/1] + [N/2] + [N/3] + ... + [N/n].$$ (a) Evaluate $f(7) - f(6)$. (b) Evaluate $f(35) - f(34)$. (c) Evaluate (with explanation) $f(1996) - f(1995)$. [b]p4.[/b] We will say that triangle $ABC$ is good if it satisfies the following conditions: $AB = 7$, the other two sides are integers, and $\cos A =\frac27$. (a) Find the sides of a good isosceles triangle. (b) Find the sides of a good scalene (i.e. non-isosceles) triangle. (c) Find the sides of a good scalene triangle other than the one you found in (b) and prove that any good triangle is congruent to one of the three triangles you have found. [b]p5.[/b] (a) A bag contains nine balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. How many of the nine balls were of one color and how many of the other color? (b) A bag contains $N$ balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. It is also found that $180 < N < 220$. Find the exact value of $N$ and determine how many of the $N$ balls were of one color and how many of the other color. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a rectangle of centre $O$, such that $\angle DAC=60^{\circ}$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$, and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

Let $ABCD$ be a square of center $O$. The parallel to $AD$ through $O$ intersects $AB$ and $CD$ at $M$ and $N$ and a parallel to $AB$ intersects diagonal $AC$ at $P$. Prove that $$OP^4 + \left(\frac{MN}{2} \right)^4 = MP^2 \cdot NP^2$$

Determine all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ so that $\forall x: x\cdot f(\frac x2) - f(\frac2x) = 1$

Each of the $15$ coaches ranked the $50$ selected football players on the places from $1$ to $50$. For each football player, the highest and lowest obtained ranks differ by at most $5$. For each of the players, the sum of the ranks he obtained is computed, and the sums are denoted by $S_1\le S_2\le\ldots\le S_{50}$. Find the largest possible value of $S_1$.

There is a set of triangles, in each of which the smallest angle does not exceed $36^o$ . A new one is formed from these triangles according to the following rule: the smallest side of the new one is equal to the sum of the smallest sides of these triangles, its middle side is equal to the sum of the middle sides, and the largest is the sum of the largest ones. Prove that the sine of the smallest angle of the resulting triangle is less than $2 \sin 18^o$ .

Compute $ \int_0^1 x^{2n\plus{}1}e^{\minus{}x^2}dx\ (n\equal{}1,\ 2,\ \cdots)$ , then use this result, prove that $ \sum_{n\equal{}0}^{\infty} \frac{1}{n!}\equal{}e$.

Let $m, n,$ and $p$ be odd positive integers. Prove that the number $\sum\limits_{k=1}^{{{(n-1)}^{p}}}{{{k}^{m}}}$ is divisible by $n$

A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet? [asy] size(250);defaultpen(linewidth(0.8)); draw(ellipse(origin, 3, 1)); fill((3,0)--(3,2)--(-3,2)--(-3,0)--cycle, white); draw((3,0)--(3,16)^^(-3,0)--(-3,16)); draw((0, 15)--(3, 12)^^(0, 16)--(3, 13)); filldraw(ellipse((0, 16), 3, 1), white, black); draw((-3,11)--(3, 5)^^(-3,10)--(3, 4)); draw((-3,2)--(0,-1)^^(-3,1)--(-1,-0.89)); draw((0,-1)--(0,15), dashed); draw((3,-2)--(3,-4)^^(-3,-2)--(-3,-4)); draw((-7,0)--(-5,0)^^(-7,16)--(-5,16)); draw((3,-3)--(-3,-3), Arrows(6)); draw((-6,0)--(-6,16), Arrows(6)); draw((-2,9)--(-1,9), Arrows(3)); label("$3$", (-1.375,9.05), dir(260), fontsize(7)); label("$A$", (0,15), N); label("$B$", (0,-1), NE); label("$30$", (0, -3), S); label("$80$", (-6, 8), W);[/asy] $ \textbf{(A)}\; 120\qquad \textbf{(B)}\; 180\qquad \textbf{(C)}\; 240\qquad \textbf{(D)}\; 360\qquad \textbf{(E)}\; 480$

All contestants at one contest are sitting in $n$ columns and are forming a "good" configuration. (We define one configuration as "good" when we don't have 2 friends sitting in the same column). It's impossible for all the students to sit in $n-1$ columns in a "good" configuration. Prove that we can always choose contestants $M_1,M_2,...,M_n$ such that $M_i$ is sitting in the $i-th$ column, for each $i=1,2,...,n$ and $M_i$ is friend of $M_{i+1}$ for each $i=1,2,...,n-1$.

