rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Let  $p$  and  $q$  be prime numbers. The sequence  $(x_n)$  is defined by  $x_1 = 1$ ,  $x_2 = p$  and  $x_{n+1} = px_n - qx_{n-1}$  for all  $n \geq 2$ .
Given that there is some  $k$  such that  $x_{3k} = -3$ , find  $p$  and  $q$ .
Answer:  p = 2
Problem: Math teacher wrote in a table a polynomial  $P(x)$  with integer coefficients and he said:
"Today my daughter have a birthday.If in polynomial  $P(x)$  we have  $x=a$  where  $a$  is the age of my daughter we have  $P(a)=a$  and  $P(0)=p$  where  $p$  is a prime number such that  $p>a$ ."
How old is the daughter of math teacher?
Answer: 1
Problem: Count the number  $N$  of all sets  $A:=\{x_1,x_2,x_3,x_4\}$  of non-negative integers satisfying  $$ x_1+x_2+x_3+x_4=36 $$  in at least four different ways.
Answer: 9139
Problem: Suppose  $a$ ,  $b$ , and  $c$  are relatively prime integers such that \[\frac{a}{b+c} = 2\qquad\text{and}\qquad \frac{b}{a+c} = 3.\] What is  $|c|$ ?
Answer: 5
Problem: Determine all triples of positive integers  $(a, b, n)$  that satisfy the following equation:  $a! + b! = 2^n$ 
Answer: (1, 1, 1), (2, 2, 2), (3, 2, 3), (2, 3, 3)
Problem: Consider the Harmonic Table 
\[\begin{array}{c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c}&&&1&&&&&\tfrac12&&\tfrac12&&&\tfrac13&&\tfrac16&&\tfrac13&\tfrac14&&\tfrac1{12}&&\tfrac1{12}&&\tfrac14&&&\vdots&&&\end{array}\] where  $a_{n,1}=1/n$  and \[a_{n,k+1}=a_{n-1,k}-a_{n,k}.\] Find the remainder when the sum of the reciprocals of the  $2007$  terms on the  $2007^\text{th}$  row gets divided by  $2008$ .
Answer: 1
Problem: One afternoon at the park there were twice as many dogs as there were people, and there were twice as many people as there were snakes. The sum of the number of eyes plus the number of legs on all of these dogs, people, and snakes was  $510$ . Find the number of dogs that were at the park.
Answer: 60
Problem: Inside the circle with center  $O$ , points  $A$  and  $B$  are marked so that  $OA = OB$ . Draw a point  $M$  on the circle from which the sum of the distances to points  $A$  and  $B$  is the smallest among all possible.
Answer:  M
Problem: There is number  $N$  on the board. Every minute Ivan makes next operation: takes any number  $a$  written on the board, 
 erases it, then writes all divisors of  $a$  except  $a$ ( Can be same numbers on the board). After some time on the board there are  $N^2$  numbers.
For which  $N$  is it possible?
Answer:  N = 1
Problem: What is the smallest positive integer  $n$  such that  $2013^n$  ends in  $001$  (i.e. the rightmost three digits of  $2013^n$  are  $001$ ?
Answer: 100
Problem: At the round table,  $10$  people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) .  It is clear thath I have at least one knight and at least one liar. 
What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ?
(A statement that is at least partially false is considered false.)
Answer: 9
Problem: If  $\zeta$  denote the Riemann Zeta Function, and  $s>1$  then   $$ \sum \limits_{k=1}^{\infty} \frac{1}{1+k^s}\geq \frac{\zeta (s)}{1+\zeta (s)} $$  
Answer: \sum_{k=1}^{\infty} \frac{1}{1+k^s} \geq \frac{\zeta(s)}{1+\zeta(s)}
Problem: There are  $1,000,000$  piles of  $1996$  coins in each of them, and in one pile there are only fake coins, and in all the others - only real ones. What is the smallest weighing number that can be used to determine a heap containing counterfeit coins if the scales used have one bowl and allow weighing as much weight as desired with an accuracy of one gram, and it is also known that each counterfeit coin weighs  $9$  grams, and each real coin weighs  $10$  grams?
Answer: 1
Problem: A total of  $n$  people compete in a mathematical match which contains  $15$  problems where  $n>12$ . For each problem,  $1$  point is given for a right answer and  $0$  is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of  $12$  people get is no less than  $36$ , then there are at least  $3$  people that got the right answer of a certain problem, among the  $n$  people. Find the least possible  $n$ .
Answer: 15
Problem: Find the number of ways that a positive integer  $n$  can be represented as a sum of one or more consecutive positive integers.
Answer:  n
Problem: In  $ISI$  club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does  $ISI$  club have????
Answer: 10
Problem: Let  $S$  be the set of positive integers  $n$  such that 
\[3\cdot
 \varphi (n)=n,\]
where  $\varphi (n)$  is the number of positive integers  $k\leq n$  such that  $\gcd (k, n)=1$ . Find
\[\sum_{n\in S} \, \frac{1}{n}.\]

