rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Let  $p = 9001$  be a prime number and let  $\mathbb{Z}/p\mathbb{Z}$  denote the additive group of integers modulo  $p$ .  Furthermore, if  $A, B \subset \mathbb{Z}/p\mathbb{Z}$ , then denote  $A+B = \{a+b \pmod{p} | a \in A, b \in B \}.$  Let  $s_1, s_2, \dots, s_8$  are positive integers that are at least  $2$ . Yang the Sheep notices that no matter how he chooses sets  $T_1, T_2, \dots, T_8\subset \mathbb{Z}/p\mathbb{Z}$  such that  $|T_i| = s_i$  for  $1 \le i \le 8,$   $T_1+T_2+\dots + T_7$  is never equal to  $\mathbb{Z}/p\mathbb{Z}$ , but  $T_1+T_2+\dots+T_8$  must always be exactly  $\mathbb{Z}/p\mathbb{Z}$ . What is the minimum possible value of  $s_8$ ?

[i]Proposed by Yang Liu
Answer: 2
Problem: A circle of length  $10$  is inscribed in a convex polygon with perimeter  $15$ . What part of the area of this polygon is occupied by the resulting circle?
Answer: \frac{2}{3}
Problem: Suppose that  $n$  is a positive integer number. Consider a regular polygon with  $2n$  sides such that one of its largest diagonals is parallel to the  $x$ -axis. Find the smallest integer  $d$  such that there is a polynomial  $P$  of degree  $d$  whose graph intersects all sides of the polygon on points other than vertices.

Proposed by Mohammad Ahmadi
Answer: n
Problem: Let  $\triangle ABC$  be an acute triangle,  $H$  is its orthocenter.  $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$  intersect  $\triangle ABC$ 's circumcircle at  $A',B',C'$  respectively. Find the range (minimum value and the maximum upper bound) of  $$ \dfrac{AH}{AA'}+\dfrac{BH}{BB'}+\dfrac{CH}{CC'} $$  
Answer:  \left[\frac{3}{2}, 2\right)
Problem: Find all the integers  $x$  in  $[20, 50]$  such that  $6x+5\equiv 19 \mod 10$ , that is,  $10$  divides  $(6x+15)+19$ .
Answer: 24, 29, 34, 39, 44, 49
Problem: Let  $p$  be a prime number. All natural numbers from  $1$  to  $p$  are written in a row in ascending order. Find all  $p$  such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum.
Answer:  p = 3
Problem: Suppose  $P(x)$  is a monic polynomial of degree  $2023$  such that  $P(k) = k^{2023}P(1-\frac{1}{k})$  for every positive integer  $1 \leq k \leq 2023$ . Then  $P(-1) = \frac{a}{b}$  where  $a$  and  $b$  are relatively prime integers. Compute the unique integer  $0 \leq n < 2027$  such that  $bn-a$  is divisible by the prime  $2027$ .
Answer: 0
Problem: All of the digits of a seven-digit positive integer are either  $7$  or  $8.$  If this integer is divisible by  $9,$  what is the sum of its digits?
Answer: 54
Problem: Consider the polynomial  $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ , with degree less than or equal to 2.

When  $ f$  varies with subject to the constrain  $ f(0) \equal{} 0,\ f(2) \equal{} 2$ , find the minimum value of  $ S\equal{}\int_0^2 |f'(x)|\ dx$ .
Answer: 2
Problem: Find all pairs  $(a,b)$  of positive integers  such that  $a!+b$  and  $b!+a$  are both powers of  $5$ .
Answer: (5, 5)
Problem: Given the positive numbers  $a$  and  $b$  and the natural number  $n$ , find the greatest among the  $n + 1$  monomials in the binomial expansion of  $\left(a+b\right)^n$ .
Answer:  a^n
Problem: Let  $P$  be the product of the first  $50$  nonzero square numbers. Find the largest integer  $k$  such that  $7^k$  divides  $P$ .
Answer: 16
Problem: In how many ways can one arrange seven white and five black balls in a line in such a way that there are no two neighboring black balls?
Answer: 56
Problem: Given that  $x$  and  $y$  are nonzero real numbers such that  $x+\frac{1}{y}=10$  and  $y+\frac{1}{x}=\frac{5}{12}$ , find all possible values of  $x$ .
Answer: 4, 6
Problem: Find all pairs of integers  $(m,n)$  such that an  $m\times n$  board can be totally covered with  $1\times 3$  and  $2 \times 5$  pieces.
Answer: 4 \times 4
Problem: Suppose we have a  $8\times8$  chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into  $1\times2$  domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers.

