rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Given a regular hexagon, a circle is drawn circumscribing it and another circle is drawn inscribing it. The ratio of the area of the larger circle to the area of the smaller circle can be written in the form  $\frac{m}{n}$  , where m and n are relatively prime positive integers. Compute  $m + n$ .
Answer:  7
Problem: Let  $ABC$  be an equilateral triangle with side length  $1$ . Points  $A_1$  and  $A_2$  are chosen on side  $BC$ , points  $B_1$  and  $B_2$  are chosen on side  $CA$ , and points  $C_1$  and  $C_2$  are chosen on side  $AB$  such that  $BA_1<BA_2$ ,  $CB_1<CB_2$ , and  $AC_1<AC_2$ . 
Suppose that the three line segments  $B_1C_2$ ,  $C_1A_2$ ,  $A_1B_2$  are concurrent, and the perimeters of triangles  $AB_2C_1$ ,  $BC_2A_1$ , and  $CA_2B_1$  are all equal. Find all possible values of this common perimeter.
Answer: 1
Problem: Let  $\lambda$  the positive root of the equation  $t^2-1998t-1=0$ .  It is defined the sequence  $x_0,x_1,x_2,\ldots,x_n,\ldots$  by  $x_0=1,\ x_{n+1}=\lfloor\lambda{x_n}\rfloor\mbox{ for }n=1,2\ldots$  Find the remainder of the division of  $x_{1998}$  by  $1998$ . 

