rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Determine all pairs  $(x, y)$  of positive integers for which the equation \[x + y + xy = 2006\] holds.
Answer: (2, 668), (668, 2), (8, 222), (222, 8)
Problem: Evaluate  $\textstyle\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$ , where  $\mathrm{Arccot}\,t$  for  $t \geq 0$  denotes the number  $\theta$  in the interval  $0 < \theta \leq \pi/2$  with  $\cot \theta = t$ .
Answer: \frac{\pi}{2}
Problem: The sequence  ${a_0, a_1, a_2, ...}$  of real numbers satisfies the recursive relation   $$ n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n $$   for every positive integer  $n$ , where  $a_0 = a_1 = 1$ . Calculate the sum   $$ \frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}} $$  .
Answer: 2009 \times 1005
Problem: Determine the prime numbers  $p, q, r$  with the property  $\frac {1} {p} + \frac {1} {q} + \frac {1} {r} \ge 1$ 
Answer: (2, 3, 5)
Problem: The sequence  $ (F_n)$  of Fibonacci numbers satisfies  $ F_1 \equal{} 1, F_2 \equal{} 1$  and  $ F_n \equal{} F_{n\minus{}1} \plus{}F_{n\minus{}2}$  for all  $ n \ge 3$ . Find all pairs of positive integers  $ (m, n)$ , such that  $ F_m . F_n \equal{} mn$ .
Answer:  (1, 1)
Problem: Claudia and Adela are betting to see which one of them will ask the boy they like for his telephone number. To decide they roll dice. If none of the dice are a multiple of 3, Claudia will do it. If exactly one die is a multiple of 3, Adela will do it. If 2 or more of the dice are a multiple of 3 neither one of them will do it. How many dice should be rolled so that the risk is the same for both Claudia and Adela?
Answer:  x = 2
Problem: Carl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving website. If, among these games, Carl wins  $5$  and loses  $0,$  James wins  $4$  and loses  $2,$  Saif wins  $1$  and loses  $6,$  and Ted wins  $4,$  how many games does Ted lose?
Answer: 6
Problem: We call a set “sum free” if no two elements of the set add up to a third element of the set. What is the maximum size of a sum free subset of  $\{ 1, 2, \ldots , 2n - 1 \}$ .
Answer: n
Problem: Let  $M$  be a  set with  $n$  elements. How many pairs  $(A, B)$  of subsets of  $M$  are there such that  $A$  is a subset of  $B?$ 
Answer: 3^n
Problem: Let  $A$  be the set of the  $16$  first positive integers. Find the least positive integer  $k$  satisfying the condition: In every  $k$ -subset of  $A$ , there exist two distinct  $a, b \in A$  such that  $a^2 + b^2$  is prime.
Answer:  k = 9
Problem: Of  $450$  students assembled for a concert,  $40$  percent were boys. After a bus containing an equal number of boys and girls brought more students to the concert,  $41$  percent of the students at the concert were boys. Find the number of students on the bus.
Answer: 50
Problem: Three real numbers  $x$ ,  $y$ , and  $z$  are such that  $(x+4)/2=(y+9)/(z-3)=(x+5)/(z-5)$ .  Determine the value of  $x/y$ .
Answer: \frac{1}{2}
Problem: Tim is thinking of a positive integer between  $2$  and  $15,$  inclusive, and Ted is trying to guess the integer. Tim tells Ted how many factors his integer has, and Ted is then able to be certain of what Tim's integer is. What is Tim's integer?
Answer: 12
Problem: Polynomial  $ P(t)$  is such that for all real  $ x$ ,
\[ P(\sin x) \plus{} P(\cos x) \equal{} 1.
\]
What can be the degree of this polynomial?
Answer: 0 or 2 + 4n for n \geq 0
Problem: Find the minimum value of the following function  $f(x) $  defined at  $0<x<\frac{\pi}{2}$ .

\[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\]
Answer: \ln\left(3 + 2\sqrt{2}\right)
Problem: Find, with explanation, the maximum value of  $f(x)=x^3-3x$  on the set of all real numbers  $x$  satisfying  $x^4+36\leq 13x^2$ .