[i]Proposed by Nathan Ramesh

Answer: \frac{1}{2}
Problem:  $100$  integers are arranged in a circle. Each number is greater than the sum of the two subsequent numbers (in a clockwise order). Determine the maximal possible number of positive numbers in such circle. *(S.Berlov)*
Answer: 49
Problem: Find all positive real numbers  $(x,y,z)$  such that:  $$ x = \frac{1}{y^2+y-1} $$    $$ y = \frac{1}{z^2+z-1} $$    $$ z = \frac{1}{x^2+x-1} $$  
Answer:  (x, y, z) = (1, 1, 1)
Problem: Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches. 
What is the largest possible number of matches?
Answer: 16
Problem: You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with  $3$  buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially  $3$  doors are closed and  $3$  mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?
Answer: 9
Problem:  $\textbf{N5.}$  Find all  $f: \mathbb{N} \rightarrow \mathbb{N}$  such that for all  $a,b,c \in \mathbb{N}$  $f(a)+f(b)+f(c)-ab-bc-ca \mid af(a)+bf(b)+cf(c)-3abc$ 
Answer:  f(a) = a^2
Problem: When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons?
Answer: \frac{7 - 3\sqrt{5}}{2}
Problem: A coin that comes up heads with probability  $ p > 0$  and tails with probability  $ 1\minus{}p > 0$  independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to  $ \frac{1}{25}$  of the probability of five heads and three tails. Let  $ p \equal{} \frac{m}{n}$ , where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m\plus{}n$ .
Answer: 11
Problem: Find all positive integers  $ a$  and  $ b$  such that  $ \frac{a^{4}\plus{}a^{3}\plus{}1}{a^{2}b^{2}\plus{}ab^{2}\plus{}1}$  is an integer.
Answer:  a = b
Problem: There are  $2020\times 2020$  squares, and at most one piece is placed in each square. Find the minimum possible number of pieces to be used when placing a piece in a way that satisfies the following conditions.
・For any square, there are at least two pieces that are on the diagonals containing that square.

Note : We say the square  $(a,b)$  is on the diagonals containing the square  $(c,d)$  when  $|a-c|=|b-d|$ .
Answer: 2020
Problem: There are 36 contestants in the CMU Puyo-Puyo Tournament, each with distinct skill levels. 
    
    The tournament works as follows:
    First, all  $\binom{36}{2}$  pairings of players are written down on slips of paper and are placed in a hat.
    Next, a slip of paper is drawn from the hat, and those two players play a match. It is guaranteed that the player with a higher skill level will always win the match.
    We continue drawing slips (without replacement) and playing matches until the results of the match completely determine the order of skill levels of all 36 contestants (i.e. there is only one possible ordering of skill levels consistent with the match results), at which point the tournament immediately finishes.
    What is the expected value of the number of matches played before the stopping point is reached?
Answer: \frac{22085}{36}
Problem: Let  $AH$  be the altitude of a given triangle  $ABC.$  The points  $I_b$  and  $I_c$  are the incenters of the triangles  $ABH$  and  $ACH$  respectively.  $BC$  touches the incircle of the triangle  $ABC$  at a point  $L.$  Find  $\angle LI_bI_c.$ 
Answer: \angle I_bLI_c = 90^\circ
Problem: A circle with diameter  $23$  is cut by a chord  $AC$ . Two different circles can be inscribed between the large circle and  $AC$ . Find the sum of the two radii.
Answer: \frac{23}{2}
Problem: Two circles  $A$  and  $B$ , both with radius  $1$ , touch each other externally. 
Four circles  $P, Q, R$  and  $S$ , all four with the same radius  $r$ , lie such that $P$  externally touches on  $A, B, Q$  and  $S$ , $Q$  externally touches on  $P, B$  and  $R$ , $R$  externally touches on  $A, B, Q$  and  $S$ ,  $S$  externally touches on  $P, A$  and  $R$ . 
Calculate the length of  $r.$ [asy]
unitsize(0.3 cm);

pair A, B, P, Q, R, S;
real r = (3 + sqrt(17))/2;