Note:  $\lfloor{x}\rfloor$  is the greatest integer less than or equal to  $x$ .
Answer: 0
Problem: For a given positive integer  $n,$  find   $$ \sum_{k=0}^{n} \left(\frac{\binom{n}{k} \cdot (-1)^k}{(n+1-k)^2} - \frac{(-1)^n}{(k+1)(n+1)}\right). $$  
Answer: 0
Problem: Two isosceles triangles with sidelengths  $x,x,a$  and  $x,x,b$  ( $a \neq b$ ) have equal areas. Find  $x$ .
Answer: \frac{\sqrt{a^2 + b^2}}{2}
Problem: Determine all positive integers  $n$  for which there exists an infinite subset  $A$  of the set  $\mathbb{N}$  of positive integers such that for all pairwise distinct  $a_1,\ldots , a_n \in A$  the numbers  $a_1+\ldots +a_n$  and  $a_1a_2\ldots a_n$  are coprime.
Answer: n = 1
Problem: The  $42$  points  $P_1,P_2,\ldots,P_{42}$  lie on a straight line, in that order, so that the distance between  $P_n$  and  $P_{n+1}$  is  $\frac{1}{n}$  for all  $1\leq n\leq41$ . What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.)
Answer: 861
Problem: A nonzero polynomial  $f(x)$  with real coefficients has the property that  $f(x)=f^\prime(x)f^{\prime\prime}(x)$ .  What is the leading coefficient of  $f(x)$ ?
Answer: \frac{1}{18}
Problem: Ancient astronaut theorist Nutter B. Butter claims that the Caloprians from planet Calop, 30 light years away and at rest with respect to the Earth, wiped out the dinosaurs. The iridium layer in the crust, he claims, indicates spaceships with the fuel necessary to travel at 30% of the speed of light here and back, and that their engines allowed them to instantaneously hop to this speed. He also says that Caloprians can only reproduce on their home planet. Call the minimum life span, in years, of a Caloprian, assuming some had to reach Earth to wipe out the dinosaurs,  $T$ . Assume that, once a Caloprian reaches Earth, they instantaneously wipe out the dinosaurs. Then,  $T$  can be expressed in the form  $m\sqrt{n}$ , where  $n$  is not divisible by the square of a prime. Find  $m+n$ .
Answer: 111
Problem: Let  $A_n$  denote the answer to the  $n$ th problem on this contest ( $n=1,\dots,30$ ); in particular, the answer to this problem is  $A_1$ . Compute  $2A_1(A_1+A_2+\dots+A_{30})$ .
Answer: 0
Problem: In equilateral triangle  $ABC$ , the midpoint of  $\overline{BC}$  is  $M$ . If the circumcircle of triangle  $MAB$  has area  $36\pi$ , then find the perimeter of the triangle.
Answer: 36
Problem: A circle of radius  $t$  is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius  $r=1+\sin \frac{\pi}{8}$ .  Find  $rt$ .
Answer: \cos^2 \frac{\pi}{8}
Problem: Find all functions  $f : R \to  R$  such that  $f(x^2)-f(y^2) \le   (f(x)+y) (x-f(y))$  for all  $x, y \in  R$ .
Answer: f(x) = x \text{ and } f(x) = -x, \forall x \in \mathbb{R}
Problem: Let  $a,b\in\mathbb{R},~a>1,~b>0.$  Find the least possible value for  $\alpha$  such that :  $$ (a+b)^x\geq a^x+b,~(\forall)x\geq\alpha. $$  
Answer: 1
Problem: Find all natural numbers  $n$  such that  $n$ ,  $n^2+10$ ,  $n^2-2$ ,  $n^3+6$ , and  $n^5+36$  are all prime numbers.
Answer:  n = 7
Problem: Let  $P$  be a non-zero polynomial with non-negative real coefficients, let  $N$  be a positive integer, and let  $\sigma$  be a permutation of the set  $\{1,2,...,n\}$ . Determine the least value the sum
\[\sum_{i=1}^{n}\frac{P(x_i^2)}{P(x_ix_{\sigma(i)})}\] may achieve, as  $x_1,x_2,...,x_n$  run through the set of positive real numbers.
Answer: n
Problem: An ant is on one face of a cube.  At every step, the ant walks to one of its four neighboring faces with equal probability.  What is the expected (average) number of steps for it to reach the face opposite its starting face?
Answer: 6
Problem: Find all quadruples  $(a, b, c, d)$  of real numbers such that
\[a + bcd = b + cda = c + dab = d + abc.\]
Answer:  (a, a, a, a)
Problem: Find all  $f:\mathbb{R}\rightarrow \mathbb{R}$  continuous functions such that  $\lim_{x\rightarrow \infty} f(x) =\infty$  and  $\forall x,y\in \mathbb{R}, |x-y|>\varphi, \exists n<\varphi^{2023}, n\in \mathbb{N}$  such that   $$ f^n(x)+f^n(y)=x+y $$  
Answer:  f(x) = x
Problem: Problem:
Find all polynomials satisfying the equation $ f(x^2) = (f(x))^2 $ for all real numbers x.
I'm not exactly sure where to start though it doesn't look too difficult. Thanks!
Answer:  f(x) = 0 and f(x) = x^k for k = 0, 1, 2, \ldots
Problem: Let  $ \left(x_{n}\right)$  be a real sequence satisfying  $ x_{0}=0$ ,  $ x_{2}=\sqrt[3]{2}x_{1}$ , and  $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$  for every integer  $ n\geq 2$ , and such that  $ x_{3}$  is a positive integer. Find the minimal number of integers belonging to this sequence.
Answer: 5
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  $ C$  and  $ A$ .
Find the ratio of the volume  $ C$  to the sum of the volumes  $ A$  and  $ B$ .
Answer: \frac{23}{12}
Problem: A ladder  $10\text{ m}$  long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of  $1\text{ m/s}$ , how fast is the top of the ladder sliding down the wall when the bottom of the ladder is  $6\text{ m}$  from the wall?
Answer: \frac{3}{4} \text{ m/s}
Problem: The graph of the equation  $x^y =y^x$  in the first quadrant consists of a straight line and a curve. Find the coordinates of the intersection of the line and the curve.
Answer: (e, e)
Problem: Determine all positive integers  $a,b,c$  such that  $ab + ac + bc$  is a prime number and  $$ \frac{a+b}{a+c}=\frac{b+c}{b+a}. $$  
Answer:  (1, 1, 1)
Problem: The real numbers  $a_1,a_2,a_3$  and  $b{}$  are given. The equation \[(x-a_1)(x-a_2)(x-a_3)=b\]has three distinct real roots,  $c_1,c_2,c_3.$  Determine the roots of the equation \[(x+c_1)(x+c_2)(x+c_3)=b.\]
Answer: -a_1, -a_2, -a_3
Problem: Let  $ \{a_n\}_{n=1}^\infty $  be an arithmetic progression  with  $ a_1 > 0 $  and  $ 5\cdot a_{13} = 6\cdot a_{19} $  . What is the smallest integer  $ n$  such that  $ a_n<0 $ ?
Answer: 50
Problem: by using the formula  $\pi cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{2z}{z^2-n^2}$  calculate values of  $\zeta(2k)$  on terms of bernoli numbers and powers of  $\pi$ .
Answer: \zeta(2k) = \frac{(2\pi)^{2k} B_{2k}}{2 (2k)!}
Problem: Let  $A$  and  $B$  be two finite sets such that there are exactly  $144$  sets which are subsets of  $A$  or subsets of  $B$ . Find the number of elements in  $A \cup B$ .
Answer: 8
Problem: What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.)