Answer: 18
Problem:  $\boxed{A2}$  Find the maximum value of  $z+x$  if  $x,y,z$  are satisfying the given conditions. $x^2+y^2=4$   $z^2+t^2=9$   $xt+yz\geq 6$ 
Answer: 5
Problem: Let  $x,y,z$  is positive. Solve: $\begin{cases}{x\left( {6 - y} \right) = 9}
{y\left( {6 - z} \right) = 9}
{z\left( {6 - x} \right) = 9}\end{cases}$ 
Answer: x = 3 , y = 3 , z = 3
Problem: For a unit circle  $O$ , arrange points  $A,B,C,D$  and  $E$  in that order evenly along  $O$ 's circumference. For each of those points, draw the arc centered at that point inside O from the point to its left to the point to its right. Denote the outermost intersections of these arcs as  $A', B', C', D'$  and  $E'$ , where the prime of any point is opposite the point. The length of  $AC'$  can be written as an expression  $f(x)$ , where  $f$  is a trigonometric function. Find this expression.
Answer: \tan 36^\circ
Problem: Done with her new problems, Wendy takes a break from math.  Still without any fresh reading material, she feels a bit antsy.  She starts to feel annoyed that Michael's loose papers clutter the family van.  Several of them are ripped, and bits of paper litter the floor.  Tired of trying to get Michael to clean up after himself, Wendy spends a couple of minutes putting Michael's loose papers in the trash.  "That seems fair to me," confirms Hannah encouragingly.

While collecting Michael's scraps, Wendy comes across a corner of a piece of paper with part of a math problem written on it.  There is a monic polynomial of degree  $n$ , with real coefficients.  The first two terms after  $x^n$  are  $a_{n-1}x^{n-1}$  and  $a_{n-2}x^{n-2}$ , but the rest of the polynomial is cut off where Michael's page is ripped.  Wendy barely makes out a little of Michael's scribbling, showing that  $a_{n-1}=-a_{n-2}$ .  Wendy deciphers the goal of the problem, which is to find the sum of the squares of the roots of the polynomial.  Wendy knows neither the value of  $n$ , nor the value of  $a_{n-1}$ , but still she finds a [greatest] lower bound for the answer to the problem.  Find the absolute value of that lower bound.
Answer: 1
Problem: This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant  $X$ , let  $t(X)$  be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true: $1)$  For any two friends  $X'$  and  $X''$ , we have  $t(X') \neq t(X''),$  $2)$  For every contestant  $X$ , the set  $\{ t(Y) : Y \text{ is a friend of } X \}$  consists of consecutive integers.
The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need?
Answer: 2
Problem: Let the triangle  $ABC$  have area  $1$ . The interior bisectors of the angles  $\angle BAC,\angle ABC, \angle BCA$  intersect the sides  $(BC), (AC), (AB) $  and the circumscribed circle of the respective triangle  $ABC$  at the points  $L$  and  $G, N$  and  $F, Q$  and  $E$ . The lines  $EF, FG,GE$  intersect the bisectors  $(AL), (CQ) ,(BN)$  respectively at points  $P, M, R$ . Determine the area of the hexagon  $LMNPR$ .
Answer: \frac{1}{2}
Problem: Find all real-valued functions on the positive integers such that  $f(x + 1019) = f(x)$  for all  $x$ , and  $f(xy) = f(x) f(y)$  for all  $x,y$ .
Answer:  f(x) = 1
Problem: One country has  $n$  cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number of flights that the worker will have to use?
Answer: 0
Problem: Jane and Josh wish to buy a candy. However Jane needs seven more cents to buy the candy, while John needs one more cent. They decide to buy only one candy together, but discover that they do not have enough money. How much does the candy cost?
Answer: 7
Problem: 
	 Dodecagon  $QWARTZSPHINX$  has all side lengths equal to  $2$ , is not self-intersecting (in particular, the twelve vertices are all distinct), and moreover each interior angle is either  $90^{\circ}$  or  $270^{\circ}$ . What are all possible values of the area of  $\triangle SIX$ ?
Answer: 2 \text{ and } 4
Problem: If  $0 < x < \frac{\pi}{2}$  and  $\frac{\sin x}{1 + \cos x} = \frac{1}{3}$ , what is  $\frac{\sin 2x}{1 + \cos 2x}$ ?
Answer: \frac{3}{4}
Problem: Marie repeatedly flips a fair coin and stops after she gets tails for the second time. What is the expected number of times Marie flips the coin?
