rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Determine all the positive real numbers  $x_1, x_2, x_3, \dots, x_{2021}$  such that  $x_{i+1}=\frac{x_i^3+2}{3x_i^2}$ for every  $i=1, 2, 3, \dots, 2020$  and  $x_{2021}=x_1$ 
Answer:  x_1 = x_2 = \dots = x_{2021} = 1
Problem: Alexis imagines a  $2008\times 2008$  grid of integers arranged sequentially in the following way:

\[\begin{array}{r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r}1,&2,&3,&\ldots,&20082009,&2010,&2011,&\ldots,&40264017,&4018,&4019,&\ldots,&6024\vdots&&&&\vdots2008^2-2008+1,&2008^2-2008+2,&2008^2-2008+3,&\ldots,&2008^2\end{array}\]
She picks one number from each row so that no two numbers she picks are in the same column.  She them proceeds to add them together and finds that  $S$  is the sum.  Next, she picks  $2008$  of the numbers that are distinct from the  $2008$  she picked the first time.  Again she picks exactly one number from each row and column, and again the sum of all  $2008$  numbers is  $S$ .  Find the remainder when  $S$  is divided by  $2008$ .
Answer: 1004
Problem: For a positive integer  $n$ , let  $d\left(n\right)$  be the number of positive divisors of  $n$  (for example  $d\left(39\right)=4$ ). Estimate the average value that  $d\left(n\right)$  takes on as  $n$  ranges from  $1$  to  $2019$ . An estimate of  $E$  earns  $2^{1-\left|A-E\right|}$  points, where  $A$  is the actual answer.
Answer: \frac{15682}{2019}
Problem: A sequence  $\{a_n\}$  is defined such that  $a_i=i$  for  $i=1,2,3\ldots,2020$  and for  $i>2020$ ,  $a_i$  is the average of the previous  $2020$  terms. What is the largest integer less than or equal to  $\displaystyle\lim_{n\to\infty}a_n$ ?
Answer: 1010
Problem: In the triangle  $ABC$  , the angle bisector  $AD$  divides the side  $BC$  into the ratio  $BD: DC = 2: 1$ . In what ratio, does the median  $CE$  divide this bisector?
Answer: 3:1
Problem: Let  $ABCD$  be an inscribed trapezoid such that the sides  $[AB]$  and  $[CD]$  are parallel. If  $m(\widehat{AOD})=60^\circ$  and the altitude of the trapezoid is  $10$ , what is the area of the trapezoid?
Answer: 100\sqrt{3}
Problem: Find all sequences of positive integers  $a_1,a_2,\dots$  such that   $$ (n^2+1)a_n = n(a_{n^2}+1) $$   for all positive integers  $n$ .
Answer: a_n = n \quad \forall n
Problem: <span style="color:darkred">Find all functions  $f:\mathbb{R}\to\mathbb{R}$  with the following property: for any open bounded interval  $I$ , the set  $f(I)$  is an open interval having the same length with  $I$  .</span>
Answer:  f(x) = \pm x + C
Problem: Consider the  $4\times4$  array of  $16$  dots, shown below.

[asy]
size(2cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((0,3));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]