Alexandru Mihalcu
Answer: 16
Problem: Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability  $\tfrac23,$  and each of the other five sides has probability  $\tfrac{1}{15}.$  Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is  $\tfrac{p}{q},$  where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .
Answer: 167
Problem: Let  $k$  be a positive integer. Lexi has a dictionary  $\mathbb{D}$  consisting of some  $k$ -letter strings containing only the letters  $A$  and  $B$ . Lexi would like to write either the letter  $A$  or the letter  $B$  in each cell of a  $k \times k$  grid so that each column contains a string from  $\mathbb{D}$  when read from top-to-bottom and each row contains a string from  $\mathbb{D}$  when read from left-to-right. 
What is the smallest integer  $m$  such that if  $\mathbb{D}$  contains at least  $m$  different strings, then Lexi can fill her grid in this manner, no matter what strings are in  $\mathbb{D}$ ?
Answer: 2^{k-1}
Problem: A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("fipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it. Student 5 then goes through and "flips" every 5th locker. This process continues with all students with odd numbers  $n < 100$  going through and "flipping" every  $n$ th locker. How many lockers are open after this process?
Answer: 10
Problem: Find the maximum number of pairwise disjoint sets of the form $S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}$ ,  $a, b \in \mathbb{Z}$ .
Answer: 2
Problem: Let  $p$  be an odd prime. Determine positive integers  $x$  and  $y$  for which  $x \le y$  and  $\sqrt{2p}-\sqrt{x}-\sqrt{y}$  is nonnegative and as small as possible.
Answer:  x = \frac{p-1}{2}
Problem: Find all triplets of real numbers  $(x,y,z)$  such that the following equations are satisfied simultaneously: 
\begin{align*} 
 x^3+y=z^2  
y^3+z=x^2  
z^3+x =y^2 
\end{align*} 
Answer: (0, 0, 0)
Problem: Find all positive integers  $ k$  with the following property: There exists an integer  $ a$  so that  $ (a\plus{}k)^{3}\minus{}a^{3}$  is a multiple of  $ 2007$ .
Answer: 669
Problem: Each lottery ticket has a 9-digit numbers, which uses only the digits  $1$ ,  $2$ ,  $3$ . Each ticket is colored <span style="color:red">red</span>,<span style="color:blue"> blue </span>or <span style="color:green">green</span>. If two tickets have numbers which differ in all nine places, then the tickets have different colors. Ticket  $122222222$  is red, and ticket  $222222222$  is <span style="color:green">green.</span> What color is ticket  $123123123$  ?
Answer: red
Problem: Let  $S=\{1,2,3,...,12\}$ . How many subsets of  $S$ , excluding the empty set, have an even sum but not an even product?
Answer: 31
Problem: Consider the set  $E = \{5, 6, 7, 8, 9\}$ . For any partition  ${A, B}$  of  $E$ , with both  $A$  and  $B$  non-empty, consider the number obtained by adding the product of elements of  $A$  to the product of elements of  $B$ . Let  $N$  be the largest prime number amonh these numbers. Find the sum of the digits of  $N$ .
Answer: 17
Problem: For an integer  $k$  let  $T_k$  denote the number of  $k$ -tuples of integers  $(x_1,x_2,...x_k)$  with  $0\le x_i < 73$  for each  $i$ , such that  $73|x_1^2+x_2^2+...+x_k^2-1$ . Compute the remainder when  $T_1+T_2+...+T_{2017}$  is divided by  $2017$ .

