rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have.

(Two cells of the grid are adjacent if they have a common vertex)
Answer: 5
Problem: A rectangle has area  $A$  and perimeter  $P$ .  The largest possible value of  $\tfrac A{P^2}$  can be expressed as  $\tfrac mn$ , where  $m$  and  $n$  are relatively prime positive integers.  Compute  $m+n$ .
Answer: 17
Problem: Consider the polynomial  $p(x)=x^{1999}+2x^{1998}+3x^{1997}+\ldots+2000$ . Find a nonzero polynomial whose roots are the reciprocal values of the roots of  $p(x)$ .
Answer: 1 + 2x + 3x^2 + dots + 2000x^{1999}
Problem: For a constant  $c$ , a sequence  $a_n$  is defined by  $a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).$ Find  $\lim_{n\to\infty} a_n$ .
Answer: 0
Problem: Determine all integers  $x$  satisfying
\[ \left[\frac{x}{2}\right] \left[\frac{x}{3}\right] \left[\frac{x}{4}\right] = x^2. \]
( $[y]$  is the largest integer which is not larger than  $y.$ )
Answer: 0 \text{ and } 24
Problem: Determine all the functions  $f:\mathbb R\mapsto\mathbb R$  satisfies the equation  $f(a^2 +ab+ f(b^2))=af(b)+b^2+ f(a^2)\,\forall a,b\in\mathbb R $ 
Answer: f(x) \equiv x \text{ or } f(x) \equiv -x
Problem: The centre of the circle passing through the midpoints of the sides of am isosceles triangle  $ABC$  lies on the circumcircle of triangle  $ABC$ . If the larger angle of triangle  $ABC$  is  $\alpha^{\circ}$  and the smaller one  $\beta^{\circ}$  then what is the value of  $\alpha-\beta$ ?
Answer: 0^	ext{circ}
Problem: Some of the  $n + 1$  cities in a country (including the capital city) are connected by one-way or two-way airlines. No two cities are connected by both a one-way airline and a two-way airline, but there may be more than one two-way airline between two cities. If  $d_A$  denotes the number of airlines from a city  $A$ , then  $d_A \le n$  for any city  $A$  other than the capital city and  $d_A + d_B \le n$  for any two cities  $A$  and  $B$  other than the capital city which are not connected by a two-way airline. Every airline has a return, possibly consisting of several connected flights. Find the largest possible number of two-way airlines and all configurations of airlines for which this largest number is attained.
Answer: \frac{n^2 + n}{2}
Problem: Let  $p_1 = 2012$  and  $p_n = 2012^{p_{n-1}}$  for  $n > 1$ . Find the largest integer  $k$  such that  $p_{2012}- p_{2011}$  is divisible by  $2011^k$ .
Answer: 1
Problem: Let  $f(x)=x^2+bx+1,$  where  $b$  is a  real number. Find the number of integer solutions to the inequality  $f(f(x)+x)<0.$ 
Answer: 2
Problem: Given an integer  $n\geq 3$ , determine the smallest positive number  $k$  such that any two points in any  $n$ -gon (or at its boundary) in the plane can be connected by a polygonal path consisting of  $k$  line segments contained in the  $n$ -gon (including its boundary). 
Answer: n-1
Problem: How many ways are there to rearrange the letters of the word RAVEN such that no two vowels are consecutive?
Answer: 72
Problem: Let  $ABC$  be a triangle with  $AB = 42$ ,  $AC = 39$ ,  $BC = 45$ . Let  $E$ ,  $F$  be on the sides  $\overline{AC}$  and  $\overline{AB}$  such that  $AF = 21, AE = 13$ . Let  $\overline{CF}$  and  $\overline{BE}$  intersect at  $P$ , and let ray  $AP$  meet  $\overline{BC}$  at  $D$ .  Let  $O$  denote the circumcenter of  $\triangle DEF$ , and  $R$  its circumradius.  Compute  $CO^2-R^2$ .
Answer: 0
Problem: Let  $P$  be a cyclic polygon with circumcenter  $O$  that does not lie on any diagonal, and let  $S$  be the set of points on 2D plane containing  $P$  and  $O$ . 

The  $\textit{Matcha Sweep Game}$  is a game between two players  $A$  and  $B$ , with  $A$  going first, such that each choosing a nonempty subset  $T$  of points in  $S$  that has not been previously chosen, and such that if  $T$  has at least  $3$  vertices then  $T$  forms a convex polygon. The game ends with all points have been chosen, with the player picking the last point wins. 