Counting the number of squares whose vertices are among the  $16$  dots and whose sides are parallel to the sides of the grid, we find that there are nine  $1\times1$  squares, four  $2\times2$  squares, and one  $3\times3$  square, for a total of  $14$  squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the  $14$  squares is missing at least one vertex?
Answer: 4
Problem: Find all nonnegative integers  $a$  and  $b$  that satisfy the equation   $$ 3 \cdot 2^a + 1 = b^2. $$  
Answer:  (a, b) = (0, 2), (3, 5), (4, 7)
Problem: Colin has  $900$  Choco Pies. He realizes that for some integer values of  $n \le 900$ , if he eats n pies a day, he will be able to eat the same number of pies every day until he runs out. How many possible values of  $n$  are there?
Answer: 27
Problem: Adam and Mada are playing a game of one-on-one basketball, in which participants may take  $2$ -point shots (worth  $2$  points) or  $3$ -point shots (worth  $3$  points). Adam makes  $10$  shots of either value while Mada makes  $11$  shots of either value. Furthermore, Adam made the same number of  $2$ -point shots as Mada made  $3$ -point shots. At the end of the game, the two basketball players realize that they have the exact same number of points! How many total points were scored in the game?
Answer: 52
Problem: Find all pairs of positive integers  $m, n$  such that  $9^{|m-n|}+3^{|m-n|}+1$  is divisible by  $m$  and  $n$  simultaneously.
Answer:  (m, n) \in \{(1, 1), (3, 3)\}
Problem: Given the polynomial   $$ P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}. $$  Express the numerical value of its derivative of order  $325$  for  $x = 0$ .
Answer: 651 \cdot 325!
Problem: Determine all positive integers  $n$  such that the decimal representation of the number  $6^n + 1$  has all its digits the same.
Answer:  n = 1 } and n = 5
Problem: In triangle  $ABC$  we have  $\angle C = 90^o$  and  $AC = BC$ . Furthermore  $M$  is an interior pont in the triangle so that  $MC = 1 , MA = 2$  and  $MB =\sqrt2$ . Determine  $AB$ 
Answer: \sqrt{10}
Problem: A circle intersects the  $y$ -axis at two points  $(0, a)$  and  $(0, b)$  and is tangent to the line  $x+100y = 100$  at  $(100, 0)$ . Compute the sum of all possible values of  $ab - a - b$ .
Answer: 10000
Problem: Given  $1962$  -digit number. It is divisible by  $9$ . Let  $x$  be the sum of its digits. Let the sum of the digits of  $x$  be  $y$ . Let the sum of the digits of  $y$  be  $z$ . Find  $z$ .
Answer: 9
Problem:  $f$  is a differentiable  function such that  $f(f(x))=x$  where  $x \in [0,1]$ .Also  $f(0)=1$ .Find the value of   $$ \int_0^1(x-f(x))^{2016}dx $$  
Answer: 0
Problem: Denote by  $ S$  the set of all positive integers. Find all functions  $ f: S \rightarrow S$  such that
\[ f (f^2(m) \plus{} 2f^2(n)) \equal{} m^2 \plus{} 2 n^2\]
for all  $ m,n \in S$ .
Answer: f(n) = n
Problem: Find all permutations  $(a_1, a_2,...,a_{2021})$  of  $(1,2,...,2021)$ , such that for every two positive integers  $m$  and  $n$  with difference bigger than  $20^{21}$ , the following inequality holds: $GCD(m+1, n+a_1)+GCD(m+2, n+a_2)+...+GCD(m+2021, n+a_{2021})<2|m-n|$ . 
Answer: a_i = i \text{ for } i = 1, 2, \ldots, 2021
Problem: Four unit circles are placed on a square of side length  $2$ , with each circle centered on one of the four corners of the square. Compute the area of the square which is not contained within any of the four circles.
Answer: 4 - \pi
Problem: What is the greatest number of balls of radius  $1/2$  that can be placed within a rectangular box of size  $10 \times 10 \times  1 \ ?$ 
Answer: 100
Problem: Find the maximum number of positive integers such that any two of them  $a, b$  (with  $a \ne b$ ) satisfy that $ |a - b| \ge \frac{ab}{100} .$ 
Answer: 10
Problem: Find all real numbers  $ a,b,c,d$  such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, 
ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]
Answer:  (5, 5, 5, 5)
Problem: In isosceles right-angled triangle  $ABC$ ,  $CA = CB = 1$ .  $P$  is an arbitrary point on the sides of  $ABC$ . Find the maximum of  $PA \cdot PB \cdot PC$ .
Answer: \frac{\sqrt{2}}{4}
Problem: A circle in the first quadrant with center on the curve  $y=2x^2-27$  is tangent to the  $y$ -axis and the line  $4x=3y$ . The radius of the circle is  $\frac{m}{n}$  where  $M$  and  $n$  are relatively prime positive integers. Find  $m+n$ .
Answer: 11
Problem: A mathematician is walking through a library with twenty-six shelves, one for each letter of the alphabet. As he walks, the mathematician will take at most one book off each shelf. He likes symmetry, so if the letter of a shelf has at least one line of symmetry (e.g., M works, L does not), he will pick a book with probability  $\frac12$ . Otherwise he has a  $\frac14$  probability of taking a book. What is the expected number of books that the mathematician will take?

