rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: As the Kubiks head out of town for vacation, Jerry takes the first driving shift while Hannah and most of the kids settle down to read books they brought along.  Tony does not feel like reading, so Alexis gives him one of her math notebooks and Tony gets to work solving some of the problems, and struggling over others.  After a while, Tony comes to a problem he likes from an old AMC 10 exam:
\begin{align*}&\text{Four distinct circles are drawn in a plane.  What is the maximum}&\quad\,\,\text{number of points where at least two of the circles intersect?}\end{align*}
Tony realizes that he can draw the four circles such that each pair of circles intersects in two points.  After careful doodling, Tony finds the correct answer, and is proud that he can solve a problem from late on an AMC 10 exam.

"Mom, why didn't we all get Tony's brain?" Wendy inquires before turning he head back into her favorite Harry Potter volume (the fifth year).

Joshua leans over to Tony's seat to see his brother's work.  Joshua knows that Tony has not yet discovered all the underlying principles behind the problem, so Joshua challenges, "What if there are a dozen circles?"

Tony gets to work on Joshua's problem of finding the maximum number of points of intersections where at least two of the twelve circles in a plane intersect.  What is the answer to this problem?
Answer: 132
Problem: Let  $\alpha,\beta$  be the distinct positive roots of the equation of  $2x=\tan x$ .
Evaluate

\[\int_0^1 \sin \alpha x\sin \beta x\ dx\]
Answer: 0
Problem: Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the attachment.

Find the "?"
Answer: ?
Problem: What is the difference between the median and the mean of the following data set:  $12,41, 44, 48, 47, 53, 60, 62, 56, 32, 23, 25, 31$ ?
Answer: \frac{38}{13}
Problem:  $6$  people each have a hat.  If they shuffle their hats and redistribute them, what is the probability that exactly one person gets their own hat back?
Answer: \frac{11}{30}
Problem: There are 2008 red cards and 2008 white cards. 2008  players sit down in circular toward the inside of the circle in situation that 2 red cards and 2 white cards from each card are delivered to each person. Each person conducts the following procedure in one turn as follows.
 $ (*)$  If you have more than one red card, then you will pass one red card to the left-neighbouring player.

If you have no red card, then you will pass one white card to the left -neighbouring player.
Find the maximum value of the number of turn required for the state such that all person will have one red card and one white card first.
Answer: 1004
Problem: Find all functions  $f$  from positive integers to themselves, such that the followings hold. $1)$ .for each positive integer  $n$  we have  $f(n)<f(n+1)<f(n)+2020$ . $2)$ .for each positive integer  $n$  we have  $S(f(n))=f(S(n))$  where  $S(n)$  is the sum of digits of  $n$  in base  $10$  representation.
Answer:  f(x) = x
Problem: Find the number of trailing zeros at the end of the base- $10$  representation of the integer  $525^{25^2}
\cdot 252^{52^5}$  .
Answer: 1250
Problem: How many  $1 \times 10\sqrt 2$  rectangles can be cut from a  $50\times 90$  rectangle using cuts parallel to its edges?
Answer: 318
Problem: Paul starts at  $1$  and counts by threes:  $1, 4, 7, 10, ... $ . At the same time and at the same speed, Penny counts backwards from  $2017$  by fives:  $2017, 2012, 2007, 2002,...$  . Find the one number that both Paul and Penny count at the same time.
Answer: 2017
Problem: Determine all possible values of integer  $k$  for which there exist positive integers  $a$  and  $b$  such that  $\dfrac{b+1}{a} + \dfrac{a+1}{b} = k$ .
Answer: 3 \text{ and } 4
Problem: A magic square is a square with side 3 consisting of 9 unit squares, such that the numbers written in the unit squares (one number in each square) satisfy the following property: the sum of the numbers in each row is equal to the sum of the numbers in each column and is equal to the sum of all the numbers written in any of the two diagonals.