[i]Proposed by Vincent Huang
Answer: 0
Problem: There is a set of control weights, each of them weighs a non-integer number of grams. Any
integer weight from  $1$  g to  $40$  g can be balanced by some of these weights (the control
weights are on one balance pan, and the measured weight on the other pan).What is the
least possible number of the control weights?
Answer: 4
Problem: A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly  $1/2$  that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
Answer: 990
Problem: Let  $ k$  be a positive integer. Find all polynomials

\[ P(x) \equal{} a_0 \plus{} a_1 x \plus{} \cdots \plus{} a_n x^n,\]

where the  $ a_i$  are real, which satisfy the equation

\[ P(P(x)) \equal{} \{ P(x) \}^k\]
Answer:  P(x) = 0
Problem: The product  $55\cdot60\cdot65$  is written as a product of 5 distinct numbers.
Find the least possible value of the largest number, among these 5 numbers.
Answer: 13
Problem: The bisector of angle  $A$  of triangle  $ABC$  meet its circumcircle  $\omega$  at point  $W$ . The circle  $s$  with diameter  $AH$  ( $H$  is the orthocenter of  $ABC$ ) meets  $\omega$  for the second time at point  $P$ . Restore the triangle  $ABC$  if the points  $A$ ,  $P$ ,  $W$  are given.
Answer: ABC
Problem: Two three-digit numbers are given. The hundreds digit of each of them is equal to the units digit of the other. Find these numbers if their difference is  $297$  and the sum of digits of the smaller number is  $23$ .
Answer: 986 \text{ and } 689
Problem: Find all ordered pairs of integers  $(x, y)$  with  $y \ge 0$  such that  $x^2 + 2xy +   y! = 131$ .
Answer: (1, 5), (-11, 5)
Problem: If  $a$  and  $b$  are positive integers such that  $\frac{1}{a}+\frac{1}{b}=\frac{1}{9}$ , what is the greatest possible value of  $a+b$ ?
Answer: 100
Problem: Sylvia has a bag of 10 coins. Nine are fair coins, but the tenth has tails on both sides. Sylvia draws a coin at random from the bag and flips it without looking. If the coin comes up tails, what is the probability that the coin she drew was the 2-tailed coin?
Answer: \frac{2}{11}
Problem: Source: 2017 Canadian Open Math Challenge, Problem B4
-----
Numbers  $a$ ,  $b$  and  $c$  form an arithmetic sequence if  $b - a = c - b$ . Let  $a$ ,  $b$ ,  $c$  be positive integers forming an arithmetic sequence with  $a < b < c$ . Let  $f(x) = ax2 + bx + c$ . Two distinct real numbers  $r$  and  $s$  satisfy  $f(r) = s$  and  $f(s) = r$ . If  $rs = 2017$ , determine the smallest possible value of  $a$ .
Answer: 1
Problem: A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest  $\ell > 0$  such that there exists a square of side  $\ell$  contained in a cube with edge  $1$ .
Answer: \frac{\sqrt{6}}{2}
Problem: Find the least positive integer  $n$  such that  $15$  divides the product
\[a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )\]
, for every positive integers  $a_1, a_2, \dots, a_{15}$ .
Answer: 4
Problem: A particle moves from  $(0, 0)$  to  $(n, n)$  directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At  $(n, y), y < n$ , it stays there if a head comes up and at  $(x, n), x < n$ , it stays there if a tail comes up. Let $ k$  be a fixed positive integer. Find the probability that the particle needs exactly  $2n+k$  tosses to reach  $(n, n).$ 
Answer: \frac{2 \cdot (2n + k)!}{n!(n + k)! \cdot 2^{2n + k}}
Problem: Triangle  $GRT$  has  $GR=5,$   $RT=12,$  and  $GT=13.$  The perpendicular bisector of  $GT$  intersects the extension of  $GR$  at  $O.$  Find  $TO.$ 
Answer: \frac{169}{10}
Problem: The edges of  $K_{2017}$  are each labeled with  $1,2,$  or  $3$  such that any triangle has sum of labels at least  $5.$  Determine the minimum possible average of all  $\dbinom{2017}{2}$  labels.
 