Answer: 4
Problem: Determine the set of real numbers  $\alpha$  that can be expressed in the form \[\alpha=\sum_{n=0}^{\infty}\frac{x_{n+1}}{x_n^3}\]
where  $x_0,x_1,x_2,\dots$   is an increasing sequence of real numbers with  $x_0=1$ .
Answer: \alpha \ge \frac{3\sqrt{3}}{2}
Problem: Find all pairs of integers  $(a,b)$  so that each of the two cubic polynomials   $$ x^3+ax+b \, \, \text{and} \, \, x^3+bx+a $$   has all the roots to be integers.
Answer: (0, 0)
Problem: Let  $n$  be an integer greater than  $3$ . A square of side length  $n$  is divided by lines parallel to each side into  $n^2$  squares of length  $1$ . Find the number of convex trapezoids which have vertices among the vertices of the  $n^2$  squares of side length  $1$ , have side lengths less than or equal  $3$  and have area equal to  $2$ Note: Parallelograms are trapezoids.
Answer: 2(n-1)^2
Problem: 8. You have been kidnapped by a witch and are stuck in the *Terrifying Tower*, which has an infinite number of floors, starting with floor 1, each initially having 0 boxes. The witch allows you to do the following two things:

- For a floor  $i$ , put 2 boxes on floor  $i+5$ , 6 on floor  $i+4$ , 13 on floor  $i+3$ , 12 on floor  $i+2$ , 8 on floor  $i+1$ , and 1 on floor  $i$ , or remove the corresponding number of boxes from each floor if possible.
- For a floor  $i$ , put 1 box on floor  $i+4$ , put 3 boxes on floor  $i+3$ , 6 on floor  $i+2$ , 5 on floor  $i+1$ , and 3 on floor  $i$ , or remove the corresponding number of boxes from each floor if possible.

At the end, suppose the witch wishes to have exactly  $n$  boxes in the tower. Specifically, she wants them to be on the first 10 floors. Let  $T(n)$  be the number of distinct distributions of these  $n$  boxes that you can make. Find  $\displaystyle\sum_{n=1}^{15} T(n)$ .
Answer: 0
Problem: Let  $ABCD$  be a tetrahedron with  $AB=CD=1300$ ,  $BC=AD=1400$ , and  $CA=BD=1500$ . Let  $O$  and  $I$  be the centers of the circumscribed sphere and inscribed sphere of  $ABCD$ , respectively. Compute the smallest integer greater than the length of  $OI$ .
Answer: 1
Problem: Let  $\triangle ABC$  be a triangle with  $AB = 7$ ,  $AC = 8$ , and  $BC = 3$ . Let  $P_1$  and  $P_2$  be two distinct points on line  $AC$  ( $A, P_1, C, P_2$  appear in that order on the line) and  $Q_1$  and  $Q_2$  be two distinct points on line  $AB$  ( $A, Q_1, B, Q_2$  appear in that order on the line) such that  $BQ_1 = P_1Q_1 = P_1C$  and  $BQ_2 = P_2Q_2 = P_2C$ . Find the distance between the circumcenters of  $BP_1P_2$  and  $CQ_1Q_2$ .
Answer: 3
Problem: Given the  $7$ -element set  $A = \{ a ,b,c,d,e,f,g \}$ , find a collection  $T$  of  $3$ -element subsets of  $A$  such that each pair of elements from  $A$  occurs exactly once on one of the subsets of  $T$ .
Answer:  T = \{\{a, b, c\}, \{a, d, e\}, \{a, f, g\}, \{b, d, f\}, \{b, e, g\}, \{c, d, g\}, \{c, e, f\}\}
Problem: Find all positive integers  $a$  and  $b$  such that  $(a + b^2)(b + a^2) = 2^m$  for some integer  $m.$ *(6 points)*
Answer: (a, b, m) = (1, 1, 2)
Problem: Inside the rectangle  $ABCD$  is taken a point  $M$  such that  $\angle BMC + \angle AMD = 180^o$ . Determine the sum of the angles  $BCM$  and  $DAM$ .
Answer: 90
Problem: Find all integer triples  $(a, b, c)$  satisfying the equation  $$     5 a^2 + 9 b^2 = 13 c^2. $$  
Answer: (0, 0, 0)
Problem: Determine the ordered systems  $(x,y,z)$  of positive rational numbers for which  $x+\frac{1}{y},y+\frac{1}{z}$  and  $z+\frac{1}{x}$  are integers.