A = (-1,0);
B = (1,0);
P = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180));
R = -P;
Q = (r + 2,0);
S = (-r - 2,0);

draw(Circle(A,1));
draw(Circle(B,1));
draw(Circle(P,r));
draw(Circle(Q,r));
draw(Circle(R,r));
draw(Circle(S,r));

label(" $A$ ", A);
label(" $B$ ", B);
label(" $P$ ", P);
label(" $Q$ ", Q);
label(" $R$ ", R);
label(" $S$ ", S);
[/asy]
Answer: 2
Problem: Find all positive integer bases  $b \ge 9$  so that the number 
\[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \]
is a perfect cube in base 10 for all sufficiently large positive integers  $n$ .
Answer:  b = 10
Problem: Find all integer pairs  $(x, y)$  such that   $$ y^2 = x^3 + 2x^2 + 2x + 1. $$  
Answer: (x, y) \in \{(-1, 0), (0, -1), (0, 1)\}
Problem: Let  $Z$  shape be a shape such that it covers  $(i,j)$ ,  $(i,j+1)$ ,  $(i+1,j+1)$ ,  $(i+2,j+1)$  and  $(i+2,j+2)$  where  $(i,j)$  stands for cell in  $i$ -th row and  $j$ -th column on an arbitrary table. At least how many  $Z$  shapes is necessary to cover one  $8 \times 8$  table if every cell of a  $Z$  shape is either cell of a table or it is outside the table (two  $Z$  shapes can overlap and  $Z$  shapes can rotate)?
Answer: 12
Problem: Point D is from AC of triangle ABC so that 2AD=DC. Let DE be perpendicular to BC and AE
intersects BD at F. It is known that triangle BEF is equilateral. Find <ADB?
Answer: 90^	extcirc
Problem: One of the boxes that Joshua and Wendy unpack has Joshua's collection of board games.  Michael, Wendy, Alexis, and Joshua decide to play one of them, a game called  $\textit{Risk}$  that involves rolling ordinary six-sided dice to determine the outcomes of strategic battles.  Wendy has never played before, so early on Michael explains a bit of strategy.

"You have the first move and you occupy three of the four territories in the Australian continent.  You'll want to attack Joshua in Indonesia so that you can claim the Australian continent which will give you bonus armies on your next turn."

"Don't tell her  $\textit{that!}$ " complains Joshua.

Wendy and Joshua begin rolling dice to determine the outcome of their struggle over Indonesia.  Joshua rolls extremely well, overcoming longshot odds to hold off Wendy's attack.  Finally, Wendy is left with one chance.  Wendy and Joshua each roll just one six-sided die.  Wendy wins if her roll is  $\textit{higher}$  than Joshua's roll.  Let  $a$  and  $b$  be relatively prime positive integers so that  $a/b$  is the probability that Wendy rolls higher, giving her control over the continent of Australia.  Find the value of  $a+b$ .
Answer: 17
Problem: There is a list of  $n$  football matches. Determine how many possible columns, with an even number of draws, there are.
Answer:  a_n
Problem: Find the least  $k$  for which the number  $2010$  can be expressed as the sum of the squares of  $k$  integers.
Answer: k=3
Problem: A finite sequence of numbers  $(a_1,\cdots,a_n)$  is said to be alternating if   $$ a_1>a_2~,~a_2<a_3~,~a_3>a_4~,~a_4<a_5~,~\cdots $$      $$ \text{or ~}~~a_1<a_2~,~a_2>a_3~,~a_3<a_4~,~a_4>a_5~,~\cdots $$   How many alternating sequences of length  $5$  , with distinct numbers  $a_1,\cdots,a_5$  can be formed such that  $a_i\in\{1,2,\cdots,20\}$  for  $i=1,\cdots,5$  ?
Answer: 32 \times \binom{20}{5}
Problem: Leo and Smilla find  $2020$  gold nuggets with masses  $1,2,\dots,2020$  gram, which they distribute to a red and a blue treasure chest according to the following rule:

First, Leo chooses one of the chests and tells its colour to Smilla. Then Smilla chooses one of the not yet distributed nuggets and puts it into this chest.