(Here  $K_{2017}$  is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.)
Answer: \frac{4033}{2017}
Problem: Given a positive integer  $ k$ , there is a positive integer  $ n$  with the property that one can obtain the sum of the first  $ n$  positive integers by writing some  $ k$  digits to the right of  $ n$ . Find the remainder of  $ n$  when dividing at  $ 9$ .
Answer: 1
Problem: For  $k=1,2,\dots$ , let  $f_k$  be the number of times
\[\sin\left(\frac{k\pi x}{2}\right)\]
attains its maximum value on the interval  $x\in[0,1]$ . Compute
\[\lim_{k\rightarrow\infty}\frac{f_k}{k}.\]
Answer:  \frac{1}{4}
Problem: Find  $3x^2 y^2$  if  $x$  and  $y$  are integers such that  $y^2 + 3x^2 y^2 = 30x^2 + 517$ .
Answer: 588
Problem: Find all positive integers  $n$  such that the number  $$ n^6 + 5n^3 + 4n + 116 $$  is the product of two or more consecutive numbers.
Answer:  n = 3
Problem: Call a positive integer *prime-simple* if it can be expressed as the sum of the squares of two distinct prime numbers. How many positive integers less than or equal to  $100$  are prime-simple?
Answer: 6
Problem: Each of the spots in a  $8\times 8$  chessboard is occupied by either a black or white “horse”. At most how many black horses can be on the chessboard so that none of the horses attack more than one black horse?**Remark:** A black horse could attack another black horse.
Answer: 16
Problem: Within a group of  $ 2009$  people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.
Answer: 0
Problem: Find all polynomials  $P(x)$  with integer coefficients, such that for all positive integers  $m, n$ ,   $$ m+n \mid P^{(m)}(n)-P^{(n)}(m). $$
Answer:  P(x) \equiv c
Problem: Let  $x$ - minimal root of equation  $x^2-4x+2=0$ . 
Find two first digits of number  $ \{x+x^2+....+x^{20} \}$  after  $0$ , where  $\{a\}$ - fractional part of  $a$ .
Answer: 41
Problem: Find all function  $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$  such that for every three real positive number  $x,y,z$  :  $$  x+f(y) , f(f(y)) + z , f(f(z))+f(x)  $$  are length of three sides of a triangle and for every postive number  $p$  , there is a triangle with these sides and perimeter  $p$ .
Answer:  f(x) = x
Problem: Find the roots of the equation   $(x-a)(x-b)=(x-c)(x-d)$ , if you know that  $a+d=b+c=2015$  and  $a \ne c$  (numbers  $a, b, c, d$  are not given).
Answer: 0
Problem: Jerry's favorite number is  $97$ .  He knows all kinds of interesting facts about  $97$ :


- $97$  is the largest two-digit prime.
- Reversing the order of its digits results in another prime.
- There is only one way in which  $97$  can be written as a difference of two perfect squares.
- There is only one way in which  $97$  can be written as a sum of two perfect squares.
- $\tfrac1{97}$  has exactly  $96$  digits in the [smallest] repeating block of its decimal expansion.
- Jerry blames the sock gnomes for the theft of exactly  $97$  of his socks.