For which polygons  $P$  can  $A$  guarantee a win?
Answer:  P
Problem: P(x) and Q(x) are polynomials of positive degree such that for all x P(P(x))=Q(Q(x)) and P(P(P(x)))=Q(Q(Q(x))). Does this necessarily mean that P(x)=Q(x)?
Answer:  P(x) = Q(x)
Problem: Determine all non-constant monic polynomials  $f(x)$  with integer coefficients for which there exists a natural number  $M$  such that for all  $n \geq M$ ,  $f(n)$  divides  $f(2^n) - 2^{f(n)}$
Answer:  f(x) = x
Problem: Let  $m$  and  $n$  be relatively prime positive integers. Determine all possible values of \[\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1).\]
Answer: 1 \text{ and } 7
Problem: Let  $ f : \mathbf {R} \to \mathbf {R} $  be a smooth function such that  $ f'(x)^2 = f(x) f''(x) $  for all  $x$ . Suppose  $f(0)=1$  and  $f^{(4)} (0) = 9$ . Find all possible values of  $f'(0)$ .
Answer: \pm \sqrt{3}
Problem: **8.** Find all integers  $a>1$  for which the least (integer) solution  $n$  of the congruence  $a^{n} \equiv 1 \pmod{p}$  differs from 6 (p is any prime number). **(N. 9)**
Answer: a = 2
Problem: Consider the part of a lattice given by the corners  $(0, 0), (n, 0), (n, 2)$  and  $(0, 2)$ . 
From a lattice point  $(a, b)$  one can move to  $(a + 1, b)$  or to  $(a + 1, b + 1)$  or to  $(a, b - 1$ ), provided that the second point is also contained in the part of the lattice.
How many ways are there to move from  $(0, 0)$  to  $(n, 2)$  considering these rules?
Answer:  a_n
Problem: Alice, Bob, Cindy, David, and Emily sit in a circle. Alice refuses to sit to the right of Bob, and Emily sits next to Cindy. If David sits next to two girls, determine who could sit immediately to the right of Alice.
Answer: Bob
Problem: A permutation  $a_1,a_2,\cdots ,a_6$  of numbers  $1,2,\cdots ,6$  can be transformed to  $1,2,\cdots,6$  by transposing two numbers exactly four times. Find the number of such permutations.
Answer: 360
Problem: Serge and Tanya want to show Masha a magic trick. Serge leaves the room. Masha writes down a sequence  $(a_1, a_2, \ldots , a_n)$ , where all  $a_k$  equal  $0$  or  $1$ . After that Tanya writes down a sequence  $(b_1, b_2, \ldots , b_n)$ , where all  $b_k$  also equal  $0$  or  $1$ . Then Masha either does nothing or says “Mutabor” and replaces both sequences: her own sequence by  $(a_n, a_{n-1}, \ldots , a_1)$ , and Tanya’s sequence by  $(1 - b_n, 1 - b_{n-1}, \ldots , 1 - b_1)$ . Masha’s sequence is covered by a napkin, and Serge is invited to the room. Serge should look at Tanya’s sequence and tell the sequence covered by the napkin. For what  $n$  Serge and Tanya can prepare and show such a trick? Serge does not have to determine whether the word “Mutabor” has been pronounced.
Answer:  n must be even
Problem: Find all positive integers  $a$  such that for any positive integer  $n\ge 5$  we have   $2^n-n^2\mid a^n-n^a$ .
Answer: a = 2 \text{ or } a = 4
Problem: Determine all pairs  $ (n,p)$  of positive integers, where  $ p$  is prime, such that  $ 3^p\minus{}np\equal{}n\plus{}p$ .
Answer:  (n, p) = (6, 3)
Problem: A set  $S$  has  $7$  elements. Several  $3$ -elements subsets of  $S$  are listed, such
that any  $2$  listed subsets have exactly  $1$  common element. What is the maximum number of subsets that can be listed?