This is repeated until all the nuggets are distributed. Finally, Smilla chooses one of the chests and wins all the nuggets from this chest.

How many gram of gold can Smilla make sure to win?
Answer: 1021110
Problem: Let  $\Delta ABC$  be an acute-angled triangle and let  $H$  be its orthocentre. Let  $G_1, G_2$  and  $G_3$  be the centroids of the triangles  $\Delta HBC , \Delta HCA$  and  $\Delta HAB$  respectively. If the area of  $\Delta G_1G_2G_3$  is  $7$  units, what is the area of  $\Delta ABC $ ? 
Answer: 63
Problem: 
	 Let  $PROBLEMZ$  be a regular octagon inscribed in a circle of unit radius. Diagonals  $MR$ ,  $OZ$  meet at  $I$ . Compute  $LI$ .
Answer: \sqrt{2}
Problem: Find all pairs  $ (p, q)$  of prime numbers such that  $ p$  divides  $ 5^q \plus{} 1$  and  $ q$  divides  $ 5^p \plus{} 1$ .
Answer: (2, 2), (2, 13), (3, 3), (3, 7), (13, 2), (7, 3)
Problem: In the game of *Winners Make Zeros*, a pair of positive integers  $(m,n)$  is written on a sheet of paper.  Then the game begins, as the players make the following legal moves:


- If  $m\geq n$ , the player choose a positive integer  $c$  such that  $m-cn\geq 0$ , and replaces  $(m,n)$  with  $(m-cn,n)$ .
- If  $m<n$ , the player choose a positive integer  $c$  such that  $n-cm\geq 0$ , and replaces  $(m,n)$  with  $(m,n-cm)$ .

When  $m$  or  $n$  becomes  $0$ , the game ends, and the last player to have moved is declared the winner.  If  $m$  and  $n$  are originally  $2007777$  and  $2007$ , find the largest choice the first player can make for  $c$  (on his first move) such that the first player has a winning strategy after that first move.
Answer: 999
Problem: There are  $100$  cards with numbers from  $1$  to  $100$  on the table.Andriy and Nick took the same number of cards in a way such that the following condition holds:if Andriy has a card with a number  $n$  then Nick has a card with a number  $2n+2$ .What is the maximal number of cards that could be taken by the two guys?
Answer: 50
Problem: Let  $A$  and  $B$  be variable points on  $x-$ axis and  $y-$ axis respectively such that the line segment  $AB$  is in the first quadrant and of a fixed length  $2d$  . Let  $C$  be the mid-point of  $AB$  and  $P$  be a point such that**(a)**  $P$  and the origin are on the opposite sides of  $AB$  and,**(b)**  $PC$  is a line segment of length  $d$  which is perpendicular to  $AB$  .

Find the locus of  $P$  .
Answer: y = x
Problem: Jae and Yoon are playing SunCraft. The probability that Jae wins the  $n$ -th game is  $\frac{1}{n+2}.$  What is the probability that Yoon wins the first six games, assuming there are no ties?
Answer: \frac{1}{4}
Problem: given a positive integer  $n$ .
the set  $\{ 1,2,..,2n \}$  is partitioned into  $a_1<a_2<...<a_n $  and  $b_1>b_2>...>b_n$ .
find the value of :  $ \sum_{i=1}^{n}|a_i - b_i| $ 
Answer: n^2
Problem: Suppose  $z^{3}=2+2i$ , where  $i=\sqrt{-1}$ . The product of all possible values of the real part of  $z$  can be written in the form  $\frac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .
Answer: 3
Problem: Find all positive integers  $n$  for which all positive divisors of  $n$  can be put into the cells of a rectangular table under the following constraints:


- each cell contains a distinct divisor;
- the sums of all rows are equal; and
- the sums of all columns are equal.

