rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Determine the smallest positive constant  $k$  such that no matter what  $3$  lattice points we choose the following inequality holds:   $$  L_{\max} - L_{\min} \ge \frac{1}{\sqrt{k} \cdot L_{max}}  $$   
where  $L_{\max}$ ,  $L_{\min}$  is the maximal and minimal distance between chosen points. 
Answer: 4
Problem: Let  $G$  be the centroid of triangle  $ABC$  with  $AB=13,BC=14,CA=15$ . Calculate the sum of the distances from  $G$  to the three sides of the triangle.

Note: The *centroid* of a triangle is the point that lies on each of the three line segments between a vertex and the midpoint of its opposite side.
Answer: \frac{2348}{195}
Problem: Determine the least positive integer  $n{}$  for which the following statement is true: the product of any  $n{}$  odd consecutive positive integers is divisible by  $45$ .
Answer: 6
Problem: All angles of  $ABC$  are in  $(30,90)$ . Circumcenter of  $ABC$  is  $O$  and circumradius is  $R$ . Point  $K$  is projection of  $O$  to angle bisector of  $\angle B$ , point  $M$  is midpoint  $AC$ . It is known, that  $2KM=R$ . Find  $\angle B$ 
Answer: 60^	extcirc}
Problem: Tommy takes a 25-question true-false test.  He answers each question correctly with independent probability  $\frac{1}{2}$ .  Tommy earns bonus points for correct streaks: the first question in a streak is worth 1 point, the second question is worth 2 points, and so on. For instance, the sequence TFFTTTFT is worth 1 + 1 + 2 + 3 + 1 = 8 points. Compute the expected value of Tommy’s score.
Answer: 50
Problem: Let  $ f(x)$  be a function such that  $ 1\minus{}f(x)\equal{}f(1\minus{}x)$  for  $ 0\leq x\leq 1$ .

Evaluate  $ \int_0^1 f(x)\ dx$ .
Answer: \frac{1}{2}
Problem: Find all triples of positive integers  $(x, y, z)$  with   $$ \frac{xy}{z}+ \frac{yz}{x}+\frac{zx}{y}= 3 $$  
Answer:  (1, 1, 1)
Problem: For each vertex of triangle  $ABC$ , the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices  $A$  and  $B$   were equal. Furthermore the angle in vertex  $C$  is greater than two remaining angles. Find angle  $C$  of the triangle.
Answer: \gamma = 120^\circ
Problem: For a permutation  $\pi$  of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where  $\pi (i)$  denotes the  $i$ th element of the permutation. Suppose that  $M$  is the maximum possible value of  $S(\pi)$  over all permutations  $\pi$  of the integers from 1 to 10. Determine the number of permutations  $\pi$  for which  $S(\pi) = M$ .
Answer: 40320
Problem: Find all prime number  $p$  such that the number of positive integer pair  $(x,y)$  satisfy the following is not  $29$ .


- $1\le x,y\le 29$
- $29\mid y^2-x^p-26$


Answer: 2 \text{ and } 7
Problem: Let  $ a,\ b$  be postive real numbers. For a real number  $ t$ , denote by  $d(t)$  the distance between the origin and the line  $ (ae^t)x \plus{} (be^{ \minus{} t})y \equal{} 1$ .