Answer:  (1, 1, 1)
Problem: Nine positive integers  $a_1,a_2,...,a_9$  have their last  $2$ -digit part equal to  $11,12,13,14,15,16,17,18$  and  $19$  respectively. Find the last  $2$ -digit part of the sum of their squares.
Answer: 85
Problem: An archery target can be represented as three concentric circles with radii  $3$ ,  $2$ , and  $1$  which split the target into  $3$  regions, as shown in the figure below. What is the area of Region  $1$  plus the area of Region  $3$ ?
Answer: 6\pi
Problem: Solve the system of equations  $ \begin{cases} x^3 \plus{} x(y \minus{} z)^2 \equal{} 2 y^3 \plus{} y(z \minus{} x)^2 \equal{} 30 z^3 \plus{} z(x \minus{} y)^2 \equal{} 16\end{cases}$ .
Answer:  (x, y, z) = (\sqrt[3]{2}, \sqrt[3]{2}, \sqrt[3]{2})
Problem: Determine all positive integers  $k$  for which there exist positive integers  $r$  and  $s$  that satisfy the equation   $$ (k^2-6k+11)^{r-1}=(2k-7)^{s}. $$  
Answer:  k \in \{2, 3, 4, 8\}
Problem: Let  $O$  be a circle with diameter  $AB = 2$ . Circles  $O_1$  and  $O_2$  have centers on  $\overline{AB}$  such that  $O$  is tangent to  $O_1$  at  $A$  and to  $O_2$  at  $B$ , and  $O_1$  and  $O_2$  are externally tangent to each other. The minimum possible value of the sum of the areas of  $O_1$  and  $O_2$  can be written in the form  $\frac{m\pi}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Compute  $m + n$ .
Answer: 3
Problem: consider a  $2008 \times 2008$  chess board. let  $M$  be the smallest no of rectangles that can be drawn on the chess board so that sides of every cell of the board is contained in the sides of one of the rectangles. find the value of  $M$ . (eg for  $2\times 3$  chessboard, the value of  $M$  is 3.)

Answer: 2009
Problem: Given a positive integer  $N$ . There are three squirrels that each have an integer. It is known that the largest integer and the least one differ by exactly  $N$ . Each time, the squirrel with the second largest integer looks at the squirrel with the largest integer. If the integers they have are different, then the squirrel with the second largest integer would be unhappy and attack the squirrel with the largest one, making its integer decrease by two times the difference between the two integers. If the second largest integer is the same as the least integer, only of the squirrels would attack the squirrel with the largest integer. The attack continues until the largest integer becomes the same as the second largest integer. What is the maximum total number of attacks these squirrels make?

Proposed by USJL, ST.
Answer: N
Problem: Let  $n\ge3$  be a positive integer. Find the real numbers  $x_1\ge0,\ldots,x_n\ge 0$ , with  $x_1+x_2+\ldots +x_n=n$ , for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value.
Answer:  n^2
Problem: Let  $a$ ,  $b$ ,  $c$  be the three roots of the equation  $x^3-(k+1)x^2+kx+12=0$ , where  $k$  is a real number. If  $(a-2)^3+(b-2)^3+(c-2)^3=-18$ , find the value of  $k$ .
Answer: 5
Problem: Compute the largest possible number of distinct real solutions for  $x$  to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where  $a$ ,  $b$ , and  $c$  are real numbers.

[i]Proposed by Tristan Shin
Answer: 4
Problem: Let  $ABC$  be a triangle with area  $92$  square centimeters. Calculate the area of another triangle whose sides have the same lengths as the medians of triangle  $ABC$ .
Answer: 69
Problem: Find all the integer solutions  $(x,y,z)$  of the equation  $(x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)$ ,
Answer: (a, -a, 0) and its permutations, where a is an integer.
Problem: Determine all prime numbers  $p, q$  and  $r$  with  $p + q^2 = r^4$ .
Answer: p = 7, q = 3, r = 2
Problem: Determine all functions  $f:\mathbb{R}\rightarrow \mathbb{R}$  such that the set 
\[\left \{   \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \}\]
is finite, and for all  $x \in \mathbb{R}$ \[f(x-1-f(x)) = f(x) - x - 1\]
Answer:  f(x) = x
Problem: Points  $A$  and  $B$  are fixed points in the plane such that  $AB = 1$ . Find the area of the region consisting of all points  $P$  such that  $\angle APB > 120^o$ 
Answer: \frac{4\pi - 3\sqrt{3}}{18}
Problem: By dividing  $2023$  by a natural number  $m$ , the remainder is  $23$ . How many numbers  $m$  are there with this property?