Answer: n = 1
Problem: Let the real parameter  $p$  be such that the system  $\begin{cases} p(x^2 - y^2) = (p^2- 1)xy  |x - 1|+ |y| = 1 \end{cases}$  has at least three different real solutions. Find  $p$  and solve the system for that  $p$ .
Answer:  p = \pm 1
Problem: Let  $k$  be a positive integer such that  $1 +\frac12+\frac13+ ... +\frac{1}{13}=\frac{k}{13!}$ . Find the remainder when  $k$  is divided by  $7$ .
Answer: 0
Problem: Let  $ABC$  be an acute triangle and  $AD$  a height. The angle bissector of  $\angle DAC$  intersects  $DC$  at  $E$ . Let  $F$  be a point on  $AE$  such that  $BF$  is perpendicular to  $AE$ . If  $\angle BAE=45º$ , find  $\angle BFC$ .
Answer: 135^	ext{o}
Problem: Susan plays a game in which she rolls two fair standard six-sided dice with sides labeled one
through six. She wins if the number on one of the dice is three times the number on the other
die. If Susan plays this game three times, compute the probability that she wins at least once.
Answer: \frac{217}{729}
Problem: Let  $S$  be the set of all natural numbers with the property: the sum of the biggest three divisors of number  $n$ , different from  $n$ , is bigger than  $n$ . Determine the largest natural number  $k$ , which divides any number from  $S$ .
(A natural number is a positive integer)
Answer: 6
Problem: Let  $ABCDEFGHIJ$  be a regular  $10$ -sided polygon that has all its vertices in one circle with center  $O$  and radius  $5$ . The diagonals  $AD$  and  $BE$  intersect at  $P$  and the diagonals  $AH$  and  $BI$  intersect at  $Q$ . Calculate the measure of the  segment  $PQ$ .
Answer: PQ = 5
Problem: Given a continuous function  $ f(x)$  such that  $ \int_0^{2\pi} f(x)\ dx \equal{} 0$ .