Answer: 7
Problem:  $f:\mathbb{R}\rightarrow \mathbb{R}$  satisfies the equation
\[f(x)f(y)-af(xy)=x+y\]
, for every real numbers  $x,y$ . Find all possible real values of  $a$ .
Answer: a = 1, -1
Problem: Let  $x$ ,  $y$ , and  $z$  be real numbers such that  $x+y+z=20$  and  $x+2y+3z=16$ . What is the value of  $x+3y+5z$ ?
Answer: 12
Problem: Let  $X$  be a set containing  $n$  elements. Find the number of ordered triples  $(A,B, C)$  of subsets of  $X$  such that  $A$  is a subset of  $B$  and  $B$  is a proper subset of  $C$ .
Answer: 4^n - 3^n
Problem: Given  $2014$  points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected points?
Answer: 2013
Problem: In a triangle two altitudes are not smaller than the sides on to which they are dropped. Find the angles of the triangle.
Answer:  45^	heta, 45^	heta, 90^	heta
Problem: Let  $\Gamma$  be a circle in the plane and  $S$  be a point on  $\Gamma$ . Mario and Luigi drive around the circle  $\Gamma$  with their go-karts. They both start at  $S$  at the same time. They both drive for exactly  $6$  minutes at constant speed counterclockwise around the circle. During these  $6$  minutes, Luigi makes exactly one lap around  $\Gamma$  while Mario, who is three times as fast, makes three laps. 

While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana.

How many points in the plane, apart from  $S$ , are marked with a banana by the end of the  $6$  minutes.

Answer: 5
Problem: Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid  $5\$ $. How much did the ball cost?
Answer: 20
Problem: Find all real solutions  $(x,y,z)$  of the system of equations  $$ \begin{cases} x^3 = 2y-1 y^3 = 2z-1 z^3 = 2x-1\end{cases} $$  
Answer: (1, 1, 1)
Problem: A student read the book with  $480$  pages two times. If he in the second time for every day read  $16$  pages more than in the first time and he finished it  $5$  days earlier than in the first time. For how many days did he read the book in the first time?
Answer: 15
Problem: Find the positive integers  $n$  that are not divisible by  $3$  if the number  $2^{n^2-10}+2133$  is a perfect cube.

<details><summary>Note</summary><span style="color:#BF0000">The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation  $x^3=2^{n^2-10}+2133$  where  $x,n \in \mathbb{N}$  and  $3\nmid n$ .</span></details>
Answer: n = 4
Problem: Let  $f(n) = \sum^n_{d=1} \left\lfloor \frac{n}{d} \right\rfloor$  and  $g(n) = f(n) -f(n - 1)$ . For how many  $n$  from  $1$  to  $100$  inclusive is  $g(n)$  even?
Answer: 90
Problem: Find all functions  $ f : Z\rightarrow Z$  for which we have  $ f (0) \equal{} 1$  and  $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$ , for every natural number  $ n$ .
Answer: f(n) = 1 - n
Problem:  $56$  lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly  $594$  points, what is the maximum number of them that could have the same slope? 
Answer: 44
Problem: Let  $ABC$  be an acute-angled triangle with  $AB =4$  and  $CD$  be the altitude through  $C$  with  $CD = 3$ . Find the distance between the midpoints of  $AD$  and  $BC$ 
Answer: 2.5
Problem: Maria and Bilyana play the following game. Maria has  $2024$  fair coins and Bilyana has  $2023$  fair coins. They toss every coin they have. Maria wins if she has strictly more heads than Bilyana, otherwise Bilyana wins. What is the probability of Maria winning this game?
Answer: \frac{1}{2}
Problem: Define a regular  $n$ -pointed star to be the union of  $n$  line segments  $P_1P_2, P_2P_3,\ldots, P_nP_1$  such that 
 $\bullet$  the points  $P_1, P_2,\ldots, P_n$  are coplanar and no three of them are collinear, $\bullet$  each of the  $n$  line segments intersects at least one of the other line segments at a point other than an endpoint, $\bullet$  all of the angles at  $P_1, P_2,\ldots, P_n$  are congruent,  $\bullet$  all of the  $n$  line segments  $P_2P_3,\ldots, P_nP_1$  are congruent, and  $\bullet$  the path  $P_1P_2, P_2P_3,\ldots, P_nP_1$  turns counterclockwise at an angle of less than 180 degrees at each vertex. 

