rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: A Geostationary Earth Orbit is situated directly above the equator and has a period equal to the Earth’s rotational pe­riod. It is at the precise distance of  $22,236$  miles above the Earth that a satellite can maintain an orbit with a period of rotation around the Earth exactly equal to  $24$  hours. Be­ cause the satellites revolve at the same rotational speed of the Earth, they appear stationary from the Earth surface. That is why most station antennas (satellite dishes) do not need to move once they have been properly aimed at a tar­ get satellite in the sky. In an international project, a total of ten stations were equally spaced on this orbit (at the precise distance of  $22,236$  miles above the equator). Given that the radius of the Earth is  $3960$  miles, find the exact straight dis­tance between two neighboring stations. Write your answer in the form  $a + b\sqrt{c}$ , where  $a, b, c$  are integers and  $c > 0$  is square-free.
Answer: -13098 + 13098\sqrt{5}
Problem: Let  $T$  be the set of ordered triples  $(x,y,z)$ , where  $x,y,z$  are integers with  $0\leq x,y,z\leq9$ . Players  $A$  and  $B$  play the following guessing game. Player  $A$  chooses a triple  $(x,y,z)$  in  $T$ , and Player  $B$  has to discover  $A$ *'s* triple in as few moves as possible.  A *move* consists of the following:  $B$  gives  $A$  a triple  $(a,b,c)$  in  $T$ , and  $A$  replies by giving  $B$  the number  $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$ . Find the minimum number of moves that  $B$  needs to be sure of determining  $A$ *'s*  triple.
Answer: 3
Problem: Find all triplets of real numbers  $(x,y,z)$  that are solutions to the system of equations $x^2+y^2+25z^2=6xz+8yz$  $ 3x^2+2y^2+z^2=240$ 
Answer: (6, 8, 2) \text{ or } (-6, -8, -2)
Problem: Find all positive integer  $k$  such that one can find a number of triangles in the Cartesian plane, the centroid of each triangle is a lattice point, the union of these triangles is a square of side length  $k$  (the sides of the square are not necessarily parallel to the axis, the vertices of the square are not necessarily lattice points), and the intersection of any two triangles is an empty-set, a common point or a common edge.
Answer:  k is a multiple of 3
Problem: In a sequence of natural numbers  $ a_1 $ ,  $ a_2 $ ,  $ \dots $ ,  $ a_ {1999} $ ,  $ a_n-a_ {n-1} -a_ {n-2} $  is divisible by  $ 100 (3 \leq n \leq 1999) $ . It is known that  $ a_1 =  19$  and  $ a_2 =  99$ . Find the remainder of  $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $  by  $8$ .
Answer: 1
Problem: How many nonzero coefficients can a polynomial  $ P(x)$  have if its coefficients are integers and  $ |P(z)| \le 2$  for any complex number  $ z$  of unit length?
Answer: 2
Problem: Compute  $\left\lceil\displaystyle\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$ . (The notation  $\left\lceil x\right\rceil$  denotes the least integer  $n$  such that  $n\geq x$ .)
Answer: 2019
Problem: 
	 Starting with an empty string, we create a string by repeatedly appending one of the letters  $H$ ,  $M$ ,  $T$  with probabilities  $\frac 14$ ,  $\frac 12$ ,  $\frac 14$ , respectively, until the letter  $M$  appears twice consecutively. What is the expected value of the length of the resulting string?
Answer: 6
Problem: Find the total number of solutions to the following system of equations:
 $ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} 
b(a + d)\equiv b \pmod{37} 
c(a + d)\equiv c \pmod{37} 
bc + d^2\equiv d \pmod{37} 
ad - bc\equiv 1 \pmod{37} \end{array}$ 
Answer: 1
Problem: Find all functions  $f:\mathbb{Q}^{+} \to \mathbb{Q}^{+}$  such that for all  $x\in \mathbb{Q}^+$ :  
- $f(x+1)=f(x)+1$ ,
- $f(x^2)=f(x)^2$ .

