rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem:  $ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$   satisfy the given conditions
 $ a)$   $ f(0,0)\equal{}1$  ,  $ f(0,1)\equal{}1$  , 
 $ b)$   $ \forall k \notin \left\{0,1\right\}$    $ f(0,k)\equal{}0$  and
 $ c)$   $ \forall n \geq 1$  and  $ k$  ,  $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$ 

find the sum  $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$ 
Answer: 2^{2008}
Problem: Kris is asked to compute  $\log_{10} (x^y)$ , where  $y$  is a positive integer and  $x$  is a positive real number. However, they misread this as  $(\log_{10} x)^y$  , and compute this value. Despite the reading error, Kris still got the right answer. Given that  $x > 10^{1.5}$  , determine the largest possible value of  $y$ .
Answer: 4
Problem: Determine the smallest integer  $k > 1$  such that there exist  $k$  distinct primes whose squares sum to a power of  $2$ .
Answer: k = 5
Problem: How many non-negative integral values of  $x$  satisfy the equation   $ \lfloor \frac{x}{5}\rfloor = \lfloor \frac{x}{7}\rfloor $ 
Answer: 0, 1, 2, 3, 4, 7, 8, 9, 14
Problem: A class consists of 26 students with two students sitting on each desk. Suddenly, the students decide to change seats, such that every two students that were previously sitting together are now apart. Find the maximum value of positive integer  $N$  such that, regardless of the students' sitting positions, at the end there is a set  $S$  consisting of  $N$  students satisfying the following property: every  two of them have never been sitting together.
Answer: 13
Problem: Let  $A$  be Abelian group of order  $p^4$ , where  $p$  is a prime number, and which has a subgroup  $N$  with order  $p$  such that  $A/N\approx\mathbb{Z}/p^3\mathbb{Z}$ . Find all  $A$  expect isomorphic.
Answer:  \mathbb{Z}/p^4\mathbb{Z} \text{ and } \mathbb{Z}/p^3\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}
Problem: An infinite rectangular stripe of width  $3$  cm is folded along a line. What is the minimum possible area of the region of overlapping?
Answer: 4.5 \text{ cm}^2
Problem: An integer  $n\ge1$  is *good* if the following property is satisfied:
If a positive integer is divisible by each of the nine numbers  $n + 1, n + 2, ..., n + 9$ , this is also divisible by  $n + 10$ .
How many good integers are  $n\ge 1$ ?
Answer: 0
Problem: Find all real solutions to: \begin{eqnarray*} 3(x^2 + y^2 + z^2) &=& 1  x^2y^2 + y^2z^2 + z^2x^2 &=& xyz(x + y + z)^3. \end{eqnarray*}
Answer: x = y = z = \pm \frac{1}{3}
Problem: On the coordinate plane is given the square with vertices  $T_1(1,0),T_2(0,1),T_3(-1,0),T_4(0,-1)$ . For every  $n\in\mathbb N$ , point  $T_{n+4}$  is defined as the midpoint of the segment  $T_nT_{n+1}$ . Determine the coordinates of the limit point of  $T_n$  as  $n\to\infty$ , if it exists.
Answer:  (0, 0)
Problem: 99 dwarfs stand in a circle, some of them wear hats. There are no adjacent dwarfs in hats and no dwarfs in hats with exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs in hats?
Answer: 33
Problem: Determine the range of  $w(w + x)(w + y)(w + z)$ , where  $x, y, z$ , and  $w$  are real numbers such that
\[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]
Answer: 0
Problem: A rope of length 10 *m* is tied tautly from the top of a flagpole to the ground 6 *m* away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 *cm/min*. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)
Answer: 2
Problem: An ant starts at vertex  $A$  in equilateral triangle  $\triangle ABC$  and walks around the perimeter of the triangle from  $A$  to  $B$  to  $C$  and back to  $A$ . When the ant is  $42$  percent of its way around the triangle, it stops for a rest. Find the percent of the way from  $B$  to  $C$  the ant is at that point
Answer: 26\%
Problem: Find all positive integers  $a,b$  such that  $b^{619}$  divides  $a^{1000}+1$  and  $a^{619}$  divides  $b^{1000}+1$ .
Answer:  (a, b) = (1, 1)
Problem: In a square  $ABCD$  of diagonals  $AC$  and  $BD$ , we call  $O$  at the center of the square. A square  $PQRS$  is constructed with sides parallel to those of  $ABCD$  with  $P$  in segment  $AO, Q$  in segment  $BO, R$  in segment  $CO, S$  in segment  $DO$ .  If area of  $ABCD$  equals two times the  area of  $PQRS$ ,  and  $M$  is the midpoint of the  $AB$  side, calculate the measure of the angle  $\angle AMP$ . 
Answer: \frac{45^{\circ}}{2}
Problem: A nine-digit number has the form  $\overline{6ABCDEFG3}$ , where every three consecutive digits sum to  $13$ . Find  $D$ .
Answer: 4
Problem: Numbers  $\frac{49}{1}, \frac{49}{2}, ... , \frac{49}{97}$  are writen on a blackboard. Each time, we can replace two numbers (like  $a, b$ ) with  $2ab-a-b+1$ . After  $96$  times doing that prenominate action, one number will be left on the board. Find all the possible values fot that number.
Answer: 1
Problem: Given an integer  $n\geq 2$ . There are  $N$  distinct circle on the plane such that any two circles have two distinct intersections and no three circles have a common intersection. Initially there is a coin on each of the intersection points of the circles. Starting from  $X$ , players  $X$  and  $Y$  alternatively take away a coin, with the restriction that one cannot take away a coin lying on the same circle as the last coin just taken away by the opponent in the previous step. The one who cannot do so will lost. In particular, one loses where there is no coin left. For what values of  $n$  does  $Y$  have a winning strategy?
Answer:  n \geq 4
Problem: Determine all triples  $(x, y, z)$  of non-zero numbers such that
\[
xy(x + y) = yz(y + z) = zx(z + x).
\]
Answer: (x, y, z) such that x = y = z or x + y + z = 0
Problem: A gambler plays the following coin-tossing game. He can bet an arbitrary positive amount of money. Then a fair coin is tossed, and the gambler wins or loses the amount he bet depending on the outcome. Our gambler, who starts playing with   $ x$  forints, where  $ 0<x<2C$ , uses the following strategy: if at a given time his capital is  $ y<C$ , he risks all of it; and if he has  $ y>C$ , he only bets  $ 2C\minus{}y$ . If he has exactly  $ 2C$  forints, he stops playing. Let  $ f(x)$  be the probability that he reaches  $ 2C$  (before going bankrupt). Determine the value of  $ f(x)$ .
Answer: \frac{x}{2C}
Problem: How many solutions of the equation  $\tan x = \tan \tan x$  are on the interval  $0 \le x \le \tan^{-1} 942$ ?  (Here  $\tan^{-1}$  means the inverse tangent function, sometimes written  $\arctan$ .)
Answer: 300
Problem: The diagram below shows a  $ 4\times4$  rectangular array of points, each of which is  $ 1$  unit away from its nearest neighbors.
[asy]unitsize(0.25inch);
defaultpen(linewidth(0.7));