Answer: 12
Problem:  $f$  is a function on the set of complex numbers such that  $f(z)=1/(z*)$ , where  $z*$  is the complex conjugate of  $z$ .  $S$  is the set of complex numbers  $z$  such that the real part of  $f(z)$  lies between  $1/2020$  and  $1/2018$ . If  $S$  is treated as a subset of the complex plane, the area of  $S$  can be expressed as  $m× \pi$  where  $m$  is an integer. What is the value of  $m$ ? 
Answer: 2019
Problem: Let  $f:[0,1]\rightarrow [0,1]$  be a continuous and bijective function.
Describe the set:
\[A=\{f(x)-f(y)\mid x,y\in[0,1]\backslash\mathbb{Q}\}\]
<details><summary>Note</summary>You are given the result that *there is no one-to-one function between the irrational numbers and  $\mathbb{Q}$ .*</details>
Answer:  (-1, 1)
Problem: Determine all (not necessarily finite) sets  $S$  of points in the plane such that given any four distinct points in  $S$ , there is a circle passing through all four or a line passing through some three.
Answer:  S
Problem: For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that **for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime.** Find the minimal possible value of the maximum of all numbers. 
Answer: 4
Problem: Real numbers  $a$  and  $b$  satisfy the system of equations   $$ \begin{cases} a^3-a^2+a-5=0  b^3-2b^2+2b+4=0  \end{cases} $$    Find the numerical value of the sum  $a+ b$ .
Answer: 1
Problem: Find the least positive integer  $n$  ( $n\geq 3$ ), such that among any  $n$  points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.
Answer: n = 7
Problem: A right circular cone has base of radius 1 and height 3.  A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone.  What is the side-length of the cube?
Answer: \frac{6}{2 + 3\sqrt{2}}
Problem:  In each cell of the table  $4 \times 4$ , in which the lines are labeled with numbers  $1,2,3,4$ , and columns with  letters  $a,b,c,d$ ,  one number is written:  $0$  or  $1$  . Such a table is called *valid* if there are exactly two units in each of its rows and in each column. Determine the number of *valid* tables.
Answer: 90
Problem: Let  $ ABC$  be an acute triangle,  $ CC_1$  its bisector,  $ O$  its circumcenter. The perpendicular from  $ C$  to  $ AB$  meets line  $ OC_1$  in a point lying on the circumcircle of  $ AOB$ . Determine angle  $ C$ .
Answer: 60^
Problem: We call a number *perfect* if the sum of its positive integer divisors(including  $1$  and  $n$ ) equals  $2n$ . Determine all *perfect* numbers  $n$  for which  $n-1$  and  $n+1$  are prime numbers.
Answer: 6
Problem: Three circles  $\Gamma_1,\Gamma_2,\Gamma_3$  are inscribed into an angle(the radius of  $\Gamma_1$  is the minimal, and the radius of  $\Gamma_3$  is the maximal) in such a way that  $\Gamma_2$  touches  $\Gamma_1$  and  $\Gamma_3$  at points  $A$  and  $B$  respectively. Let  $\ell$  be a tangent to  $A$  to  $\Gamma_1$ . Consider circles  $\omega$  touching  $\Gamma_1$  and  $\ell$ . Find the locus of meeting points of common internal tangents to  $\omega$  and  $\Gamma_3$ .
Answer: \Gamma_2 \cap AB \setminus \{A, B\}
Problem: Triangle  $ABC$  has  $\angle A=90^\circ$ , side  $BC=25$ ,  $AB>AC$ , and area  $150$ .  Circle  $\omega$  is inscribed in  $ABC$ , with  $M$  its point of tangency on  $AC$ .  Line  $BM$  meets  $\omega$  a second time at point  $L$ .  Find the length of segment  $BL$ .
Answer: 22.5
Problem: Find all pairs of  $ (a, n) $  natural numbers such that  $ \varphi (a ^ n + n) = 2 ^ n. $  
( $ \varphi (n) $  is the Euler function, that is, the number of integers from  $1$  up to  $ n $ , relative prime to  $ n $ )
Answer:  (2, 1)
Problem: Triangle  $ABC$  has  $BC=1$  and  $AC=2$ . What is the maximum possible value of  $\hat{A}$ .