Answer:  f(x) = x
Problem:  Let  $ABCDEF$  be a regular hexagon with side length two. Extend  $FE$  and  $BD$  to meet at  $G$ . Compute the area of  $ABGF$ .
Answer: 3\sqrt{3}
Problem: Let the set of all bijective functions taking positive integers to positive integers be  $\mathcal B.$  Find all functions  $\mathbf F:\mathcal B\to \mathbb R$  such that   $$ (\mathbf F(p)+\mathbf F(q))^2=\mathbf F(p \circ p)+\mathbf F(p\circ q)+\mathbf F(q\circ p)+\mathbf F(q\circ q) $$   for all  $p,q \in \mathcal B.$
Answer: \mathbf{F}(p) = 1
Problem: What is the maximum number of possible change of directions in a path traveling on the edges of a rectangular array of  $2004 \times 2004$ , if the path does not cross the same place twice?.
Answer: 2004 \times 2005 - 1
Problem: Draw the perpendicular to the tangent line of the ellipse  $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\ (a>0,\ b>0)$  from the origin  $O(0,\ 0).$ 

Let  $\theta$  be the angle between the perpendicular and the positive direction of  $x$  axis. Denote the length of the perpendicular by  $r(\theta).$ 

Calculate  $\int_{0}^{2\pi}r(\theta )^{2}\ d\theta.$ 
Answer: \pi (a^{2} + b^{2})
Problem: A number  $N$  is in base 10,  $503$  in base  $b$  and  $305$  in base  $b+2$  find product of digits of  $N$ 
Answer: 64
Problem: There are  $100$  white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.
Answer: 50
Problem: Given a rhombus  $ABCD$ , find the locus of the points  $P$  lying inside the rhombus and satisfying  $\angle APD+\angle BPC=180^{\circ}$ .
Answer: AC \text{ and } BD
Problem: Estimate the number of primes among the first thousand primes divide some term of the sequence
    \[2^0+1,2^1+1,2^2+1,2^3+1,\ldots.\]
    An estimate of  $E$  earns  $2^{1-0.02|A-E|}$  points, where  $A$  is the actual answer.
Answer: 10
Problem: Niki and Kyle play a triangle game. Niki first draws  $\triangle ABC$  with area  $1$ , and Kyle picks a point  $X$  inside  $\triangle ABC$ . Niki then draws segments  $\overline{DG}$ ,  $\overline{EH}$ , and  $\overline{FI}$ , all through  $X$ , such that  $D$  and  $E$  are on  $\overline{BC}$ ,  $F$  and  $G$  are on  $\overline{AC}$ , and  $H$  and  $I$  are on  $\overline{AB}$ . The ten points must all be distinct. Finally, let  $S$  be the sum of the areas of triangles  $DEX$ ,  $FGX$ , and  $HIX$ . Kyle earns  $S$  points, and Niki earns  $1-S$  points. If both players play optimally to maximize the amount of points they get, who will win and by how much?
Answer: \frac{2}{3}
Problem: For each  $k$  , find the least  $n$  in terms of  $k$  st the following holds:
There exists  $n$  real numbers  $a_1 , a_2 ,\cdot \cdot \cdot , a_n$  st for each  $i$  :  $$ 0 < a_{i+1} - a_{i} < a_i - a_{i-1} $$   And , there exists  $k$  pairs  $(i,j)$  st  $a_i - a_j = 1$ .
Answer: 2k
Problem: Janet played a song on her clarinet in 3 minutes and 20 seconds. Janet’s goal is to play the song at a 25%
faster rate. Find the number of seconds it will take Janet to play the song when she reaches her goal.