int i, j;
for(i = 0; i < 4; ++i)
	for(j = 0; j < 4; ++j)
		dot(((real)i, (real)j));[/asy]Define a *growing path* to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let  $ m$  be the maximum possible number of points in a growing path, and let  $ r$  be the number of growing paths consisting of exactly  $ m$  points. Find  $ mr$ .
Answer: 240
Problem: If  $f$  is a monic cubic polynomial with  $f(0)=-64$ , and all roots of  $f$  are non-negative real numbers, what is the largest possible value of  $f(-1)$ ? (A polynomial is monic if it has a leading coefficient of  $1$ .)
Answer: -125
Problem: Find all real-valued functions  $f$  on the reals such that  $f(-x) = -f(x)$ ,  $f(x+1) = f(x) + 1$  for all  $x$ , and  $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$  for  $x \not = 0$ .
Answer: f(x) = x
Problem: Circle  $\Gamma$  with radius  $1$  is centered at point  $A$  on the circumference of circle  $\omega$  with radius  $7$ . Suppose that point  $P$  lies on  $\omega$  with  $AP=4$ . Determine the product of the distances from  $P$  to the two intersections of  $\omega$  and  $\Gamma$ .
Answer: 15
Problem: Let  $f(x)$  be a continuous function satisfying  $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$ 