(Day 1, 3rd problem
author: Michal Rolínek)
Answer: 4
Problem: Volume  $A$  equals one fourth of the sum of the volumes  $B$  and  $C$ , while volume  $B$  equals one sixth of the sum of the volumes  $A$  and  $C$ . There are relatively prime positive integers  $m$  and  $n$  so that the ratio of volume  $C$  to the sum of the other two volumes is  $\frac{m}{n}$ . Find  $m+n$ .
Answer: 35
Problem: The area of a circle (in square inches) is numerically larger than its circumference (in inches). What is the smallest possible integral area of the circle, in square inches?
Answer: 29
Problem: Find all positive integer  $n$  and prime number  $p$  such that  $p^2+7^n$  is a perfect square 
Answer: (n, p) = (1, 3)
Problem: For any  $0\leq x_1,x_2,\ldots,x_{2011} \leq 1$ , Find the maximum value of \begin{align*} \sum_{k=1}^{2011}(x_k-m)^2 \end{align*} where  $m$  is the arithmetic mean of  $x_1,x_2,\ldots,x_{2011}$ .
Answer: \frac{1005 \cdot 1006}{2011}
Problem: Find all positive integers  $n$  for which both numbers \[1\;\;\!\!\!\!\underbrace{77\ldots 7}_{\text{ $n$  sevens}}\!\!\!\!\quad\text{and}\quad 3\;\; \!\!\!\!\underbrace{77\ldots 7}_{\text{ $n$  sevens}}\] are prime.
Answer:  n = 1
Problem: The increasing sequence  $1; 3; 4; 9; 10; 12; 13; 27; 28; 30; 31, \ldots$  is formed with positive integers which are powers of  $3$  or sums of different powers of  $3$ . Which number is in the  $100^{th}$  position?
Answer: 981
Problem: Find all integers  $ n \ge 3$  such that there are  $ n$  points in space, with no three on a line and no four on a circle, such that all the circles pass through three points between them are congruent.
Answer:  n = 4
Problem: In the expansion of  $(2x +3y)^{20}$ , find the number of coefficients divisible by  $144$ .
Answer: 15
Problem: What is the largest number of squares on  $9 \times 9$  square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?
Answer: 21
Problem: In the cells of an  $8\times 8$  board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.
Answer: 36
Problem: Determine all positive integers  $n$  for which there exists an integer  $m$  such that  ${2^{n}-1}$  is a divisor of  ${m^{2}+9}$ .
Answer: all powers of 2
Problem: find all  $k$  distinct integers  $a_1,a_2,...,a_k$  such that there exists an injective function  $f$  from reals to themselves such that for each positive integer  $n$  we have  $$ \{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\} $$  .
Answer: \{0\}
Problem: The product of two positive numbers is equal to  $50$  times their sum and  $75$  times their difference.  Find their sum.
Answer: 360
Problem: Fix a sequence  $a_1,a_2,a_3\ldots$  of integers satisfying the following condition:for all prime numbers  $p$  and all positive integers  $k$ ,we have  $a_{pk+1}=pa_k-3a_p+13$ .Determine all possible values of  $a_{2013}$ .
Answer: 13
Problem: Let  $x$ ,  $y$ ,  $z$  be positive integers satisfying  $x<y<z$  and  $x+xy+xyz=37$ . Find the greatest possible value of  $x+y+z$ .
Answer: 20
Problem: Six members of the team of Fatalia for the International Mathematical Olympiad are selected from  $13$  candidates. At the TST the candidates got  $a_1,a_2, \ldots, a_{13}$  points with  $a_i \neq a_j$  if  $i \neq j$ .

The team leader has already  $6$  candidates and now wants to see them and nobody other in the team. With that end in view he constructs a polynomial  $P(x)$  and finds the creative potential of each candidate by the formula  $c_i = P(a_i)$ .