It is known that there are polyhedrons whose faces are more numbered than the vertices. Find the smallest number of triangular faces that such a polyhedron can have.

For an integer $n$, denote $A =\sqrt{n^{2}+24}$ and $B =\sqrt{n^{2}-9}$. Find all values of $n$ for which $A-B$ is an integer.

Two bees start at the same spot and fly at the same rate in the following directions. Bee $A$ travels $1$ foot north, then $1$ foot east, then $1$ foot upwards, and then continues to repeat this pattern. Bee $B$ travels $1$ foot south, then $1$ foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly $10$ feet away from each other? $\textbf{(A) }A \text{ east}, B \text{ west} \qquad \textbf{(B) } A\text{ north}, B\text{ south} \qquad \textbf{(C) } A\text{ north}, B\text{ west} \qquad \textbf{(D) } A\text{ up}, B\text{ south} \qquad \textbf{(E) } A\text{ up}, B\text{ west}$

Let $p$ be a prime number and $n$ a natural one. How many natural numbers are between $1$ and $p^n$ that are relatively prime with $p^n$?

If $\log_{10}2=a$ and $\log_{10}3=b$, then $\log_{5}12=?$ ${{ \textbf{(A)}\ \frac{a+b}{a+1} \qquad\textbf{(B)}\ \frac{2a+b}{a+1} \qquad\textbf{(C)}\ \frac{a+2b}{1+a} \qquad\textbf{(D)}\ \frac{2a+b}{1-a} }\qquad\textbf{(E)}\ \frac{a+2b}{1-a}} $

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$n!+f(m)!|f(n)!+f(m!)$$ for all $m,n\in\mathbb{N}$ [i]Proposed by Valmir Krasniqi and Dorlir Ahmeti, Albania[/i]

Find all continuous functions $f(x)$ defined for all $x>0$ such that for every $x$, $y > 0$ \[ f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)= f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) . \] [i]Proposed by F. Petrov[/i]

(a) Let there be given a rectangular parallelepiped. Show that some four of its vertices determine a tetrahedron whose all faces are right triangles. (b) Conversely, prove that every tetrahedron whose all faces are right triangles can be obtained by selecting four vertices of a rectangular parallelepiped. (c) Now investigate such tetrahedra which also have at least two isosceles faces. Given the length $a$ of the shortest edge, compute the lengths of the other edges.

Find all functions $f : R_{>0} \to R_{>0}$, which for all $x > y > z > 0$ is the following equation holds $$f(x - y + z) = f(x) + f(y) + f(z) - xy - yz + xz.$$

[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer. [b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$. [b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer. [b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$. [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a social network, no user has more than ten friends ( the state "friendship" is symmetrical). The network is connected: if, upon learning interesting news a user starts sending it to its friends, and these friends to their own friends and so on, then at the end, all users hear about the news. Prove that the network administration can divide users into groups so that the following conditions are met: [list] [*] each user is in exactly one group [*] each group is connected in the above sense [*] one of the groups contains from $1$ to $100$ members and the remaining from $100$ to $900$. [/list]

Find all positive real numbers \((a,b,c) \leq 1\) which satisfy \[ \huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\]

How many triangles are in this figure? [asy] import olympiad; pair A = (0,0); pair B = (0,1); pair C = (0,2); pair D = (0,3); pair E = (0,4); pair F = (1,0); pair G = (2,0); pair H = (3,0); pair I = (4,0); pair J = (1,4); pair K = (2,4); pair L = (3,4); pair M = (4,4); pair N = (4,3); pair O = (4,2); pair P = (4,1); draw(A--E--I--A); draw(M--E--I--M); draw(B--F); draw(C--G); draw(D--H); draw(L--N); draw(O--K); draw(P--J); draw(B--F); draw(B--F); draw(H--P); draw(G--O); draw(F--N); draw(B--L); draw(C--K); draw(D--J); draw(A--M); [/asy] $(A) 56$ $(B) 60$ $(C) 64$ $(D) 68$