Let  $ S(x) \equal{} A_0 \plus{} A_1\cos x \plus{} B_1\sin x$ , find constant numbers  $ A_0,\ A_1$  and  $ B_1$  for which  $ \int_0^{2\pi} \{f(x) \minus{} S(x)\}^2\ dx$  is minimized.
Answer:  A_0 = 0, A_1 = \frac{1}{\pi} \int_0^{2\pi} f(x) \cos x \, dx, B_1 = \frac{1}{\pi} \int_0^{2\pi} f(x) \sin x \, dx
Problem: Forty teams play a tournament in which every team plays every other team exactly once.  No ties occur, and each team has a  $50 \%$  chance of winning any game it plays.  The probability that no two teams win the same number of games is  $m/n,$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $\log_2 n.$ 
Answer: 742
Problem: If  $a$  and  $b$  are each randomly and independently chosen in the interval  $[-1, 1]$ , what is the probability that  $|a|+|b|<1$ ?
Answer: \frac{1}{2}
Problem: Let  $ABCD$  be a square of side length  $4$ . Points  $E$  and  $F$  are chosen on sides  $BC$  and  $DA$ , respectively, such that  $EF = 5$ . Find the sum of the minimum and maximum possible areas of trapezoid  $BEDF$ .
Answer: 16
Problem: Let  $ABC$  be a triangle with sides  $AB=19$ ,  $BC=21$ , and  $AC=20$ . Let  $\omega$  be the incircle of  $ABC$  with center  $I$ . Extend  $BI$  so that it intersects  $AC$  at  $E$ . If  $\omega$  is tangent to  $AC$  at the point  $D$ , then compute the length of  $DE$ .
Answer: \frac{1}{2}
Problem: Find all continuous functions   $f : R \rightarrow R$  such, that  $f(xy)= f\left(\frac{x^2+y^2}{2}\right)+(x-y)^2$  for any real numbers  $x$  and  $y$ 
Answer:  f(x) = c - 2x
Problem: Let  $\triangle XOY$  be a right-angled triangle with  $\angle XOY=90^\circ$ .  Let  $M$  and  $N$  be the midpoints of legs  $OX$  and  $OY$ , respectively.  Find the length  $XY$  given that  $XN=22$  and  $YM=31$ .
Answer: 34
Problem: We define the function  $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$ . Then choose any distinct  $a, b, c \in \mathbb{R}$  such that the following holds:  $f(a,b)=f(b,c)=f(c,a)$ . Over all such choices of  $a, b, c$ , what is the maximum value achieved by
\[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\]
Answer: 1
Problem: In a school tennis tournament with  $ m \ge 2$  participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. During the tournament, each participant plays exactly one match against each remaining player. Find the least number of participants m for which it is possible that some participant wins more sets than any other participant but obtains less points than any other participant.
Answer: 6
Problem: Find all continuous functions  $f: \mathbb R \to \mathbb R$  such that
\[f(x^2-y^2)=f(x)^2 + f(y)^2, \quad \forall x,y \in \mathbb R.\]
Answer: f(x) = 0 \text{ or } f(x) = \frac{1}{2}
Problem: All possible sequences of numbers  $-1$  and  $+1$   of length  $100$  are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.
Answer: 100
Problem: The fourth-degree equation  $x^4-x-504=0$  has  $4$  roots  $r_1$ ,  $r_2$ ,  $r_3$ ,  $r_4$ . If  $S_x$  denotes the value of  ${r_1}^4+{r_2}^4+{r_3}^4+{r_4}^4$ , compute  $S_4$ .
Answer: 2016
Problem: Determine the number of angles  $\theta$  between  $0$  and  $2 \pi$ , other than integer multiples of  $\pi /2$ , such that the quantities  $\sin \theta, \cos \theta, $  and  $\tan \theta$  form a geometric sequence in some order.
Answer: 4
Problem: In the triangle  $ABC$  on the sides  $AB$  and  $AC$  outward constructed equilateral triangles  $ABD$  and  $ACE$ . The segments  $CD$  and  $BE$  intersect at point  $F$ . It turns out that point  $A$  is the center of the circle inscribed in triangle  $ DEF$ . Find the angle  $BAC$ .
Answer: 60^	extcirc}
Problem: Let  $\alpha$  be the unique real root of the polynomial  $x^3-2x^2+x-1$ . It is known that  $1<\alpha<2$ . We define the sequence of polynomials  $\left\{{p_n(x)}\right\}_{n\ge0}$  by taking  $p_0(x)=x$  and setting
\begin{align*}
p_{n+1}(x)=(p_n(x))^2-\alpha
\end{align*}
How many distinct real roots does  $p_{10}(x)$  have?
Answer: 1
Problem: Find value of   $$ \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx} $$   if  $x$ ,  $y$  and  $z$  are real numbers usch that  $xyz=1$ 
Answer: 1
Problem: Let  $d(n)$  denote the number of positive divisors of  $n$ .  Find all triples  $(n,k,p)$ , where  $n$  and  $k$  are positive integers and  $p$  is a prime number, such that
\[n^{d(n)} - 1 = p^k.\]
Answer: (n, k, p) = (2, 1, 3), (3, 3, 2)
Problem: Let  $N$  denote the number of all natural numbers  $n$  such that  $n$  is divisible by a prime  $p> \sqrt{n}$  and  $p<20$ . What is the value of  $N$  ?
Answer: 69
Problem: The intramural squash league has 5 players, namely Albert, Bassim, Clara, Daniel, and Eugene. Albert has played one game, Bassim has played two games, Clara has played 3 games, and Daniel has played 4 games. Assuming no two players in the league play each other more than one time, how many games has Eugene played?
Answer: 4
Problem: Five rays  $\overrightarrow{OA}$ , $\overrightarrow{OB}$ ,  $\overrightarrow{OC}$ ,  $\overrightarrow{OD}$ , and  $\overrightarrow{OE}$   radiate in a clockwise order from  $O$  forming four non-overlapping angles such that  $\angle EOD = 2\angle COB$ ,  $\angle COB = 2\angle BOA$ , while  $\angle DOC = 3\angle BOA$ . If  $E$ ,  $O$ ,  $A$  are collinear with  $O$  between  $A$  and  $E$ , what is the degree measure of  $\angle DOB?$ 
Answer: 90^{\circ}
Problem: Given a positive integer  $n$ , find the proportion of the subsets of  $\{1,2, \ldots, 2n\}$  such that their smallest element is odd.
Answer: \frac{1 - 4^{-n}}{3}
Problem: Let  $N$  be the number of ordered 5-tuples  $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$  of positive integers satisfying $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$ Is  $N$  even or odd?

Oh and **HINTS ONLY**, please do not give full solutions. Thanks.
Answer: even