For what minimum  $n$  can he always find a polynomial  $P(x)$  of degree not exceeding  $n$  such that the creative potential of all  $6$  candidates is strictly more than that of the  $7$  others?
Answer: 12
Problem: The graphs of  $y=3(x-h)^2+j$  and  $y=2(x-h)^2+k$  have  $y$ -intercepts of  $2013$  and  $2014$ , respectively, and each graph has two positive integer  $x$ -intercepts. Find  $h$ .
Answer:  h = 36
Problem: Let  $a$ ,  $b$ ,  $c$ , and  $d$  be positive real numbers such that 
\[\begin{array}{c@{\hspace{3pt}} c@{\hspace{3pt}} c@{\hspace{3pt}} c@{\hspace{3pt}}c}a^2+b^2&=&c^2+d^2&=&2008, ac&=&bd&=&1000.\end{array}\]If  $S=a+b+c+d$ , compute the value of  $\lfloor S\rfloor$ .
Answer: 126
Problem: Find all  $ a, b, c \in \mathbb{Z} $ ,  $ c \ge 0 $  such that  $ a^n + 2^n | b^n + c $  for all positive integers  $ n $  where  $ 2ab $  is non-square.
Answer:  c = 0 \text{ or } c = 1
Problem: In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B.
Answer:  \angle B = 45^	au
Problem: A number is between  $500$  and  $1000$  and has a remainder of  $6$  when divided by  $25$  and a remainder of  $7$  when divided by  $9$ . Find the only odd number to satisfy these requirements.
Answer: 781
Problem: The squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers.

[i]Proposed by Clive Chan
Answer: 1008
Problem: The cheering team of Ubon Ratchathani University sits on the amphitheater that has  $441$  seats arranged into a  $21\times 21$  grid. Every seat is occupied by exactly one person, and each person has a blue sign and a yellow sign.

Count the number of ways for each person to raise one sign so that each row and column has an odd number of people raising a blue sign.
Answer: 2^{400}
Problem: Triangle  $ABC$  has an incenter  $I$ . Let points  $X$ ,  $Y$  be located on the line segments  $AB$ ,  $AC$  respectively, so that  $BX \cdot AB = IB^2$  and  $CY \cdot AC = IC^2$ . Given that the points  $X, I, Y$  lie on a straight line, find the possible values of the measure of angle  $A$ .
Answer: 90^	extcirc
Problem: The point  $(a,b)$  lies on the circle  $x^2+y^2=1$ . The tangent to the circle at this point meets the parabola  $y=x^2+1$  at exactly one point. Find all such points  $(a,b)$ .
Answer:  (0, 1)
Problem: I have five different pairs of socks. Every day for five days, I pick two socks at random without replacement to wear for the day. Find the probability that I wear matching socks on both the third day and the fifth day.
Answer: \frac{1}{63}
Problem: There is a fence that consists of  $n$  planks arranged in a line. Each plank is painted with one of the available  $100$  colors. Suppose that for any two distinct colors  $i$  and  $j$ , there is a plank with color  $i$  located to the left of a (not necessarily adjacent) plank with color  $j$ . Determine the minimum possible value of  $n$ .
Answer: 199
Problem: Suppose you have a  $6$  sided dice with  $3$  faces colored red,  $2$  faces colored blue, and  $1$  face colored green. You roll this dice  $20$  times and record the color that shows up on top. What is the expected value of the product of the number of red faces, blue faces, and green faces?
Answer: 31.6667
Problem: Find all functions  $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$  that satisfy the following two conditions:
[list] $\bullet\ f(n)$  is a perfect square for all  $n\in\mathbb{Z}_{>0}$  $\bullet\ f(m+n)=f(m)+f(n)+2mn$  for all  $m,n\in\mathbb{Z}_{>0}$ .[/list]
Answer: f(n) = n^2
Problem: Let  $n$  be a positive integer, and let  $a>0$  be a real number. Consider the equation: 
\[ \sum_{i=1}^{n}(x_i^2+(a-x_i)^2)= na^2 \]
How many solutions ( $x_1, x_2 \cdots , x_n$ ) does this equation have, such that:
\[ 0 \leq x_i \leq a, i \in N^+ \]
Answer: 2^n
Problem: Find all functions  $f : \mathbb{N} \to \mathbb{N}$  such that for any positive integers  $m, n$  the number  $$ (f(m))^2+ 2mf(n) + f(n^2) $$  is the square of an integer.
Answer:  f(n) = n
Problem: Two cubes  $A$  and  $B$  have different side lengths, such that the volume of cube  $A$  is numerically equal to the surface area of cube  $B$ . If the surface area of cube  $A$  is numerically equal to six times the side length of cube  $B$ , what is the ratio of the surface area of cube  $A$  to the volume of cube  $B$ ?
Answer: 7776
Problem: Michelle has a word with  $2^n$  letters, where a word can consist of letters from any alphabet. Michelle performs a swicheroo on the word as follows: for each  $k = 0, 1, \ldots, n-1$ , she switches the first  $2^k$  letters of the word with the next  $2^k$  letters of the word. For example, for  $n = 3$ , Michelle changes \[
ABCDEFGH \to BACDEFGH \to CDBAEFGH \to EFGHCDBA
\] in one switcheroo.