There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?
Answer: 199
Problem: Given a random string of 33 bits (0 or 1), how many (they can overlap) occurrences of two consecutive 0's would you expect? (i.e. "100101" has 1 occurrence, "0001" has 2 occurrences)
Answer: 8
Problem: In this problem, the symbol  $0$  represents the number zero, the symbol  $1$  represents the number seven, the symbol  $2$  represents the number five, the symbol  $3$  represents the number three, the symbol  $4$  represents the number four, the symbol  $5$  represents the number two, the symbol  $6$  represents the number nine, the symbol  $7$  represents the number one, the symbol  $8$  represents an arbitrarily large positive integer, the symbol  $9$  represents the number six, and the symbol  $\infty$  represents the number eight. Compute the value of  $\left|0-1+2-3^4-5+6-7^8\times9-\infty\right|$ .
Answer: 90
Problem: Find all functions  $f: \mathbb N \cup \{0\} \to \mathbb N\cup \{0\}$  such that  $f(1)>0$  and 
\[f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2 \quad \forall m,n \in \mathbb N\cup \{0\}.\]
Answer:  f(n) = n
Problem: Let  $ p > 2$  be a prime number. Find the least positive number  $ a$  which can be represented as
\[ a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X),
\]
where  $ f(X)$  and  $ g(X)$  are integer polynomials.

*Mircea Becheanu*.
Answer: p
Problem: A rectangle has an area of  $16$  and a perimeter of  $18$ ; determine the length of the diagonal of the rectangle.
Answer: 7
Problem: Sally rolls an  $8$ -sided die with faces numbered  $1$  through  $8$ . Compute the probability that she gets a power of  $2$ .
Answer: \frac{1}{2}
Problem: In one class in the school, number of abscent students is  $\frac{1}{6}$  of number of students who were present. When teacher sent one student to bring chalk, number of abscent students was  $\frac{1}{5}$  of number of students who were present. How many students are in that class?
Answer: 7
Problem: Let  $ n>1$  and for  $ 1 \leq k \leq n$  let  $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$  be the sum of the products of all possible combinations of k of the numbers  $ a_1,a_2,...,a_n$ . Furthermore let  $ P \equal{} P(a_1, a_2, . . . , a_n)$  be the sum of all  $ p_k$  with odd values of  $ k$  less than or equal to  $ n$ .

How many different values are taken by  $ a_j$  if all the numbers  $ a_j (1 \leq j \leq n)$  and  $ P$  are prime?
Answer: 2
Problem: There were three times as many red candies as blue candies on a table. After Darrel took the same number of red candies and blue candies, there were four times as many red candies as blue candies left on the table. Then after Cloe took  $12$  red candies and  $12$  blue candies, there were five times as many red candies as blue candies left on the table. Find the total number of candies that Darrel took.
Answer: 48
Problem: Find the number of pairs of integers  $x$  and  $y$  such that  $x^2 + xy + y^2 = 28$ .
Answer: 4
Problem: Find all the two-digit numbers  $\overline{ab}$  that squared give a result where the last two digits are  $\overline{ab}$ .
Answer: 25 \text{ and } 76
Problem: **Q12.** Find all positive integers  $P$  such that the sum and product of all its divisors are  $2P$  and  $P^2$ , respectively.
Answer: 6
Problem: Let  $ABC$  be a triangle. Choose points  $A'$ ,  $B'$  and  $C'$  independently on side segments  $BC$ ,  $CA$  and  $AB$  respectively with a uniform distribution. For a point  $Z$  in the plane, let  $p(Z)$  denote the probability that  $Z$  is contained in the triangle enclosed by lines  $AA'$ ,  $BB'$  and  $CC'$ . For which interior point  $Z$  in triangle  $ABC$  is  $p(Z)$  maximised?
Answer: Z is the centroid of the triangle ABC.
Problem: Call a positive integer  $N$  a  $\textit{7-10 double}$  if the digits of the base-7 representation of  $N$  form a base-10 number that is twice  $N.$   For example, 51 is a 7-10 double because its base-7 representation is 102.  What is the largest 7-10 double?
Answer: 315
Problem: Let  $A$  be a set containing  $4k$  consecutive positive integers, where  $k \geq 1$  is an
integer. Find the smallest  $k$  for which the set A can be partitioned into two subsets
having the same number of elements, the same sum of elements, the same sum
of the squares of elements, and the same sum of the cubes of elements.
Answer:  k = 4
Problem: Draw the graph of the function  $y = \frac{| x^3 - x^2 - 2x | }{3} - | x + 1 |$ .
Answer:  y = \frac{| x^3 - x^2 - 2x | }{3} - | x + 1 |
Problem: Let  $p=2017$  be a prime.  Find the remainder when \[\left\lfloor\dfrac{1^p}p\right\rfloor + \left\lfloor\dfrac{2^p}p\right\rfloor+\left\lfloor\dfrac{3^p}p\right\rfloor+\cdots+\left\lfloor\dfrac{2015^p}p\right\rfloor \] is divided by  $p$ .  Here  $\lfloor\cdot\rfloor$  denotes the greatest integer function.
Answer: 0
Problem: Let  $n$  be a five digit number (whose first digit is non-zero) and let  $m$  be the four digit number formed from  $n$  by removing its middle digit. Determine all  $n$  such that  $n/m$  is an integer.
Answer: n = \overline{ab000}
Problem: Given that  $r$  and  $s$  are relatively prime positive integers such that  $\dfrac{r}{s}=\dfrac{2(\sqrt{2}+\sqrt{10})}{5\left(\sqrt{3+\sqrt{5}}\right)}$ , find  $r$  and  $s$ .
Answer:  (r, s) = (4, 5)
Problem: Find all non-negative integers  $ x,y,z$  such that  $ 5^x \plus{} 7^y \equal{} 2^z$ .
 :lol: 
(*Daniel Kohen, University of Buenos Aires - Buenos Aires,Argentina*)
Answer:  (0, 0, 1), (0, 1, 3), (2, 1, 5)
Problem: Find all triples  $(a,b,c)$  of real numbers all different from zero that satisfies: 