Answer: \frac{41}{4}
Problem: In an  $m\times n$  table there is a policeman in cell  $(1,1)$ , and there is a thief in cell  $(i,j)$ . A move is going from a cell to a neighbor (each cell has at most four neighbors). Thief makes the first move, then the policeman moves and ... For which  $(i,j)$  the policeman can catch the thief?
Answer: (i,j) have the same parity
Problem: Let  $a,b$  be positive reals such that  $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$  and  $(a-b)^2=4(ab)^3$ . Find  $\log_a b$ .
Answer: -1
Problem: Let  $k$  be a positive integer. A sequence of integers  $a_1, a_2, \cdots$  is called  $k$ -pop if the following holds: for every  $n \in \mathbb{N}$ ,  $a_n$  is equal to the number of distinct elements in the set  $\{a_1, \cdots , a_{n+k} \}$ . Determine, as a function of  $k$ , how many  $k$ -pop sequences there are.
Answer: 2^k
Problem: Find all positive integer solutions  $ x, y, z$  of the equation  $ 3^x \plus{} 4^y \equal{} 5^z.$ 
Answer: (2, 2, 2)
Problem: Find the real number  $\alpha$  such that the curve  $f(x)=e^x$  is tangent to the curve  $g(x)=\alpha x^2$ .
Answer: \frac{e^2}{4}
Problem: Let  $ x \equal{} \sqrt[3]{\frac{4}{25}}\,$ .  There is a unique value of  $ y$  such that  $ 0 < y < x$  and  $ x^x \equal{} y^y$ .  What is the value of  $ y$ ?  Express your answer in the form  $ \sqrt[c]{\frac{a}{b}}\,$ , where  $ a$  and  $ b$  are relatively prime positive integers and  $ c$  is a prime number.
Answer: \sqrt[3]{\frac{32}{3125}}
Problem: Find for which values of  $n$ , an integer larger than  $1$  but smaller than  $100$ , the following expression has its minimum value: $S = |n-1| + |n-2| + \ldots + |n-100|$ 
Answer:  n = 50 } or n = 51
Problem: An integer  $n\ge2$  is called *resistant*, if it is coprime to the sum of all its divisors (including  $1$  and  $n$ ).
Determine the maximum number of consecutive resistant numbers.

For instance: 
*   $n=5$  has sum of divisors  $S=6$  and hence is resistant.
*   $n=6$  has sum of divisors  $S=12$  and hence is not resistant.
*   $n=8$  has sum of divisors  $S=15$  and hence is resistant.
*   $n=18$  has sum of divisors  $S=39$  and hence is not resistant.

Answer: 5
Problem: A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly  $ k $  rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly  $\ell$  rectangles. How much the number of rectangles can be?
Answer: k\ell
Problem: The wording is just ever so slightly different, however the problem is identical.

Problem 3. Determine all functions  $f: \mathbb{N} \to \mathbb{N}$  such that  $n^2 + f(n)f(m)$  is a multiple of  $f(n) + m$  for all natural numbers  $m, n$ .
Answer:  f(n) = n
Problem: Let  $ABC$  be an equilateral triangle.  $N$  is a point on the  side  $AC$  such that  $\vec{AC} = 7\vec{AN}$ ,  $M$  is a point on the side   $AB$   such that  $MN$  is parallel to  $BC$  and  $P$  is a point on the side  $BC$   such that  $MP$  is parallel to  $AC$ . Find the ratio of areas  $\frac{ (MNP)}{(ABC)}$ 
Answer: \frac{1}{49}
Problem: 3 points  $ O(0,\ 0),\ P(a,\ a^2), Q( \minus{} b,\ b^2)\ (a > 0,\ b > 0)$  are on the parabpla  $ y \equal{} x^2$ .
Let  $ S_1$  be the area bounded by the line  $ PQ$  and the parabola and let  $ S_2$  be the area of the triangle  $ OPQ$ .