A rectangle with sides  $m\ge3$  and  $n\ge3$  consists of  $mn$  unit squares. If in each of those unit squares exactly one number is written, such that any square with side  $3$  is a magic square, then find the number of most different numbers that can be written in that rectangle.
Answer:  9
Problem: Let  $ABC$  be a triangle with  $\angle A = 120^\circ$ ,  $I$  be the incenter, and  $M$  be the midpoint of  $BC$ . The line passing through  $M$  and parallel to  $AI$  meets the circle with diameter  $BC$  at points  $E$  and  $F$  ( $A$  and  $E$  lie on the same semiplane with respect to  $BC$ ). The line passing through  $E$  and perpendicular to  $FI$  meets  $AB$  and  $AC$  at points  $P$  and  $Q$  respectively. Find the value of  $\angle PIQ$ .
Answer: \angle PIQ = 90^\circ
Problem: The sequence  $\{c_{n}\}$  is determined by the following equation. \[c_{n}=(n+1)\int_{0}^{1}x^{n}\cos \pi x\ dx\ (n=1,\ 2,\ \cdots).\] Let  $\lambda$  be the limit value  $\lim_{n\to\infty}c_{n}.$   Find  $\lim_{n\to\infty}\frac{c_{n+1}-\lambda}{c_{n}-\lambda}.$ 
Answer: 1
Problem: We took a non-equilateral acute-angled triangle and marked  $4$  wonderful points in it: the centers of the inscribed and circumscribed circles, the center of gravity (the point of intersection of the medians) and the intersection point of altitudes. Then the triangle itself was erased. It turned out that it was impossible to establish which of the centers corresponds to each of the marked points. Find the angles of the triangle
Answer: 45^
\circ, 45^
\circ, 90^
\circ
Problem: If the sum of the slope and the  $y$ -intercept of a line is  $3$ , then through which point is the line guaranteed to pass?
Answer:  (1, 3)
Problem: Anne-Marie has a deck of  $16$  cards, each with a distinct positive factor of  $2002$  written on it. She shuffles the deck and begins to draw cards from the deck without replacement. She stops when there exists a nonempty subset of the cards in her hand whose numbers multiply to a perfect square. What is the expected number of cards in her hand when she stops?
Answer: 4.0625
Problem: Given any two positive real numbers  $x$  and  $y$ , then  $x\Diamond y$  is a positive real number defined in terms of  $x$  and  $y$  by some fixed rule.  Suppose the operation  $x\Diamond y$  satisfies the equations  $(x\cdot y)\Diamond y=x(y\Diamond y)$  and  $(x\Diamond 1)\Diamond x=x\Diamond 1$  for all  $x,y>0$ .  Given that  $1\Diamond 1=1$ , find  $19\Diamond 98$ .
Answer: 19
Problem: Find the max. value of  $ M$ ,such that for all  $ a,b,c>0$ :
    $ a^{3}+b^{3}+c^{3}-3abc\geq M(|a-b|^{3}+|a-c|^{3}+|c-b|^{3})$ 
Answer: \frac{1}{2}
Problem: Solve the system   $\begin{cases} x+ y +z = a 
 x^2 + y^2 + z^2 = a^2 
 x^3 + y^3 +z^3 = a^3
\end{cases}$ 
Answer: (x, y, z) \in \{(a, 0, 0), (0, a, 0), (0, 0, a)\}
Problem: The numbers  $2^{1989}$  and  $5^{1989}$  are written out one after the other (in decimal notation). How many digits are written altogether?
Answer: 1990
Problem: A cake has a shape of triangle with sides  $19,20$  and  $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't  stick out of the plate. What is the minimal diameter of the plate?
Answer: 21
Problem: There are 1000 rooms in a row along a long corridor. Initially the first room contains 1000 people and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different
rooms will have people in them?
Answer: 61
Problem: How many noncongruent triangles are there with one side of length  $20,$  one side of length  $17,$  and one  $60^{\circ}$  angle?
Answer: 1
Problem: Determine all positive integers  $n{}$  for which there exist pairwise distinct integers  $a_1,\ldots,a_n{}$  and  $b_1,\ldots, b_n$  such that \[\prod_{i=1}^n(a_k^2+a_ia_k+b_i)=\prod_{i=1}^n(b_k^2+a_ib_k+b_i)=0, \quad \forall k=1,\ldots,n.\]
Answer: 2
Problem: Let  $b$  be a real number randomly sepected from the interval  $[-17,17]$ .  Then,  $m$  and  $n$  are two relatively prime positive integers such that  $m/n$  is the probability that the equation \[x^4+25b^2=(4b^2-10b)x^2\] has  $\textit{at least}$  two distinct real solutions.  Find the value of  $m+n$ .
Answer: 63
Problem: Find all positive integers  $n$  such that the equation  $y^2 + xy + 3x = n(x^2 + xy + 3y)$  has at least a solution  $(x, y)$  in positive integers.
Answer: n = 1
Problem: Consider the parabola  $C: y^{2}=4 x$  and the straight line  $L: y=x+2$ . Let  $P$  be a variable point on  $L$ . Draw the two tangents from  $P$  to  $C$  and let  $Q_{1}$  and  $Q_{2}$  denote the two points of contact on  $C$ . Let  $Q$  be the mid-point of the line segment joining  $Q_{1}$  and  $Q_{2}$ . Find the locus of  $Q$  as  $P$  moves along  $L$ .
Answer: (y - 1)^2 = 2 \left( x - \frac{3}{2} \right)
Problem: Determine the largest positive integer  $n$  such that the following statement is true:

There exists  $n$  real polynomials,  $P_1(x),\ldots,P_n(x)$  such that the sum of any two of them have no real roots but the sum of any three does.
Answer:  n = 3
Problem: A rigid (solid) cylinder is put at the top of a frictionless  $25^\circ$ -to-the-horizontal incline that is  $3.0 \, \text{m}$  high. It is then released so that it rolls down the incline. If  $v$  is the speed at the bottom of the incline, what is  $v^2$ , in  $\text{m}^2/\text{s}^2$ ? 

*(B. Dejean and Ahaan Rungta, 3 points)***Note**: Since there is no friction, the cylinder cannot roll, and thus the problem is flawed. Two answers were accepted and given full credit.
Answer: 58.8 \, 	ext{m}^2/\text{s}^2
Problem: Let  $z=z(x,y)$  be implicit function with two variables from  $2sin(x+2y-3z)=x+2y-3z$ . Find  $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$ .
Answer: 1
Problem: Suppose 2012 reals are selected independently and at random from the unit interval  $[0,1]$ , and then written in nondecreasing order as  $x_1\le x_2\le\cdots\le x_{2012}$ . If the probability that  $x_{i+1} - x_i \le \frac{1}{2011}$  for  $i=1,2,\ldots,2011$  can be expressed in the form  $\frac{m}{n}$  for relatively prime positive integers  $m,n$ , find the remainder when  $m+n$  is divided by 1000.
Answer: 1
Problem: Determine all values of  $n$  satisfied the following condition: there's exist a cyclic  $(a_1,a_2,a_3,...,a_n)$  of  $(1,2,3,...,n)$  such that  $\left\{ {{a_1},{a_1}{a_2},{a_1}{a_2}{a_3},...,{a_1}{a_2}...{a_n}} \right\}$  is a complete residue systems modulo  $n$ .
Answer:  n is a prime number.
Problem: The number 1 is a solution of the equation   $(x + a)(x + b)(x + c)(x + d) = 16$ ,  

where  $a, b, c, d$  are positive real numbers. Find the largest value of  $abcd$ .
Answer: 1
Problem: Find the product of all values of  $d$  such that  $x^{3} +2x^{2} +3x +4 = 0$  and  $x^{2} +dx +3 = 0$  have a common root.
Answer: 0
Problem: Using  $600$  cards,  $200$  of them having written the number  $5$ ,  $200$  having a  $2$ , and the other  $200$  having a  $1$ , a student wants to create groups of cards such that the sum of the card numbers in each group is  $9$ . What is the maximum amount of groups that the student may create?
Answer: 100
Problem: Let  $p$  and  $q$  be two given positive integers. A set of  $p+q$  real numbers  $a_1<a_2<\cdots <a_{p+q}$  is said to be balanced iff  $a_1,\ldots,a_p$  were an arithmetic progression with common difference  $q$  and  $a_p,\ldots,a_{p+q}$  where an arithmetic progression with common difference  $p$ . Find the maximum possible number of balanced sets, so that any two of them have nonempty intersection. 