Find the value of  $k$  for which  $\int_0^{\pi} f(x)dx$  is maximized.
Answer:  k = \frac{2}{\pi}
Problem: For which  $n$  do the equations have a solution in integers:  \begin{eqnarray*}x_1 ^2 + x_2 ^2 + 50 &=& 16x_1 + 12x_2  x_2 ^2 + x_3 ^2 + 50 &=& 16x_2 + 12x_3  \cdots \quad \cdots \quad \cdots & \cdots & \cdots \quad \cdots  x_{n-1} ^2 + x_n ^2 + 50 &=& 16x_{n-1} + 12x_n  x_n ^2 + x_1 ^2 + 50 &=& 16x_n + 12x_1 \end{eqnarray*}
Answer: n \text{ must be even}
Problem: Given a parallelogram with area  $1$  and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the  $8$  lines formed we have  a octagon inside of the parallelogram. Determine the area of this octagon
Answer: \frac{1}{2}
Problem: Determine the maximum possible real value of the number  $ k$ , such that
\[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\]
for all real numbers  $ a,b,c\ge 0$  with  $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$ .
Answer: 1
Problem: For each integer  $n \ge 2$ , let  $A(n)$  be the area of the region in the coordinate plane defined by the inequalities  $1\le x \le n$  and  $0\le y \le x \left\lfloor \sqrt x \right\rfloor$ , where  $\left\lfloor \sqrt x \right\rfloor$  is the greatest integer not exceeding  $\sqrt x$ . Find the number of values of  $n$  with  $2\le n \le 1000$  for which  $A(n)$  is an integer.
Answer: 483
Problem: Find all odd natural numbers  $n$  such that  $d(n)$  is the largest divisor of the number  $n$  different from  $n$ .
( $d(n)$  is the number of divisors of the number n including  $1$  and  $n$  ).
Answer: 9
Problem: A right rectangular prism  $P$  (i.e., a rectangular parallelpiped) has sides of integral length  $a, b, c,$  with  $a\le b\le c.$   A plane parallel to one of the faces of  $P$  cuts  $P$  into two prisms, one of which is similar to  $P,$  and both of which have nonzero volume.  Given that  $b=1995,$  for how many ordered triples  $(a, b, c)$  does such a plane exist?
Answer: 40
Problem: A triangle of numbers is constructed as follows. The first row consists of the numbers from  $1$  to  $2000$  in increasing order, and under any two consecutive numbers their sum is written. (See the example corresponding to  $5$  instead of  $2000$  below.) What is the number in the lowermost row?

1 2 3 4 5
  3 5 7 9
   8 12 16
    20 28
       4
Answer: 2001 \cdot 2^{1998}
Problem: Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his  $20$  friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag contains over  $500$  candies, what is the fewest number of candies the bag can contain?
Answer: 509
Problem: There are  $n$  pawns on  $n$  distinct squares of a  $19\times 19$  chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in two successive moves. What is largest number of pawns can a player place on the board (being able to arrange them freely) so as to be able to continue the game indefinitely?
Answer: 361
Problem: Denote by  $\mathbb{Q}^+$  the set of positive rational numbers. A function  $f : \mathbb{Q}^+ \to \mathbb{Q}$  satisfies

•  $f(p) = 1$  for all primes  $p$ , and

•  $f(ab) = af(b) + bf(a)$  for all  $ a,b \in \mathbb{Q}^+ $ .