Find the minimum value of  $ \frac {S_1}{S_2}$ .
Answer:  \frac{4}{3}
Problem: On a plane, there are  $7$  seats. Each is assigned to a passenger. The passengers walk on the plane one at a time. The first passenger sits in the wrong seat (someone else's). For all the following people, they either sit in their assigned seat, or if it is full, randomly pick another. You are the last person to board the plane. What is the probability that you sit in your own seat?
Answer: \frac{1}{7}
Problem: Find all sets of positive integers  $\{x_1, x_2, \dots, x_{20}\}$  such that   $$ x_{i+2}^2=lcm(x_{i+1}, x_{i})+lcm(x_{i}, x_{i-1}) $$   for  $i=1, 2, \dots, 20$  where  $x_0=x_{20}, x_{21}=x_1, x_{22}=x_2$ .
Answer: x_i = 2 \text{ for all } i \in \{1, 2, \ldots, 20\}
Problem: Given  $8$  coins, at most one of them is counterfeit. A counterfeit coin is lighter than a real coin. You have a free weight balance. What is the minimum number of weighings necessary to determine the identity of the counterfeit coin if it exists
Answer: 2
Problem: Alice is counting up by fives, starting with the number  $3$ . Meanwhile, Bob is counting down by fours, starting with the number  $2021$ . How many numbers between  $3$  and  $2021$ , inclusive, are counted by both Alice and Bob?
Answer: 101
Problem: How many integers  $n$  are there such that  $(n+1!)(n+2!)(n+3!)\cdots(n+2013!)$  is divisible by  $210$  and  $1 \le n \le 210$ ?
Answer: 1
Problem: For  $0\leq a\leq 2$ , find the minimum value of  $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$ *2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.*
Answer: \ln \left( \frac{2+2e^2}{1+2e+e^2} \right)
Problem: Two years ago Tom was  $25\%$  shorter than Mary. Since then Tom has grown  $20\%$  taller, and Mary has grown  $4$  inches taller. Now Mary is  $20\%$  taller than Tom. How many inches tall is Tom now?
Answer: 45
Problem: Suppose that  $20^{21} = 2^a5^b = 4^c5^d = 8^e5^f$  for positive integers  $a,b,c,d,e,$  and  $f$ . Find  $\frac{100bdf}{ace}$ .
Answer: 75
Problem: Let  $f:\mathbb{N} \rightarrow \mathbb{N},$   $f(n)=n^2-69n+2250$  be a function. Find the prime number  $p$ , for which the sum of the digits of the number  $f(p^2+32)$  is as small as possible.
Answer: 3
Problem: Determine all positive real  $M$  such that for any positive reals  $a,b,c$ , at least one of  $a + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca}$  is greater than or equal to  $1+M$ .
Answer:  M = \frac{1}{2}
Problem: On Lineland there are 2018 bus stations numbered 1 through 2018 from left to right. A self-driving bus that can carry at most  $N$  passengers starts from station 1 and drives all the way to station 2018, while making a stop at each bus station. Each passenger that gets on the bus at station  $i$  will get off at station  $j$  for some  $j>i$  (the value of  $j$  may vary over different passengers). Call any group of four distinct stations  $i_1, i_2, j_1, j_2$  with  $i_u< j_v$  for all  $u,v\in \{1,2\}$  a *good* group. Suppose that in any good group  $i_1, i_2, j_1, j_2$ , there is a passenger who boards at station  $i_1$  and de-boards at station  $j_1$ , or there is a passenger who boards at station  $i_2$  and de-boards at station  $j_2$ , or both scenarios occur. Compute the minimum possible value of  $N$ .
Answer: 1009
Problem: A jar was filled with jelly beans so that  $54\%$  of the beans were red,  $30\%$  of the beans were green, and  $16\%$  of the beans were blue. Alan then removed the same number of red jelly beans and green jelly beans from the jar so that now  $20\%$  of the jelly beans in the jar are blue. What percent of the jelly beans in the jar are now red?
Answer: 55\%
Problem: Compute the largest integer  $N$  such that one can select  $N$  different positive integers, none of which is larger than  $17$ , and no two of which share a common divisor greater than  $1$ .
Answer: 8
Problem: Let  $a\neq 1$  be a positive real number. Find all real solutions to the equation  $a^x=x^x+\log_a(\log_a(x)).$
Answer: x = a
Problem: Find the largest  $K$  satisfying the following:

Given any closed intervals  $A_1,\ldots, A_N$  of length  $1$  where  $N$  is an arbitrary positive integer. If their union is  $[0,2021]$ , then we can always find  $K$  intervals from  $A_1,\ldots, A_N$  such that the intersection of any two of them is empty.
Answer: K = 1011
Problem: Jonah recently harvested a large number of lychees and wants to split them into groups. Unfortunately, for all  $n$  where  $3\leq n\leq8$ , when the lychees are distributed evenly into  $n$  groups,  $n-1$  lychees remain. What is the smallest possible number of lychees that Jonah could have?
Answer: 839
Problem: We consider sports tournaments with  $n \ge 4$  participating teams and where every pair of teams plays against one another at most one time. We call such a tournament *balanced* if any four participating teams play exactly three matches between themselves. So, not all teams play against one another.
Determine the largest value of  $n$  for which a balanced tournament with  $n$  teams exists.
Answer: 5
Problem: How many ways are there to choose distinct positive integers  $a, b, c, d$  dividing  $15^6$  such that none of  $a, b, c,$  or  $d$  divide each other? (Order does not matter.)
Answer: 1225
Problem: Compute \[\left\lfloor \dfrac{2007!+2004!}{2006!+2005!}\right\rfloor.\] (Note that  $\lfloor x \rfloor$  denotes the greatest integer less than or equal to  $x$ .)
Answer: 2006
Problem: Hamilton Avenue has eight houses. On one side of the street are the houses
numbered 1,3,5,7 and directly opposite are houses 2,4,6,8 respectively. An
eccentric postman starts deliveries at house 1 and delivers letters to each of
the houses, finally returning to house 1 for a cup of tea. Throughout the
entire journey he must observe the following rules. The numbers of the houses
delivered to must follow an odd-even-odd-even pattern throughout, each house
except house 1 is visited exactly once (house 1 is visited twice) and the postman
at no time is allowed to cross the road to the house directly opposite. How
many different delivery sequences are possible?
Answer: 12
Problem: The sum of the first  $n$  terms of the sequence   $$ 1,1+2,1+2+2^2,\ldots,1+2+\cdots+2^{k-1},\ldots $$   is of the form  $2^{n+R}+Sn^2+Tn+U$  for all  $n>0.$  Find  $R,S,T,$  and  $U.$ 
Answer:  R = 1, S = 0, T = -1, U = -2
Problem: Ben is throwing darts at a circular target with diameter 10.  Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target.  Ben gets  $\lceil 5-x \rceil$  points for having the dart land  $x$  units away from the center of the target.  What is the expected number of points that Ben can earn from throwing a single dart?  (Note that  $\lceil y \rceil$  denotes the smallest integer greater than or equal to  $y$ .)
Answer: \frac{11}{5}
Problem: Find the number of arithmetic sequences  $a_1,a_2,a_3$  of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence.
Answer: 4
Problem: Sarah buys  $3$  gumballs from a gumball machine that contains  $10$  orange,  $6$  green, and  $9$  yellow gumballs. What is the probability that the first gumball is orange, the second is green or yellow, and the third is also orange?
Answer: \frac{9}{92}
Problem: For which real numbers  $c$  is  $$ \frac{e^x +e^{-x} }{2} \leq e^{c x^2 } $$  for all real  $x?$ 
Answer: c \geq \frac{1}{2}
Problem: Source: 2017 Canadian Open Math Challenge, Problem A4
-----
Three positive integers  $a$ ,  $b$ ,  $c$  satisfy  $$ 4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}. $$  Determine the sum of  $a + b + c$ .

Answer: 36
Problem: The trinomial  $f(x)$  is such that  $(f(x))^3-f(x)=0$  has three real roots. Find the y-coordinate of the vertex of  $f(x)$ .
Answer: 0
Problem: Find all functions  $f: (1,\infty)\text{to R}$  satisfying $f(x)-f(y)=(y-x)f(xy)$  for all  $x,y>1$ .
<details><summary>hint</summary>you may try to find  $f(x^5)$  by two ways and then continue the solution.
I have also solved by using this method.By finding  $f(x^5)$  in two ways
I found that  $f(x)=xf(x^2)$  for all  $x>1$ .</details>
Answer:  f(x) = \frac{k}{x}
Problem: suppose that  $A$  is the set of all Closed intervals  $[a,b] \subset \mathbb{R}$ . Find all functions  $f:\mathbb{R} \rightarrow A$  such that $\bullet$   $x \in f(y) \Leftrightarrow y \in f(x)$  $\bullet$   $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$  $\bullet$  For all real numbers  $0\leq r\leq 1$ ,  $f(r)=[r^2-1,r^2+1]$ Proposed by Matin Yousefi
Answer:  f(x) = [g(x), h(x)]
Problem: Determine all positive integers  $n$  such that  $n$  divides  $\phi(n)^{d(n)}+1$  but  $d(n)^5$  does not divide  $n^{\phi(n)}-1$ .
Answer:  n = 2
Problem: A straight line connects City A at  $(0, 0)$  to City B, 300 meters away at  $(300, 0)$ . At time  $t=0$ , a bullet train instantaneously sets out from City A to City B while another bullet train simultaneously leaves from City B to City A going on the same train track. Both trains are traveling at a constant speed of  $50$  meters/second. 