In terms of  $n$ , what is the minimum positive integer  $m$  such that after Michelle performs the switcheroo operation  $m$  times on any word of length  $2^n$ , she will receive her original word?

Answer: 2^n
Problem: A function  $f$  defined on integers such that $f (n) =n + 3$  if  $n$  is odd $f (n) = \frac{n}{2}$  if  $n$  is even

If  $k$  is an odd integer, determine the values for which  $f (f (f (k))) = k$ .
Answer: k = 1
Problem: A  triangle with perimeter  $7$  has integer sidelengths. What is the maximum possible area of such a triangle?
Answer: \frac{3\sqrt{7}}{4}
Problem: In an isosceles triangle, we drew one of the angle bisectors. At least one of the resulting two smaller ones  triangles is similar to the original. What can be the leg of the original triangle if the length of its base is  $1$  unit?
Answer: \frac{\sqrt{2}}{2} \text{ or } \frac{\sqrt{5} + 1}{2}
Problem: Find the smallest positive integer  $n$  for which we can find an integer  $m$  such that  $\left[\frac{10^n}{m}\right] = 1989$ .
Answer: n = 7
Problem: Given is a non-isosceles triangle  $ABC$  with  $\angle ABC=60^{\circ}$ , and in its interior, a point  $T$  is selected such that  $\angle ATC= \angle BTC=\angle BTA=120^{\circ}$ . Let  $M$   the intersection point of the medians in  $ABC$ . Let  $TM$  intersect  $(ATC)$  at  $K$ . Find  $TM/MK$ .
Answer: \frac{TM}{MK} = 2
Problem: Find the largest value that the expression can take  $a^3b + b^3a$  where  $a, b$  are non-negative real numbers, with  $a + b = 3$ .
Answer: \frac{81}{4}
Problem:  Let  $A_n$  be the number of arranged n-tuples of natural numbers  $(a_1,a_2…a_n)$ , such that $\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_n} =1$ .
Find the parity of  $A_{68}$ .

Answer: odd
Problem:  $A$  and  $B$  are on a circle of radius  $20$  centered at  $C$ , and  $\angle ACB = 60^\circ$ .  $D$  is chosen so that  $D$  is also on the circle,  $\angle ACD = 160^\circ$ , and  $\angle DCB = 100^\circ$ . Let  $E$  be the intersection of lines  $AC$  and  $BD$ . What is  $DE$ ?
Answer: 20
Problem: Let  $x$  be in the interval  $\left(0, \frac{\pi}{2}\right)$  such that  $\sin x - \cos x = \frac12$  . Then  $\sin^3 x + \cos^3 x = \frac{m\sqrt{p}}{n}$  , where  $m, n$ , and  $p$  are relatively prime positive integers, and  $p$  is not divisible by the square of any prime. Find  $m + n + p$ .
Answer: 28
Problem: Find  $a$  if  $a$  and  $b$  are integers such that  $x^2 - x - 1$  is a factor of  $ax^{17} + bx^{16} + 1$ .
Answer: 987
Problem: Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in  $3$ , Anne and Carl in  $5$ .  How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?
Answer: \frac{59}{20}
Problem: If \[2011^{2011^{2012}} = x^x\] for some positive integer  $x$ , how many positive integer factors does  $x$  have?
Answer: 2012
Problem: For every point on the plane, one of  $ n$  colors are colored to it such that:
 $ (1)$  Every color is used infinitely many times.
 $ (2)$  There exists one line such that all points on this lines are colored exactly by one of two colors.