\begin{eqnarray}  a^4+b^2c^2=16a\nonumber  b^4+c^2a^2=16b \nonumber c^4+a^2b^2=16c \nonumber \end{eqnarray}

Answer: (2, 2, 2)
Problem: Suppose that for some positive integer  $n$ , the first two digits of  $5^n$  and  $2^n$  are identical. Suppose the first two digits are  $a$  and  $b$  in this order. Find the two-digit number  $\overline{ab}$ .
Answer: 31
Problem: Determine all triples  $\left( x,y,z\right)$  of positive integers for which the number \[ \sqrt{\frac{2005}{x+y}}+\sqrt{\frac{2005}{y+z}}+\sqrt{\frac{2005}{z+x}} \] is an integer .
Answer:  (2005 \cdot 2, 2005 \cdot 2, 2005 \cdot 14)
Problem: Given a right-angled triangle with perimeter  $18$ . The sum of the squares
of the three side lengths is  $128$ . What is the area of the triangle?

Answer: 9
Problem: There exist two positive numbers  $ x$  such that  $ \sin(\arccos(\tan(\arcsin x)))\equal{}x$ . Find the product of the two possible  $ x$ .
Answer: 1
Problem: Jeffrey jogs  $3$  miles at  $4$  miles per hour. Then, he walks  $4$  miles at  $3$  miles an hour. What was Jeffrey's average speed in miles per hour for the entire trip? Express your answer as a decimal to the nearest hundredth.
Answer: 3.36
Problem: Determine all trios of integers  $(x, y, z)$  which are solution of system of equations $\begin{cases} x - yz = 1  xz + y = 2 \end{cases}$ 
Answer: (x, y, z) \in \{(1, 0, 2), (1, 2, 0)\}
Problem: Suppose that  $x^{10} + x + 1 = 0$  and  $x^100 = a_0 + a_1x +... + a_9x^9$ . Find  $a_5$ .
Answer: -252
Problem: It is given the function  $f:\mathbb{R} \to \mathbb{R}$  such that it holds  $f(\sin x)=\sin (2011x)$ . Find the value of  $f(\cos x)$ .
Answer: \cos(2011x)
Problem: Given is a number with  $1998$  digits which is divisible by  $9$ . Let  $x$  be the sum of its digits, let  $y$  be the sum of the digits of  $x$ , and  $z$  the sum of the digits of  $y$ . Find  $z$ .
Answer: 9
Problem: Apples cost  $2$  dollars. Bananas cost  $3$  dollars. Oranges cost  $5$  dollars. Compute the number of distinct baskets of fruit such that there are  $100$  pieces of fruit and the basket costs  $300$  dollars. Two baskets are distinct if and only if, for some type of fruit, the two baskets have differing amounts of that fruit.
Answer: 34
Problem: Jamie, Linda, and Don bought bundles of roses at a ﬂower shop, each paying the same price for each
bundle. Then Jamie, Linda, and Don took their bundles of roses to a fair where they tried selling their
bundles for a ﬁxed price which was higher than the price that the ﬂower shop charged. At the end of the
fair, Jamie, Linda, and Don donated their unsold bundles of roses to the fair organizers. Jamie had bought
20 bundles of roses, sold 15 bundles of roses, and made  $60$  proﬁt. Linda had bought 34 bundles of roses,
sold 24 bundles of roses, and made $69 proﬁt. Don had bought 40 bundles of roses and sold 36 bundles of
roses. How many dollars proﬁt did Don make?