Also, at  $t=0$ , a super 
y stationed at  $(150, 0)$  and restricted to move only on the train tracks travels towards City B. The 
y always travels at 60 meters/second, and any time it hits a train, it instantaneously reverses its direction and travels at the same speed. At the moment the
trains collide, what is the total distance that the y will have traveled? Assume each train is a point
and that the trains travel at their same respective velocities before and after collisions with the y
Answer: 180
Problem: Find the largest positive integer  $n$  for which the inequality 
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\]
holds true for all  $a, b, c \in [0,1]$ . Here we make the convention  $\sqrt[1]{abc}=abc$ .
Answer: 3
Problem: Six points  $A, B, C, D, E, F$  are chosen on a circle anticlockwise. None of  $AB, CD, EF$  is a diameter. Extended $AB$  and  $DC$  meet at  $Z, CD$  and  $FE$  at  $X, EF$  and  $BA$  at  $Y.  AC$  and  $BF$  meets at  $P, CE$  and  $BD$  at  $Q$  and  $AE$  and  $DF$  at  $R.$  If  $O$  is the point of intersection of  $YQ$  and  $ZR,$  find the  $\angle XOP.$ 
Answer: 90^
Problem: Thirty nine nonzero numbers are written in a row. The sum of any two neighbouring numbers is positive, while the sum of all the numbers is negative. Is the product of all these numbers negative or positive?    (4 points)
                                                                                                                           
                                                       Boris Frenkin
Answer: positive
Problem: A point  $P$  is given on the curve  $x^4+y^4=1$ . Find the maximum distance from the point  $P$  to the origin.
Answer: \sqrt{2}
Problem: In a scalene triangle one angle is exactly two times as big as another one and some angle in this triangle is  $36^o$ . Find all possibilities, how big the angles of this triangle can be.
Answer:  (36^
\circ, 48^
\circ, 96^
\circ)
Problem: Aaron takes a square sheet of paper, with one corner labeled  $A$ . Point  $P$  is chosen at random inside of the square and Aaron folds the paper so that points  $A$  and  $P$  coincide. He cuts the sheet along the crease and discards the piece containing  $A$ . Let  $p$  be the probability that the remaining piece is a pentagon. Find the integer nearest to  $100p$ .
Answer: 25
Problem: For a permutation  $P = (p_1, p_2, ... , p_n)$  of  $(1, 2, ... , n)$  define  $X(P)$  as the number of  $j$  such that  $p_i < p_j$  for every  $i < j$ . What is the expected value of  $X(P)$  if each permutation is equally likely?
Answer: H_n
Problem: Find all triples of positive integers  $(a,b,p)$  with  $a,b$  positive integers and  $p$  a prime number such that  $2^a+p^b=19^a$ 
Answer: (a, b, p) = (1, 1, 17)
Problem: A classroom has  $30$  seats arranged into  $5$  rows of  $6$  seats. Thirty students of distinct heights come to class every day, each sitting in a random seat. The teacher stands in front of all the rows, and if any student seated in front of you (in the same column) is taller than you, then the teacher cannot notice that you are playing games on your phone. What is the expected number of students who can safely play games on their phone?
Answer: \frac{163}{10}
Problem: For a positive integer  $k$ , let  $d(k)$  denote the number of divisors of  $k$  and let  $s(k)$  denote the digit sum of  $k$ . A positive integer  $n$  is said to be *amusing* if there exists a positive integer  $k$  such that  $d(k)=s(k)=n$ . What is the smallest amusing odd integer greater than  $1$ ?
Answer: 9