Answer: 160 \text{ seconds}
Problem: A regular  $ n\minus{}$ gon  $ A_1A_2A_3 \cdots A_k \cdots A_n$  inscribed in a circle of radius  $ R$  is given. If  $ S$  is a point on the circle, calculate \[ T \equal{} \sum^n_{k\equal{}1} SA^2_k.\]
Answer: 2nR^2
Problem: The function  $f_n (x)\ (n=1,2,\cdots)$  is defined  as follows.

\[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\]

Evaluate 

\[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]
Answer: e^{1/2}
Problem: Find all pairs of nonnegative integers  $(x, y)$  for which  $(xy + 2)^2 = x^2 + y^2 $ .
Answer: (0, 2) \text{ and } (2, 0)
Problem: Let  $k$  be a constant number larger than  $1$ . Find all polynomials  $P(x)$  such that  $P({x^k}) = {\left( {P(x)} \right)^k}$  for all real  $x$ .
Answer:  P(x) = 0 or P(x) = 1
Problem: The number  $N$  is the product of two primes. The sum of the positive divisors of  $N$  that are less than  $N$  is  $2014$ . Find  $N$ .
Answer: 4022
Problem: In a coordinate system, a circle with radius  $7$  and center is on the y-axis placed inside the parabola with equation  $y = x^2$  , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.

Answer: (0, \frac{197}{4})
Problem: David, when submitting a problem for CMIMC, wrote his answer as  $100\tfrac xy$ , where  $x$  and  $y$  are two positive integers with  $x<y$ . Andrew interpreted the expression as a product of two rational numbers, while Patrick interpreted the answer as a mixed fraction. In this case, Patrick's number was exactly double Andrew's!  What is the smallest possible value of  $x+y$ ?
Answer: 299
Problem: Find the positive integer  $n$  such that  $10^n$  cubic centimeters is the same as 1 cubic kilometer.

Answer: 15
Problem:  $1.$ A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high.

Find the height of the bottle.
Answer: 10
Problem: n \geq 6 is an integer. evaluate the minimum of f(n) s.t: any graph with n vertices and f(n) edge contains two cycle which are distinct( also they have no comon vertice)?
Answer: 3(n-2)
Problem: Suppose that  $a$  and  $b$  are positive real numbers such that  $3\log_{101}\left(\frac{1,030,301-a-b}{3ab}\right) = 3 - 2 \log_{101}(ab)$ . Find  $101 - \sqrt[3]{a}-  \sqrt[3]{b}$ .
Answer: 0
Problem: Find the set of all real numbers  $k$  with the following property: For any positive, differentiable function  $f$  that satisfies  $f^{\prime}(x) > f(x)$  for all  $x,$  there is some number  $N$  such that  $f(x) > e^{kx}$  for all  $x > N.$ 
Answer: k < 1
Problem: Let  $ABC$  be an acute triangle. Let  $P$  be a point on the circle  $(ABC)$ , and  $Q$  be a point on the segment  $AC$  such that  $AP\perp BC$  and  $BQ\perp AC$ . Lot  $O$  be the circumcenter of triangle  $APQ$ . Find the angle  $OBC$ .
Answer: \angle OBC = 90^\circ
Problem: If  $x$  and  $y$  are positive integers such that  $(x-4)(x-10)=2^y$ , then Find maximum value of  $x+y$ 
Answer: 16
Problem: find all pairs of relatively prime natural numbers  $ (m,n) $  in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b)  > 1 \]
for every natural numbers  $ a $  and  $ b $ 
Answer:  (m, n) = (1, 1)
Problem: The first triangle number is  $1$ ; the second is  $1 + 2 = 3$ ; the third is  $1 + 2 + 3 = 6$ ; and so on. Find the sum of the first  $100$  triangle numbers.
Answer: 171700
Problem: Let  $\mathcal{P}$  be a regular  $2022$ -gon with area  $1$ . Find a real number  $c$  such that, if points  $A$  and  $B$  are chosen independently and uniformly at random on the perimeter of  $\mathcal{P}$ , then the probability that  $AB \geq c$  is  $\tfrac{1}{2}$ .
Answer: \sqrt{\frac{2}{\pi}}
Problem: Let  $ABC$  be a triangle with  $AB = 3$ ,  $BC = 4$ , and  $AC =5$ . If  $D$  is the projection from  $B$  onto  $AC$ ,  $E$  is the projection from  $D$  onto  $BC$ , and  $F$  is the projection from  $E$  onto  $AC$ , compute the length of the segment  $DF$ .
Answer: 0
Problem: Let  $\alpha$  be a solution satisfying the equation  $|x|=e^{-x}.$  Let  $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$ Find  $\lim_{n\to\infty} n^2I_n.$ 
Answer: 1
Problem: A  $8\times 8$  board is given. Seven out of  $64$  unit squares are painted black. Suppose that there exists a positive  $k$  such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area  $k$  containing no black squares. Find the maximum value of  $k$ . 