Answer: 252
Problem: Consider a regular pentagon  $ABCDE$ , and let the intersection of diagonals  $\overline{CA}$  and  $\overline{EB}$  be  $F$ . Find  $\angle AFB$ .
Answer: 108^	extcirc
Problem:  $(HUN 6)$  Find the positions of three points  $A,B,C$  on the boundary of a unit cube such that  $min\{AB,AC,BC\}$  is the greatest possible.
Answer: \sqrt{2}
Problem: Given that there are  $24$  primes between  $3$  and  $100$ , inclusive, what is the number of ordered pairs  $(p, a)$  with  $p$  prime,  $3 \le p < 100$ , and  $1 \le a < p$  such that  $p \mid (a^{p-2} - a)$ ?
Answer: 48
Problem: How many different values can  $ \angle ABC$  take, where  $ A,B,C$  are distinct vertices of a cube?
Answer: 3
Problem: Let  $ABC$  is a triangle with  $\angle BAC=\frac{\pi}{6}$  and the circumradius equal to 1. If  $X$  is a point inside or in its boundary let  $m(X)=\min(AX,BX,CX).$  Find all the angles of this triangle if  $\max(m(X))=\frac{\sqrt{3}}{3}.$ 
Answer:  \angle A = 30^
\circ , \angle B = 120^
\circ , \angle C = 30^
\circ
Problem: The graphs of the equations \[ y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k, \] are drawn in the coordinate plane for  $k=-10,-9,-8,\ldots,9,10.$   These 63 lines cut part of the plane into equilateral triangles of side  $2/\sqrt{3}.$   How many such triangles are formed?
Answer: 660
Problem: According to Moor's Law, the number of shoes in Moor's room doubles every year.  In 2013, Moor's room starts out having exactly one pair of shoes.  If shoes always come in unique, matching pairs, what is the earliest year when Moor has the ability to wear at least 500 mismatches pairs of shoes?  Note that left and right shoes are distinct, and Moor must always wear one of each.
Answer: 2018
Problem: Let  $m$  be a fixed integer greater than  $1$ . The sequence  $x_0$ ,  $x_1$ ,  $x_2$ ,  $\ldots$  is defined as follows: 
\[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\]
Find the greatest  $k$  for which the sequence contains  $k$  consecutive terms divisible by  $m$  .
Answer: m-1
Problem: Find all prime numbers  $p$  such that  $ p = m^2 + n^2$  and  $p\mid m^3+n^3-4$ .
Answer: 2
Problem: Trains  $A$  and  $B$  are on the same track a distance  $100$  miles apart heading towards one another, each at a speed of  $50$  miles per hour. A fly starting out at the front of train  $A$  flies towards train  $B$  at a speed of  $75$  miles per hour. Upon reaching train  $B$ , the fly turns around and flies towards train  $A$ , again at  $75$  miles per hour. The fly continues flying back and forth between the two trains at  $75$  miles per hour until the two trains hit each other. How many minutes does the fly spend closer to train  $A$  than to train  $B$  before getting squashed?
Answer: \frac{130}{3}
Problem: Determine the greatest positive integer  $ n$  such that in three-dimensional space, there exist n points  $ P_{1},P_{2},\cdots,P_{n},$  among  $ n$  points no three points are collinear, and for arbitary  $ 1\leq i < j < k\leq n$ ,  $ P_{i}P_{j}P_{k}$  isn't obtuse triangle.
Answer: 8
Problem: Every member, starting from the third one, of two sequences  $\{a_n\}$  and  $\{b_n\}$  equals to the sum of two preceding ones. First members are:  $a_1 = 1, a_2 = 2, b_1 = 2, b_2 = 1$ . How many natural numbers are encountered in both sequences (may be on the different places)?
Answer: 3