Comment: The intended problem also had " $p$  and  $q$  are coprime" in the hypothesis. A typo when the problems where written made it appear like that in the exam (as if it were the only typo in the olympiad). Fortunately, the problem can be solved even if we didn't suppose that and it can be further generalized: we may suppose that a balanced set has  $m+n$  reals  $a_1<\cdots <a_{m+n-1}$  so that  $a_1,\ldots,a_m$  is an arithmetic progression with common difference  $p$  and  $a_m,\ldots,a_{m+n-1}$  is an arithmetic progression with common difference  $q$ .
Answer: p + q
Problem: Let  $n$  and  $k$  be integers satisfying  $\binom{2k}{2} + n = 60$ .  It is known that  $n$  days before Evan's 16th birthday, something happened.  Compute  $60-n$ .
Answer: 45
Problem: 4. Let  $v$  and  $w$  be two randomly chosen roots of the equation  $z^{1997} -1 = 0$  (all roots are equiprobable). Find the probability that  $\sqrt{2+\sqrt{3}}\le |u+w|$ 
Answer: \frac{333}{1997}
Problem: Determine the number of real roots of the equation \[x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x +\frac{5}{2}= 0.\]
Answer: 0
Problem: A triangle has a perimeter of  $4$  *yards* and an area of  $6$  square *feet*.  If one of the angles of the triangle is right, what is the length of the largest side of the triangle, in feet?
Answer: 5
Problem: We build a tower of  $2\times 1$  dominoes in the following way. First, we place  $55$  dominoes on the table such that they cover a  $10\times 11$  rectangle; this is the first story of the tower. We then build every new level with  $55$  domioes above the exact same  $10\times 11$  rectangle. The tower is called *stable* if for every non-lattice point of the  $10\times 11$  rectangle, we can find a domino that has an inner point above it. How many stories is the lowest *stable* tower?
Answer: 5
Problem: Let  $ABC$  be a triangle (right in  $B$ ) inscribed in a semi-circumference of diameter  $AC=10$ . Determine the distance of the vertice  $B$  to the side  $AC$  if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle.
Answer: \frac{5}{2}
Problem: Let  $a,b,c$  be positive real numbers such that  $a+b+c=3$ . Determine, with certainty, the largest possible value of the expression   $$  \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b} $$  
Answer: 1
Problem: Find all positive integers  $ m,n $   that satisfy the equation  \[ 3.2^m +1 = n^2 \]
Answer: (m, n) \in \{(3, 5), (4, 7)\}
Problem: We say a positive integer  $ n$  is *good* if there exists a permutation  $ a_1, a_2, \ldots, a_n$  of  $ 1, 2, \ldots, n$  such that  $ k \plus{} a_k$  is perfect square for all  $ 1\le k\le n$ . Determine all the good numbers in the set  $ \{11, 13, 15, 17, 19\}$ .
Answer: \{13, 15, 17, 19\}
Problem: Let  $ABC$  be an acute triangle.  $PQRS$  is a rectangle with  $P$  on  $AB$ ,  $Q$  and  $R$  on  $BC$ , and  $S$  on  $AC$  such that  $PQRS$  has the largest area among all rectangles  $TUVW$  with  $T$  on  $AB$ ,  $U$  and  $V$  on  $BC$ , and  $W$  on  $AC$ . If  $D$  is the point on  $BC$  such that  $AD\perp BC$ , then  $PQ$  is the harmonic mean of  $\frac{AD}{DB}$  and  $\frac{AD}{DC}$ . What is  $BC$ ?

Note: The harmonic mean of two numbers  $a$  and  $b$  is the reciprocal of the arithmetic mean of the reciprocals of  $a$  and  $b$ .
Answer: 4
Problem:  $2000$  people are standing on a line. Each one of them is either a *liar*, who will always lie, or a *truth-teller*, who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line.
Answer: 1000
Problem: For any  $n\in\mathbb N$ , denote by  $a_n$  the sum  $2+22+222+\cdots+22\ldots2$ , where the last summand consists of  $n$  digits of  $2$ . Determine the greatest  $n$  for which  $a_n$  contains exactly  $222$  digits of  $2$ .
Answer: 222
Problem: Regular hexagon  $NOSAME$  with side length  $1$  and square  $UDON$  are drawn in the plane such that  $UDON$  lies outside of  $NOSAME$ . Compute  $[SAND] + [SEND]$ , the sum of the areas of quadrilaterals  $SAND$  and  $SEND$ .