Answer: 30^
Problem: Let  $ AB$  be the diameter of a given circle with radius  $ 1$  unit, and let  $ P$  be a given point on  $ AB$ . A line through  $ P$  meets the circle at points  $ C$  and  $ D$ , so a convex quadrilateral  $ ABCD$  is formed. Find the maximum possible area of the quadrilateral.
Answer: 1
Problem: The number  $2^{1997}$  has  $m$  decimal digits, while the number  $5^{1997}$  has  $n$  digits. Evaluate  $m+n$ .
Answer: 1998
Problem: In chess, a king threatens another king if, and only if, they are on neighboring squares, whether horizontally, vertically, or diagonally . Find the greatest amount of kings that can be placed on a  $12 \times 12$  board such that each king threatens just another king. Here, we are not considering part colors, that is, consider that the king are all, say, white, and that kings of the same color can threaten each other.
Answer: 72
Problem: Find the sum of all four-digit integers whose digits are a rearrangement of the digits  $1$ ,  $2$ ,  $3$ ,  $4$ , such as  $1234$ ,  $1432$ , or  $3124$ .
Answer: 66660
Problem: A triangle  $ABC$   $(a>b>c)$  is given. Its incenter  $I$  and the touching points  $K, N$  of the incircle with  $BC$  and  $AC$  respectively are marked. Construct a segment with length  $a-c$  using only a ruler and drawing at most three lines.
Answer: a - c
Problem: Each cell of a  $29 \times 29$  table contains one of the integers  $1, 2, 3, \ldots , 29$ , and each of these integers appears  $29$  times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.
Answer: 15
Problem: On each non-boundary unit segment of an  $8\times 8$  chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers?
Answer: 0
Problem: Call a polynomial  $P$  *prime-covering* if for every prime  $p$ , there exists an integer  $n$  for which  $p$  divides  $P(n)$ . Determine the number of ordered triples of integers  $(a,b,c)$ , with  $1\leq a < b < c \leq 25$ , for which  $P(x)=(x^2-a)(x^2-b)(x^2-c)$  is prime-covering.
Answer: 0
Problem: Mark's teacher is randomly pairing his class of  $16$  students into groups of  $2$  for a project. What is the probability that Mark is paired up with his best friend, Mike? (There is only one Mike in the class)
Answer: \frac{1}{15}
Problem: In the game of Avalon, there are  $10$  players,  $4$  of which are bad. A *quest* is a subset of those players. A quest fails if it contains at least one bad player. A randomly chosen quest of  $3$  players happens to fail. What is the probability that there is exactly one bad player in the failed quest?
Answer: \frac{3}{5}
Problem: Let  $ S \equal{}\{1,2,3, \ldots, 2n\}$  ( $ n \in \mathbb{Z}^\plus{}$ ). Ddetermine the number of subsets  $ T$  of  $ S$  such that there are no 2 element in  $ T$   $ a,b$  such that  $ |a\minus{}b|\equal{}\{1,n\}$ 
Answer:  a_n - b_n
Problem: The sum of the (decimal) digits of a natural number  $n$  equals  $100$ , and the sum of digits of  $44n$  equals  $800$ .  Determine the sum of digits of  $3n$ .
Answer: 300
Problem: A sequence  $\{a_n\}_{n\geq0}$  obeys the recurrence  $a_n=1+a_{n-1}+\alpha a_{n-2}$  for all  $n\geq2$  and for some  $\alpha>0$ . Given that  $a_0=1$  and  $a_1=2$ , compute the value of  $\alpha$  for which  $$ \sum_{n=0}^{\infty}\frac{a_n}{2^n}=10 $$  
Answer: \frac{6}{5}
Problem: Given a cube of unit side. Let  $A$  and  $B$  be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point  $A$  and tangent to the three sides with common point  $B$ .
Answer: \frac{1}{2}
Problem:  $\textbf{Problem 4.}$  What is the largest number such that dividing  $17$  or  $30$  by this number, the same remainder is obtained in both cases
Answer: 13
Problem: Find all pairs  $ (m, n)$  of positive integers such that  $ m^n \minus{} n^m \equal{} 3$ .
Answer:  (m, n) = (4, 1)