Answer: 8
Problem: In an attempt to copy down from the board a sequence of six positive integers in arithmetic progression, a student wrote down the five numbers, \[ 113,137,149,155,173,  \] accidentally omitting one. He later discovered that he also miscopied one of them. Can you help him and recover the original sequence?
Answer: 113, 125, 137, 149, 161, 173
Problem: Positive integers m and n are both greater than 50, have a least common multiple equal to 480, and have a
greatest common divisor equal to 12. Find m + n.

Answer: 156
Problem: At a mathematical competition  $n$  students work on  $6$  problems each one with three possible answers. After the competition, the Jury found that for every two students the number of the problems, for which these students have the same answers, is  $0$  or  $2$ .  Find the maximum possible value of  $n$ .
Answer:  n = 18
Problem: A  $4$ -dimensional hypercube of edge length  $1$  is constructed in  $4$ -space with its edges parallel to
the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given
by  $(a, b, c, d)$ , where each of  $a, b, c,$  and  $d$  is either  $0$  or  $1$ . The  $3$ -dimensional hyperplane given
by  $x + y + z + w = 2$  intersects the hypercube at  $6$  of its vertices. Compute the  $3$ -dimensional
volume of the solid formed by the intersection.
Answer: 1
Problem: Let  $n$  be a positive integer. A panel of dimenisions  $2n\times2n$  is divided in  $4n^2$  squares with dimensions  $1\times1$ . What is the highest possible number of diagonals that can be drawn in  $1\times1$  squares, such that each two diagonals have no common points. 
Answer: 2n^2
Problem: Let  $p$  be a given odd prime. Find the largest integer  $k'$  such that it is possible to partition  $\{1,2,\cdots,p-1\}$  into two sets  $X,Y$  such that for any  $k$  with  $0 \le k \le k'$ ,  $$ \sum_{a \in X}a^k \equiv \sum_{b \in Y}b^k \pmod p $$  *houkai*
Answer: k' = \frac{p-3}{2}
Problem: There are seven cards in a hat, and on the card  $k$  there is a number  $2^{k-1}$ ,  $k=1,2,...,7$ . Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds  $124$ . What is the most probable sum he can get?
Answer: 127
Problem: In an acute triangle  $ABC$ , the bisector  $AL$ , the altitude  $BH$ , and the perpendicular bisector of the side  $AB$  intersect at one point. Find the value of the angle  $BAC$ .
Answer: 60^	extcirc
Problem: A Christmas tree must be erected inside a convex rectangular garden and attached to the posts at the corners of the garden with four ropes running at the same height from the ground. At what point should the Christmas tree be placed, so that the sum of the lengths of these four cords is as small as possible?
Answer: The point P should be placed at the center of the rectangle.
Problem: For each of the  $9$  positive integers  $n,2n,3n,\dots , 9n$  Alice take the first decimal digit (from the left) and writes it onto a blackboard. She selected  $n$  so that among the nine digits on the blackboard there is the least possible number of different digits. What is this number of different digits equals to?
Answer: 4
Problem: A *root of unity* is a complex number that is a solution to  $ z^n \equal{} 1$  for some positive integer  $ n$ . Determine the number of roots of unity that are also roots of  $ z^2 \plus{} az \plus{} b \equal{} 0$  for some integers  $ a$  and  $ b$ .
Answer: 4
Problem: Lynnelle and Moor love toy cars, and together, they have  $27$  red cars,  $27$  purple cars, and  $27$  green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has  $17$  more cars of any color than Moor has of any color. How many purple cars does Lynnelle have?
Answer: 22
Problem: Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle?
Answer: 52 - 2\sqrt{235}
Problem: Determine the smallest positive number  $a$  such that the number of all integers belonging to  $(a, 2016a]$  is  $2016$ .
Answer: \frac{2017}{2016}
Problem: Find all real-valued functions  $f$  defined on pairs of real numbers, having the following property: for all real numbers  $a, b, c$ , the median of  $f(a,b), f(b,c), f(c,a)$  equals the median of  $a, b, c$ .