Find the least value of  $ n$  such that there exist four concyclic points with pairwise distinct colors.
Answer:  n = 5
Problem:  $ k$  is a given natural number. Find all functions  $ f: \mathbb{N}\rightarrow\mathbb{N}$  such that for each  $ m,n\in\mathbb{N}$  the following holds:	\[ f(m)\plus{}f(n)\mid (m\plus{}n)^k\]
Answer: f(m) = m \text{ for all } m \in \mathbb{N}
Problem: Let  $n$  be a positive integer. Find the greatest possible integer  $m$ , in terms of  $n$ , with the following property: a table with  $m$  rows and  $n$  columns can be filled with real numbers in such a manner that for any two different rows  $\left[ {{a_1},{a_2},\ldots,{a_n}}\right]$  and  $\left[ {{b_1},{b_2},\ldots,{b_n}} \right]$  the following holds: \[\max\left( {\left| {{a_1} - {b_1}} \right|,\left| {{a_2} - {b_2}} \right|,...,\left| {{a_n} - {b_n}} \right|} \right) = 1\]
Answer: 2^n
Problem: A positive number  $\dfrac{m}{n}$  has the property that it is equal to the ratio of  $7$  plus the number’s reciprocal and  $65$  minus the number’s reciprocal.  Given that  $m$  and  $n$  are relatively prime positive integers, find  $2m + n$ .
Answer: 7
Problem: Peter is chasing after Rob.  Rob is running on the line  $y=2x+5$  at a speed of  $2$  units a second, starting at the point  $(0,5)$ .  Peter starts running  $t$  seconds after Rob, running at  $3$  units a second.  Peter also starts at  $(0,5)$  and catches up to Rob at the point  $(17,39)$ .  What is the value of t?
Answer: \frac{\sqrt{1445}}{6}
Problem: Find all pairs  $(p,q)$  of prime numbers which  $p>q$  and  $$ \frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1} $$  is an integer.
Answer: (3, 2)
Problem: We have a  $100\times100$  garden and we’ve plant  $10000$  trees in the  $1\times1$  squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree.
Answer: 2500
Problem: Let  $a, b, c, d, e \in \mathbb{R}^+$  and  $f:\{(x, y) \in (\mathbb{R}^+)^2|c-dx-ey > 0\}\to \mathbb{R}^+$  be given by  $f(x, y) = (ax)(by)(c- dx- ey)$ . 
Find the maximum value of  $f$ .
Answer: \frac{a}{d} \frac{b}{e} \left( \frac{c}{3} \right)^3
Problem: What is the maximal dimension of a linear subspace  $ V$  of the vector space of real  $ n \times n$  matrices such that for all  $ A$  in  $ B$  in  $ V$ , we have  $ \text{trace}\left(AB\right) \equal{} 0$  ?
Answer: \frac{n(n-1)}{2}
Problem: Find all pairs of positive integers  $(s, t)$ , so that for any two different positive integers  $a$  and  $b$  there exists some positive integer  $n$ , for which   $$ a^s + b^t | a^n + b^{n+1}. $$
Answer:  (s, t) such that |s - t| = 1
Problem: You have 2 six-sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 ﬁves. You pick a die and roll it. Because of some secret magnetic attraction of the unfair die, you have a 75% chance of picking the unfair die and a 25% chance of picking the fair die. If you roll a three, what is the probability that you chose the fair die?
Answer: \frac{1}{7}
Problem: There are  $n$  cities,  $2$  airline companies in a country. Between any two cities, there is exactly one  $2$ -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of  $n$ .
Answer: 4
Problem: In a group of  $2020$  people, some pairs of people are friends (friendship is mutual). It is known that no two people (not necessarily friends) share a friend. What is the maximum number of unordered pairs of people who are friends?
Answer: 1010
Problem: Let  $ABC$  be a triangle with  $AB=13$ ,  $BC=14$ , and  $CA=15$ . Points  $P$ ,  $Q$ , and  $R$  are chosen on segments  $BC$ ,  $CA$ , and  $AB$ , respectively, such that triangles  $AQR$ ,  $BPR$ ,  $CPQ$  have the same perimeter, which is  $\frac{4}{5}$  of the perimeter of  $PQR$ . What is the perimeter of  $PQR$ ?
Answer: 30
Problem: Find all triples  $(a, b, c)$  of positive integers for which  $a + (a, b) = b + (b, c) = c + (c, a)$ .