For which positive integers  $n$  does the equation  $nf(c) = c$  have at least one solution  $c$  in  $\mathbb{Q}^+$ ?
Answer:  n is square-free.
Problem: Let  $n$  be a integer  $1<n<2010$ , where we have a polygon with  $2010$  sides and  $n$  coins, we have to paint the vertices of this polygon with  $n$  colors and we've to put the  $n$  coins in  $n$  vertices of the polygon. In each second the coins will go to the neighbour vertex, going in the clockwise.
Determine the values of  $n$  such that is possible paint and choose the initial position of the coins where in each second the  $n$  coins are in vertices of distinct colors
Answer:  n divides 2010.
Problem: An elephant writes a sequence of numbers on a board starting with 1. Each minute, it doubles the sum of all the numbers on the board so far, and without erasing anything, writes the result on the board. It stops after writing a number greater than one billion. How many distinct prime factors does the largest number on the board have?
Answer: 2
Problem: Black and white checkers are placed on an  $8 \times 8$  chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?
Answer: 48
Problem: Hello Everyone, 

i'm trying to make a strong marathon for number theory .. which will be in Pre-Olympiad level

Please if you write any problem don't forget to indicate its number and if you write a solution please indicate for what problem also to prevent the confusion that happens in some marathons. 

it will be preferred to write the source of the problem.
please , show detailed solutions , and please post some challenging Pre-Olympiad problems..
remember ,, different approaches are most welcome :)

now .. let's get our hands dirty :lol: :


let  $ f(n)$  denote the sum of the digits of  $ n$ . Now let  $ N \equal{} {4444^{4444}}$ .
Find  $ f\left( {f\left( {f\left( N \right)} \right)} \right)$ .
Answer: 7
Problem: One day students in school organised a exchange between them such that :  $11$  strawberries change for  $14$  raspberries,  $22$  cherries change for  $21$  raspberries,  $10$  cherries change for  $3$  bananas and  $5$  pears for  $2$  bananas. How many pears has Amila to give to get  $7$  strawberries
Answer: 7 \ \text{pears}
Problem: Let  $n$  be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses.  At the HMPT,  $2^{n+1}$  competitors, numbered  $1$  through  $2^{n+1}$  clockwise, stand in a circle.  They are equals in Pushover: whenever two of them face off, each has a  $50\%$  probability of victory.  The tournament unfolds in  $n+1$  rounds.  In each rounjd, the referee randomly chooses one of the surviving players, and the players pair off going clockwise, starting from the chosen one.  Each pair faces off in Pushover, and the losers leave the circle.  What is the probability that players  $1$  and  $2^n$  face each other in the last round?  Express your answer in terms of  $n$ .
Answer: \frac{1}{4^n}
Problem: There is a complex number  $K$  such that the quadratic polynomial  $7x^2 +Kx + 12 - 5i$  has exactly one root, where  $i =\sqrt{-1}$ . Find  $|K|^2$ .
Answer: 364
Problem:  $ M$  is a subset of  $ \{1, 2, 3, \ldots, 15\}$  such that the product of any three distinct elements of  $ M$  is not a square. Determine the maximum number of elements in  $ M.$ 
Answer: 10
Problem: Solve in  $\mathbb{R}$  the equation : $(x+1)^5 + (x+1)^4(x-1) + (x+1)^3(x-1)^2 +$   $ (x+1)^2(x-1)^3 + (x+1)(x-1)^4 + (x-1)^5 =$   $ 0$ .
Answer: x = 0
Problem: On an east-west shipping lane are ten ships sailing individually.  The first five from the west are sailing eastwards while the other five ships are sailing westwards.  They sail at the same constant speed at all times.  Whenever two ships meet, each turns around and sails in the opposite direction.  When all ships have returned to port, how many meetings of two ships have taken place?
Answer: 25
Problem: You are given a positive integer  $n$ . What is the largest possible number of numbers that can be chosen from the set $\{1, 2, \ldots, 2n\}$  so that there are no two chosen numbers  $x > y$  for which  $x - y = (x, y)$ ?

Here  $(x, y)$  denotes the greatest common divisor of  $x, y$ .
Answer: n
Problem: Monica and Bogdan are playing a game, depending on given integers  $n, k$ . First, Monica writes some  $k$  positive numbers. Bogdan wins, if he is able to find  $n$  points on the plane with the following property: for any number  $m$  written by Monica, there are some two points chosen by Bogdan with distance exactly  $m$  between them. Otherwise, Monica wins.