(The *median* of three real numbers, not necessarily distinct, is the number that is in the middle when the three numbers are arranged in nondecreasing order.)
Answer:  f(x, y) = x \text{ or } f(x, y) = y
Problem: find all functions from the reals to themselves. such that for every real  $x,y$ .  $$ f(y-f(x))=f(x)-2x+f(f(y)) $$  
Answer:  f(x) = x
Problem: In order to save money on gas and use up less fuel, Hannah has a special battery installed in the family van.  Before the installation, the van averaged  $18$  miles per gallon of gas.  After the conversion, the van got  $24$  miles per gallong of gas.

Michael notes, "The amount of money we will save on gas over any time period is equal to the amount we would save if we were able to convert the van to go from  $24$  miles per gallon to  $m$  miles per gallon.  It is also the same that we would save if we were able to convert the van to go from  $m$  miles per gallon to  $n$  miles per gallon."

Assuming Michael is correct, compute  $m+n$ .  In this problem, assume that gas mileage is constant over all speeds and terrain and that the van gets used the same amount regardless of its present state of conversion.
Answer: 108
Problem: Alice is thinking of a positive real number  $x$ , and Bob is thinking of a positive real number  $y$ . Given that  $x^{\sqrt{y}}=27$  and  $(\sqrt{x})^y=9$ , compute  $xy$ .
Answer: 16\sqrt[4]{3}
Problem: Find all real numbers  $a$  for which the following equation has a unique real solution:  $$ |x-1|+|x-2|+\ldots+|x-99|=a. $$  
Answer: 2450
Problem: Elmo is drawing with colored chalk on a sidewalk outside. He first marks a set  $S$  of  $n>1$  collinear points. Then, for every unordered pair of points  $\{X,Y\}$  in  $S$ , Elmo draws the circle with diameter  $XY$  so that each pair of circles which intersect at two distinct points are drawn in different colors. Count von Count then wishes to count the number of colors Elmo used. In terms of  $n$ , what is the minimum number of colors Elmo could have used?
Answer: \left\lceil \frac{n}{2} \right\rceil
Problem: Find the greatest  $n$  such that  $(z+1)^n = z^n + 1$  has all its non-zero roots in the unitary circumference, e.g.  $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$  implies  $|\alpha| = 1.$ 
Answer:  n = 7
Problem: Let  $k$  be a positive real number. Suppose that the set of real numbers  $x$  such that  $x^2+k|x| \leq 2019$  is an interval of length  $6$ . Compute  $k$ .
Answer: 670
Problem: Let  $M$  be the intersection of the diagonals of a cyclic quadrilateral  $ABCD$ . Find the length of  $AD$ , if it is known that  $AB=2$  mm ,   $BC = 5$  mm,  $AM = 4$  mm, and  $\frac{CD}{CM}= 0.6$ .
Answer: 2 \text{ mm}
Problem: A class of  $10$  students took a math test. Each problem was solved by exactly  $7$  of the students. If the first nine students each solved  $4$  problems, how many problems did the tenth student solve?
Answer: 6
Problem: Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to  $1.$ 
Answer: \frac{1}{4}
Problem: A point  $P$  is given in the square  $ABCD$  such that  $\overline{PA}=3$ ,  $\overline{PB}=7$  and  $\overline{PD}=5$ . Find the area of the square.
Answer: 58
Problem: Let  $m$  and  $n$  be positive integers and  $p$  be a prime number. Find the greatest positive integer  $s$  (as a function of  $m,n$  and  $p$ ) such that from a random set of  $mnp$  positive integers we can choose  $snp$  numbers, such that they can be partitioned into  $s$  sets of  $np$  numbers, such that the sum of the numbers in every group gives the same remainder when divided by  $p$ .
Answer: m
Problem: If   $$ \sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c} $$  where  $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$  then what is the value of  $a+ b ?$ 