Here  $(a, b)$  denotes the greatest common divisor of integers  $a, b$ .
Answer: (k, k, k) \text{ for all natural numbers } k
Problem: **Problem 1** : Find the smallest positive integer  $n$ , such that  $\sqrt[5]{5n}$ ,   $\sqrt[6]{6n}$  ,  $\sqrt[7]{7n}$  are integers.
Answer: 2^5 \cdot 3^5 \cdot 5^4 \cdot 7^6
Problem: Suppose three boba drinks and four burgers cost  $28$  dollars, while two boba drinks and six burgers cost  $\$  37.70$. If you paid for one boba drink using only pennies, nickels, dimes, and quarters, determine the least number of coins you could use.
Answer: 10
Problem: Given  $\triangle{ABC}$  with  $\angle{B}=60^{\circ}$  and  $\angle{C}=30^{\circ}$ , let  $P,Q,R$  be points on the sides  $BA,AC,CB$  respectively such that  $BPQR$  is an isosceles trapezium with  $PQ \parallel BR$  and  $BP=QR$ .
Find the maximum possible value of  $\frac{2[ABC]}{[BPQR]}$  where  $[S]$  denotes the area of any polygon  $S$ .
Answer: 4
Problem: The integers from  $1$  through  $9$  inclusive, are placed in the squares of a  $3 \times 3$  grid. Each square contains a different integer. The product of the integers in the first and second rows are  $60$  and  $96$  respectively. Find the sum of the integers in the third row.
Answer: 17
Problem: Each of the  $10$  dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?
Answer: 4
Problem: [4] Let  $x_1,x_2,\ldots,x_{100}$  be defined so that for each  $i$ ,  $x_i$  is a (uniformly) random integer between  $1$  and  $6$  inclusive. Find the expected number of integers in the set  $\{x_1,x_1+x_2,\ldots,x_1+x_2+\cdots+x_{100}\}$  that are multiples of  $6$ .
Answer: \frac{50}{3}
Problem: Given a prime  $p$  and positive integer  $k$ , an integer  $n$  with  $0 \le n < p$  is called a  $(p, k)$ -Hofstadterian residue if there exists an infinite sequence of integers  $n_0, n_1, n_2, \ldots$  such that  $n_0 \equiv n$  and  $n_{i + 1}^k \equiv n_i \pmod{p}$  for all integers  $i \ge 0$ . If  $f(p, k)$  is the number of  $(p, k)$ -Hofstadterian residues, then compute  $\displaystyle \sum_{k = 1}^{2016} f(2017, k)$ .
Answer: 1162656
Problem: Find all polynomials  $P(x)$  with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.
Answer:  P(x) = 0
Problem: A tetrahedron is a polyhedron composed of four triangular faces. Faces  $ABC$  and  $BCD$  of a tetrahedron  $ABCD$  meet at an angle of  $\pi/6$ . The area of triangle  $\triangle ABC$  is  $120$ . The area of triangle  $\triangle BCD$  is  $80$ , and  $BC = 10$ . What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.
Answer: 320
Problem: Positive integers  $ a$ ,  $ b$ ,  $ c$ , and  $ d$  satisfy  $ a > b > c > d$ ,  $ a \plus{} b \plus{} c \plus{} d \equal{} 2010$ , and  $ a^2 \minus{} b^2 \plus{} c^2 \minus{} d^2 \equal{} 2010$ . Find the number of possible values of  $ a$ .
Answer: 501
Problem: Let  $x$  and  $y$  be integers with  $x + y \ne 0$ . Find all pairs  $(x, y)$  such that   $$ \frac{x^2 + y^2}{x + y}= 10. $$  (Walther Janous)
Answer:  (12, 6), (-2, 6), (12, 4), (-2, 4), (10, 10), (0, 10), (10, 0)
Problem: Let  $a,\ b$  are real numbers such that  $a+b=1$ .

Find the minimum value of the following integral.
\[\int_{0}^{\pi}(a\sin x+b\sin 2x)^{2}\ dx \]
Answer: \frac{\pi}{4}