Determine who has a winning strategy depending on  $n, k$ .
Answer:  n \leq k
Problem: Elmo bakes cookies at a rate of one per 5 minutes. Big Bird bakes cookies at a rate of one per 6 minutes. Cookie Monster *consumes* cookies at a rate of one per 4 minutes. Together Elmo, Big Bird, Cookie Monster, and Oscar the Grouch produce cookies at a net rate of one per 8 minutes. How many minutes does it take Oscar the Grouch to bake one cookie?
Answer: 120
Problem: For positive real numbers  $a,b,c,d$ , with  $abcd = 1$ , determine all values taken by the expression
\[\frac {1+a+ab} {1+a+ab+abc} + \frac {1+b+bc} {1+b+bc+bcd} +\frac {1+c+cd} {1+c+cd+cda} +\frac {1+d+da} {1+d+da+dab}.\]
Answer: 2
Problem: Define a sequence of positive integers  $\{a_n\}$  through the recursive formula: $a_{n+1}=a_n+b_n(n\ge 1)$ ,where  $b_n$  is obtained by rearranging the digits of  $a_n$  (in decimal representation) in reverse order (for example,if  $a_1=250$ ,then  $b_1=52,a_2=302$ ,and so on). Can  $a_7$  be a prime?
Answer: a_7 \text{ cannot be a prime number}
Problem: There are  $2024$  people, which are knights and liars and some of them are friends. Every person is asked for the number of their friends and the answers were  $0,1, \ldots, 2023$ . Every knight answered truthfully, while every liar changed the real answer by exactly  $1$ . What is the minimal number of liars? 
Answer: 1012
Problem: A circle of radius  $2$  is inscribed in equilateral triangle  $ABC$ . The altitude from  $A$  to  $BC$  intersects the circle at a point  $D$  not on  $BC$ .  $BD$  intersects the circle at a point  $E$  distinct from  $D$ . Find the length of  $BE$ .
Answer: 2\sqrt{3}
Problem: A farmer wants to build a rectangular region, using a river as one side and some fencing as the other three sides. He has 1200 feet of fence which he can arrange to different dimensions. He creates the rectangular region with length  $ L$  and width  $ W$  to enclose the greatest area. Find  $ L\plus{}W$ .
Answer: 900
Problem: A sequence of  $N$  consecutive positive integers is called *good* if it is possible to choose two of these numbers so that their product is divisible by the sum of the other  $N-2$  numbers. For which  $N$  do there exist infinitely many *good* sequences?
Answer: All even numbers \geq 4
Problem: The sum 
\[ \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}\]
can be written in the form  $\frac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .
Answer: 5
Problem: Find all real numbers  $x,y,z$  such that satisfied the following equalities at same time: $\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$ 
Answer: x = y = z = 1
Problem: Determine the smallest positive integer  $a$  for which there exist a prime number  $p$  and a positive integer  $b \ge 2$  such that   $$ \frac{a^p -a}{p}=b^2. $$  
Answer: 9
Problem: It takes  $3$  rabbits  $5$  hours to dig  $9$  holes. It takes  $5$  beavers  $36$  minutes to build  $2$  dams. At this rate, how many more minutes does it take  $1$  rabbit to dig  $1$  hole than it takes  $1$  beaver to build  $1$  dam?
Answer: 10
Problem: Numbers  $1, 2, \dots , n$  are written around a circle in some order. What is the smallest possible sum of the absolute differences of adjacent numbers?
Answer: 2n - 2
Problem: Find all pairs of integers,  $n$  and  $k$ ,  $2 < k < n$ , such that the binomial coefficients
\[\binom{n}{k-1}, \binom{n}{k}, \binom{n}{k+1}\]
form an increasing arithmetic series.
Answer:  (n, k) = (4p^2 - 2, 2p^2 - 1 \pm p)
Problem: Let  $n$  be positive numbers. Consider the following region on  $xy$  plane.

\[ D_n=\left\{(x,y) |\ \frac{x}{n+\frac{1}{2}}\leq y\leq [x+1]-x,\ x\geq 0 \right\}. \]