Answer: \frac{3}{2} + \frac{3\sqrt{3}}{2}
Problem: Real numbers  $a,b,c$  with  $a\neq b$  verify   $$ a^2(b+c)=b^2(c+a)=2023. $$   Find the numerical value of  $E=c^2(a+b)$ .
Answer: 2023
Problem: Suppose  $xy-5x+2y=30$ , where  $x$  and  $y$  are positive integers. Find the sum of all possible values of  $x$ 
Answer: 31
Problem: The distances from a certain point inside a regular hexagon to three of its consecutive vertices are equal to  $1, 1$  and  $2$ , respectively. Determine the length of this hexagon's side.
Answer: \sqrt{3}
Problem: We call the polynomial  $P (x)$  simple if the coefficient of each of its members belongs to the set  $\{-1, 0, 1\}$ .
Let  $n$  be a positive integer,  $n> 1$ . Find the smallest possible number of terms with a non-zero coefficient in a simple  $n$ -th degree polynomial with all values at integer places are divisible by  $n$ .

Answer: 2
Problem: Given a positive integer  $n$ , determine the maximum number of lattice points in the plane a square of side length  $n +\frac{1}{2n+1}$  may cover.
Answer: (n+1)^2
Problem: What integers  $a, b, c$  satisfy  $a^2 + b^2 + c^2 + 3 < ab + 3b + 2c$  ?
Answer:  (1, 2, 1)
Problem: Find all monotonous functions  $ f: \mathbb{R} \to \mathbb{R}$  that satisfy the following functional equation:
\[f(f(x)) \equal{} f( \minus{} f(x)) \equal{} f(x)^2.\]
Answer:  f(x) = 0 or f(x) = 1
Problem: Find the number of integers  $n$  for which  $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$  is a real number.
Answer: 90
Problem: Find all quadrilaterals  $ABCD$  such that all four triangles  $DAB$ ,  $CDA$ ,  $BCD$  and  $ABC$  are similar to one-another.
Answer: ABCD is a rectangle
Problem: In the  Mathematical  Competition  of  HMS (Hellenic  Mathematical  Society)  take part  boys and  girls  who are divided  into  two  groups : *Juniors*  and  *seniors.*The number of the  boys  taking  part  of this year  competition  is  55%  of the  number  of all participants. The  ratio  of the number  of juniors  boys  to  the number of senior  boys  is equal  to the  ratio of the  number of juniors  to the number of seniors. Find  the ratio  of the number of junior  boys  to the number  of  junior  girls.
Answer: \frac{11}{9}
Problem: Find all natural numbers  $n$  for which both  $n^n + 1$  and  $(2n)^{2n} + 1$  are prime numbers.
Answer:  n = 2
Problem: Determine all functions  $f : N_0 \to R$  satisfying  $f (x+y)+ f (x-y)= f (3x)$  for all  $x,y$ .
Answer: f(x) = 0
Problem:  $ 2n\plus{}1$  lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?
Answer: \frac{1}{4} \binom{2n + 2}{3}
Problem: The set Φ consists of a finite number of points on the plane. The distance between any two points from Φ is at least  $\sqrt{2}$ . It is known that a regular triangle with side lenght  $3$  cut out of paper can cover all points of Φ. What is the greatest number of points that Φ can consist of?
Answer: 3
Problem: Consider a sequence of eleven squares that have side lengths  $3, 6, 9, 12,\ldots, 33$ . Eleven copies of a single square each with area  $A$  have the same total area as the total area of the eleven squares of the sequence. Find  $A$ .
Answer: 414
Problem: Find all pairs of odd integers  $a$  and  $b$  for which there exists a natural number $ c$  such that  the number  $\frac{c^n+1}{2^na+b}$  is integer for all natural  $n$ .
Answer:  (a, b) = (1, 1)
Problem: For how many integers  $k$  does the following system of equations has a solution other than  $a=b=c=0$  in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b), b^2+c^2 = ka(b+c), c^2+a^2=kb(c+a).\end{cases}\end{align*}
Answer: 2
Problem: Find all prime numbers  $p, q$  and  $r$  such that  $p>q>r$  and the numbers  $p-q, p-r$  and  $q-r$  are also prime.
Answer:  (p, q, r) = (7, 5, 2)
Problem: Find the maximal possible value of  $n$  such that there exist points  $P_1,P_2,P_3,\ldots,P_n$  in the plane and real numbers  $r_1,r_2,\ldots,r_n$  such that the distance between any two different points  $P_i$  and  $P_j$  is  $r_i+r_j$ .
Answer:  n = 4
Problem: A train traveling at  $ 80$  mph begins to cross a  $ 1$  mile long bridge. At this moment, a man begins to walk from the front of the train to the back of the train at a speed of  $5$  mph. The man reaches the back of the train as soon as the train is completely off the bridge. What is the length of the train (as a fraction of a mile)?
Answer: \frac{1}{16} \text{ miles}
Problem: There are 1000 doors  $D_1, D_2, . . . , D_{1000}$  and 1000 persons  $P_1, P_2, . . . , P_{1000}$ .
Initially all the doors were closed. Person  $P_1$  goes and opens all the doors.
Then person  $P_2$  closes door  $D_2, D_4, . . . , D_{1000}$   and leaves the odd numbered doors open. Next  $P_3$  changes the state of every third door, that
is,  $D_3, D_6, . . . , D_{999}$  . (For instance,  $P_3$  closes the open door  $D_3$  and opens
the closed door D6, and so on). Similarly,  $P_m$  changes the state of the
the doors  $D_m, D_{2m}, D_{3m}, . . . , D_{nm}, . . .$  while leaving the other doors untouched. Finally,  $P_{1000}$  opens  $D_{1000}$  if it was closed or closes it if it were
open. At the end, how many doors will remain open?
Answer: 31
Problem: Let  $P(x) = x^2 - 20x - 11$ . If  $a$  and  $b$  are natural numbers such that  $a$  is composite,  $\gcd(a, b) = 1$ , and  $P(a) = P(b)$ , compute  $ab$ .