A repunit is a natural number whose digits are all  $1$ .  For instance, \begin{align*}&1,&11,&111,&1111,&\vdots\end{align*} are the four smallest repunits.  How many digits are there in the smallest repunit that is divisible by  $97?$ 
Answer: 96
Problem: Let  $n$  be a given positive integer. Determine all positive divisors  $d$  of  $3n^2$  such that  $n^2 + d$  is the square of an integer.
Answer: 3n^2
Problem: 2018 little ducklings numbered 1 through 2018 are standing in a line, with each holding a slip of paper with a nonnegative number on it; it is given that ducklings 1 and 2018 have the number zero.  At some point, ducklings 2 through 2017 change their number to equal the average of the numbers of the ducklings to their left and right.  Suppose the new numbers on the ducklings sum to 1000.  What is the maximum possible sum of the original numbers on all 2018 slips?
Answer: 2000
Problem: Determine all natural numbers  $n$ , such that it's possible to insert one digit at the right side of  $n$  to obtain  $13n$ .
Answer: 1, 2, 3
Problem: Let  $n$  be positive integers. Two persons play a game in which they are calling a integer  $m\ (1\leq m\leq n)$  alternately. Note that you may not call the number which have already said. The game is over when no one can call numbers, if the sum of the numbers that the lead have said  is divisible by 3, then the lead wins, otherwise the the second move wins. Find  $n$  for which there exists the way of forestalling.
Answer:  n \equiv 0 \pmod{3}
Problem: In a certain triangle  $ABC$ , there are points  $K$  and  $M$  on sides  $AB$  and  $AC$ , respectively, such that if  $L$  is the intersection of  $MB$  and  $KC$ , then both  $AKLM$  and  $KBCM$  are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle  $ABC$ .
Answer:  \angle BAC = 45^ullet
Problem: In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag.  Tony then pours all the jelly beans out of the five bags and begins making patterns with them.  One of the patterns that he makes has one jelly bean in a top row, three jelly beans in the next row, five jelly beans in the row after that, and so on:

\[\begin{array}{ccccccccc}&&&&*&&&&&&&*&*&*&&&&&*&*&*&*&*&&&*&*&*&*&*&*&*& *&*&*&*&*&*&*&*&*&&&&\vdots&&&&\end{array}\]

Continuing in this way, Tony finishes a row with none left over.  For instance, if Tony had exactly  $25$  jelly beans, he could finish the fifth row above with no jelly beans left over.  However, when Tony finishes, there are between  $10$  and  $20$  rows.  Tony then scoops all the jelly beans and puts them all back into the five bags so that each bag once again contains the same number.  How many jelly beans are in each bag?  (Assume that no marble gets put inside more than one bag.)
Answer: 45
Problem: Points  $A_1, A_2, \ldots, A_{2022}$  are chosen on a plane so that no three of them are collinear. Consider all angles  $A_iA_jA_k$  for distinct points  $A_i, A_j, A_k$ . What largest possible number of these angles can be equal to  $90^\circ$ ?
Answer: 2{,}042{,}220
Problem: Cells of a  $n*n$  square are filled with positive integers in the way that in the intersection of the  $i-$ th column and  $j-$ th row, the number  $i+j$  is written. In every step, we can choose two non-intersecting equal rectangles with one dimension equal to  $n$  and swap all the numbers inside these two rectangles with one another. ( without reflection or rotation ) Find the minimum number of moves one should do to reach the position where the intersection of the  $i-$ th column and  $j-$ row is written  $2n+2-i-j$ .
Answer: n-1
Problem: Find the sum of all integers  $x$  satisfying  $1 + 8x \le 358 - 2x \le 6x + 94$ .
Answer: 102
Problem: Given a positive integer  $n$ , suppose that  $P(x,y)$  is a real polynomial such that 
\[P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all  $x,y\in\{0,1,2,\dots,n\}$ } \] What is the minimum degree of  $P$ ?
Answer: 2n
Problem: [4] Let  $D$  be the set of divisors of  $100$ . Let  $Z$  be the set of integers between  $1$  and  $100$ , inclusive. Mark chooses an element  $d$  of  $D$  and an element  $z$  of  $Z$  uniformly at random. What is the probability that  $d$  divides  $z$ ?
Answer: \frac{217}{900}
Problem: What fraction of the Earth's volume lies above the  $45$  degrees north parallel? You may assume the Earth is a perfect sphere. The volume in question is the smaller piece that we would get if the sphere were sliced into two pieces by a plane.
Answer: \frac{8-5\sqrt{2}}{16}
Problem: Let  $ABC$  be an equilateral triangle.  $A $  point  $P$  is chosen at random within this triangle. What is the probability that the sum of the distances from point  $P$  to the sides of triangle  $ABC$  are measures of the sides of a triangle?
Answer: \frac{1}{4}