Here  $\ [x]$  is the greatest integer  $\ \leq x.$  Find the area of  $D_n.$ 
Answer: \frac{1}{2} \left(\frac{n+\frac{3}{2}}{n+\frac{1}{2}}\right)
Problem: Given three non-negative real numbers  $a$ ,  $b$  and  $c$ . The sum of the modules of their pairwise differences is equal to  $1$ , i.e.  $|a- b| + |b -c| + |c -a| = 1$ . What can the sum  $a + b + c$  be equal to?
Answer: \frac{1}{2}
Problem: Triangle  $ABC$  with  $\measuredangle A = 90^\circ$  has incenter  $I$ . A circle passing through  $A$  with center  $I$  is drawn, intersecting  $\overline{BC}$  at  $E$  and  $F$  such that  $BE < BF$ . If  $\tfrac{BE}{EF} = \tfrac{2}{3}$ , then  $\tfrac{CF}{FE} = \tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .
Answer: 7
Problem: Find the number of digit of  $\sum_{n=0}^{99} 3^n$ .
You may use  $\log_{10} 3=0.4771$ .

2012 Tokyo Institute of Technology entrance exam, problem 2-A
Answer: 48
Problem: A function  $f$  from the positive integers to the nonnegative integers is defined recursively by  $f(1) = 0$  and  $f(n+1) = 2^{f(n)}$  for every positive integer  $n$ .  What is the smallest  $n$  such that  $f(n)$  exceeds the number of atoms in the observable universe (approximately  $10^{80}$ )?
Answer: 7
Problem: A square plate has side length  $L$  and negligible thickness. It is laid down horizontally on a table and is then rotating about the axis  $\overline{MN}$  where  $M$  and  $N$  are the midpoints of two adjacent sides of the square. The moment of inertia of the plate about this axis is  $kmL^2$ , where  $m$  is the mass of the plate and  $k$  is a real constant. Find  $k$ . 

<span style="color:red">Diagram will be added to this post very soon. If you want to look at it temporarily, see the PDF.</span>
Answer: k = \frac{5}{12}
Problem: A point  $M$  is chosen inside the square  $ABCD$  in such a way that  $\angle MAC = \angle MCD = x$  . Find  $\angle ABM$ .
Answer: 45^\circ
Problem: The number  $123454321$  is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by  $9$ . What is the fewest number of digits he could have erased?
Answer: 2
Problem: Let  $a$  be a fixed real number. Consider the equation  $$ (x+2)^{2}(x+7)^{2}+a=0, x \in R $$  where  $R$  is the set of real numbers. For what values of  $a$ , will the equ have exactly one double-root?
Answer:  a = -39.0625
Problem: Suppose that  $PQ$  and  $RS$  are two chords of a circle intersecting at a point  $O$ . It is given that  $PO=3 \text{cm}$  and  $SO=4 \text{cm}$ . Moreover, the area of the triangle  $POR$  is  $7 \text{cm}^2$ . Find the area of the triangle  $QOS$ .
Answer: \frac{112}{9} \text{ cm}^2
Problem: The Devil and the Man play a game. Initially, the Man pays some cash  $s$  to the Devil. Then he lists some  $97$  triples  $\{i,j,k\}$  consisting of positive integers not exceeding  $100$ . After that, the Devil draws some convex polygon  $A_1A_2...A_{100}$  with area  $100$  and pays to the Man, the sum of areas of all triangles  $A_iA_jA_k$ . Determine the maximal value of  $s$  which guarantees that the Man receives at least as much cash as he paid.
Answer: 0
Problem: Find  $n$  such that  $n - 76$  and  $n + 76$  are both cubes of positive integers.
Answer: 140
Problem: Let be an odd natural number  $ n\ge 3. $  Find all continuous functions  $ f:[0,1]\longrightarrow\mathbb{R} $  that satisfy the following equalities.  $$  \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n,\quad\forall k\in\{ 1,2,\ldots ,n-1\}  $$  *Titu Andreescu*
Answer:  f(t) = t
Problem: Find all real numbers  $a, b, c$  that satisfy  $$  2a - b =a^2b, \qquad 2b-c = b^2 c, \qquad 2c-a= c^2 a. $$  
Answer:  (-1, -1, -1), (0, 0, 0), (1, 1, 1)