Answer: 80
Problem: Let  $P(X,Y)=X^2+2aXY+Y^2$  be a real polynomial where  $|a|\geq 1$ . For a given positive integer  $n$ ,  $n\geq 2$  consider the system of equations:  \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions  $(x_1,x_2,\ldots,x_n)$  and  $(y_1,y_2,\ldots,y_n)$  of the system to be equivalent if there exists a real number  $\lambda \neq 0$ ,  $x_1=\lambda y_1$ ,  $\ldots$ ,  $x_n= \lambda y_n$ . How many nonequivalent solutions does the system have?
Answer: 1
Problem: Let  $x_1, x_2, . . . , x_{2022}$  be nonzero real numbers. Suppose that  $x_k + \frac{1}{x_{k+1}} < 0$  for each  $1 \leq k \leq 2022$ , where  $x_{2023}=x_1$ . Compute the maximum possible number of integers  $1 \leq n \leq 2022$  such that  $x_n > 0$ .
Answer: 1011
Problem: In four years Kay will be twice as old as Gordon. Four years after that Shaun will be twice as old as Kay. Four years after that Shaun will be three times as old as Gordon. How many years old is Shaun now?
Answer: 48
Problem: Let  $a_1, a_2, \cdots, a_{2022}$  be nonnegative real numbers such that  $a_1+a_2+\cdots +a_{2022}=1$ . Find the maximum number of ordered pairs  $(i, j)$ ,  $1\leq i,j\leq 2022$ , satisfying   $$ a_i^2+a_j\ge \frac 1{2021}. $$  
Answer: 2021 \times 2022
Problem: A three-digit number  $x$  in base  $10$  has a units-digit of  $6$ . When  $x$  is written is base  $9$ , the second digit of the number is  $4$ , and the first and third digit are equal in value. Compute  $x$  in base  $10$ .
Answer: 446
Problem: Find all integes  $a,b,c,d$  that form an arithmetic progression satisfying  $d-c+1$  is prime number and  $a+b^2+c^3=d^2b$ 
Answer: (a, b, c, d) = (n, n+1, n+2, n+3) where n is an arbitrary integer.
Problem: The decimal digits of a natural number  $A$  form an increasing sequence (from left to right).  Find the sum of the digits of  $9A$ .
Answer: 9
Problem: Starting with a positive integer  $M$  written on the board , Alice plays the following game: in each move, if  $x$  is the number on the board, she replaces it with  $3x+2$ .Similarly, starting with a positive integer  $N$  written on the board, Bob plays the following game: in each move, if  $x$  is the number on the board, he replaces it with  $2x+27$ .Given that Alice and Bob reach the same number after  playing  $4$  moves each, find the smallest value of  $M+N$ 
Answer: 10
Problem: In triangle  $ ABC$ , one has marked the incenter, the foot of altitude from vertex  $ C$  and the center of the excircle tangent to side  $ AB$ . After this, the triangle was erased. Restore it.
Answer:  \triangle ABC
Problem: Let  $N>1$  be an integer. We are adding all remainders when we divide  $N$  by all positive integers less than  $N$ . If this sum is less than  $N$ , find all possible values of  $N$ .
Answer: 2
Problem: If  $1<k_1<k_2<...<k_n$  and  $a_1,a_2,...,a_n$  are integers such that for every integer  $N,$   $k_i \mid N-a_i$  for some  $1 \leq i \leq n,$  find the smallest possible value of  $n.$ 
Answer: 5
Problem: After a typist has written ten letters and had addressed the ten corresponding envelopes, a careless mailing clerk inserted the letters in the envelopes at random, one letter per envelope. What is the probability that **exactly** nine letters were inserted in the proper envelopes?
Answer: 0
Problem: What is the last two digits of the number  $(11^2 + 15^2 + 19^2 +  ...  + 2007^2)^2$ ?
Answer: 0
Problem: Bob and Bill's history class has  $32$  people in it, but only  $30$  people are allowed per class. Two people will be randomly selected for transfer to a random one of two history classes. What is the probability that Bob and Bill are both transferred, and that they are placed in the same class? Write your answer as a fraction in lowest terms.
Answer: \frac{1}{992}