Note:  $\gcd(m, n)$  denotes the greatest common divisor of  $m$  and  $n$ .
Answer: 99
Problem: The positive numbers  $a, b, c,d,e$  are such that the following identity hold for all real number  $x$ : $(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$ . Find the smallest value of  $d$ .
Answer: 1
Problem:  $\textbf{Problem 4.}$  The number of perfect inhabitants of a city was a perfect square, in other words, a whole number squared. with  $100$  people plus the new number of inhabitants turned out to be a perfect square plus one. Now, with another increase of  $100$  people, the number of inhabitants is again a perfect square. What was the number of inhabitants original city?
Answer: 49^2
Problem: A square of side  $n$  is formed from  $n^2$  unit squares, each colored in red, yellow or green. Find minimal  $n$ , such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column).
Answer:  n = 7
Problem: On a chess board ( $8*8$ ) there are written the numbers  $1$  to  $64$ : on the first line, from left to right, there are the numbers  $1, 2, 3, ... , 8$ ; on the second line, from left to right, there are the numbers  $9, 10, 11, ... , 16$ ;etc. The  $\"+\"$  and  $\"-\"$  signs are put to each number such that, in each line and in each column, there are  $4$   $\"+\"$  signs and  $4$   $\"-\"$  signs. Then, the  $64$  numbers are added. Find all the possible values of this sum.
Answer: 0
Problem: Solve the equation
  $mn =$  (gcd( $m,n$ )) $^2$  + lcm( $m, n$ ) 
in positive integers, where gcd( $m, n$ ) – greatest common divisor of  $m,n$ , and lcm( $m, n$ ) – least common multiple of  $m,n$ .
Answer: (m, n) = (2, 4) \text{ or } (4, 2)
Problem: A paper equilateral triangle with area 2019 is folded over a line parallel to one of its sides.  What is the greatest possible area of the overlap of folded and unfolded parts of the triangle?
Answer: 673
Problem: Source: 1976 Euclid Part B Problem 1
-----
Triangle  $ABC$  has  $\angle{B}=30^{\circ}$ ,  $AB=150$ , and  $AC=50\sqrt{3}$ . Determine the length of  $BC$ .
Answer: 50\sqrt{3}
Problem: Let  $x_1, x_2,... , x_{84}$  be the roots of the equation  $x^{84} + 7x - 6 = 0$ . Compute  $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$ .
Answer: \frac{77}{2}
Problem: Let  $a_{i}$  and  $b_{i}$  ( $i=1,2, \cdots, n$ ) be rational numbers such that for any real number  $x$  there is:
\[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\]
Find the least possible value of  $n$ .
Answer: 5