Let  $ a,\ b$  vary with  $ ab \equal{} 1$ , find the minimum value of  $ \int_0^1 \frac {1}{d(t)^2}\ dt$ .
Answer: e - \frac{1}{e}
Problem: Find all real solutions of the equation:   $$ x=\frac{2z^2}{1+z^2} $$     $$ y=\frac{2x^2}{1+x^2} $$     $$ z=\frac{2y^2}{1+y^2} $$  
Answer: (0, 0, 0) or (1, 1, 1)
Problem: Let  $a \geq 0$  be a natural number. Determine all rational  $x$ , so that
\[\sqrt{1+(a-1)\sqrt[3]x}=\sqrt{1+(a-1)\sqrt x}\]
All occurring square roots, are not negative.**Note.** It seems the set of natural numbers =  $\mathbb N = \{0,1,2,\ldots\}$  in this problem.
Answer: x = 0, 1
Problem: A circle contains the points  $(0, 11)$  and  $(0, -11)$  on its circumference and contains all points  $(x, y)$  with  $x^2+y^2<1$  in its interior. Compute the largest possible radius of the circle.
Answer: \sqrt{122}
Problem:   $$ Problem 4  $$   Straight lines  $k,l,m$  intersecting each other in three different points are drawn on a classboard. Bob remembers that in some coordinate system the lines $ k,l,m$  have the equations  $y = ax, y = bx$  and  $y = c +2\frac{ab}{a+b}x$  (where  $ab(a + b)$  is non zero).
 Misfortunately, both axes are erased. Also, Bob remembers that there is missing a line  $n$  ( $y = -ax + c$ ), but he has forgotten  $a,b,c$ . How can he reconstruct the line  $n$ ?
Answer: y = -ax + c
Problem: Given an integer  $n \ge 2$ , solve in real numbers the system of equations \begin{align*}
\max\{1, x_1\} &= x_2 
\max\{2, x_2\} &= 2x_3 
&\cdots 
\max\{n, x_n\} &= nx_1.  
\end{align*}
Answer: (x_1, x_2, \, \ldots, \, x_n) = (1, 1, \, \ldots, \, 1)
Problem: Let  ${(a_n)_{n\ge1}} $  be a sequence with   ${a_1 = 1} $   and   ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$  for all  ${n \ge 1}$ , where  ${\lfloor x \rfloor}$  denotes the greatest integer less than or equal to  ${x}$ . Find all  ${n \le 2013}$  such that  ${a_n}$  is a perfect square
Answer: n = 1
Problem: Let  $n$  a positive integer. In a  $2n\times 2n$  board,   $1\times n$  and  $n\times 1$  pieces are arranged without overlap. 
Call an arrangement **maximal** if it is impossible to put a new piece in the board without overlapping the previous ones.
Find the least  $k$  such that there is a **maximal** arrangement that uses  $k$  pieces.
Answer: 2n + 1
Problem: The sequence  $(x_n)_{n\geqslant 0}$  is defined as such:  $x_0=1, x_1=2$  and  $x_{n+1}=4x_n-x_{n-1}$ , for all  $n\geqslant 1$ . Determine all the terms of the sequence which are perfect squares.
Answer:  x_0 = 1
Problem: Let us have  $6050$  points in the plane, no three collinear. Find the maximum number  $k$  of non-overlapping triangles without common vertices in this plane. 
Answer: 2016
Problem: Let  $N$  be the number of ordered triples  $(a,b,c) \in \{1, \ldots, 2016\}^{3}$  such that  $a^{2} + b^{2} + c^{2} \equiv 0 \pmod{2017}$ . What are the last three digits of  $N$ ?
Answer: 000
Problem: Let x; y; z be real numbers, satisfying the relations $x \ge 20$  $y \ge 40$  $z \ge 1675$ x + y + z = 2015
Find the greatest value of the product P =  $xy z$ 
Answer: \frac{721480000}{27}
Problem: There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number  $k$  such that no matter how I select and color  $k$  points, you can always color the remaining  $100-k$  points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect.
Answer: k = 50
Problem: Let  $n$  be a given positive integer. Solve the system 
\[x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,\]
\[x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}\]
in the set of nonnegative real numbers.
Answer:  (1, 1, \, \ldots, 1)
Problem: Find all triples  $ \left(p,x,y\right)$  such that  $ p^x\equal{}y^4\plus{}4$ , where  $ p$  is a prime and  $ x$  and  $ y$  are natural numbers.
Answer: (5, 1, 1)
Problem: Find all  $(m,n) \in \mathbb{Z}^2$  that we can color each unit square of  $m \times n$  with the colors black and white that for each unit square number of unit squares that have the same color with it and have at least one common vertex (including itself) is even.
Answer: (m, n) \in \mathbb{Z}^2 \text{ such that } 2 \mid mn
Problem: Let  $m$  be a positive integer, and let  $a_0, a_1,\ldots,a_m$  be a sequence of reals such that  $a_0=37$ ,  $a_1=72$ ,  $a_m=0$ , and \[a_{k+1}=a_{k-1}-\frac{3}{a_k}\] for  $k=1,2, \dots, m-1$ . Find  $m$ .
Answer: 889
Problem: Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \]
Answer: 330
Problem: A large candle is  $119$  centimeters tall.  It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom.  Specifically, the candle takes  $10$  seconds to burn down the first centimeter from the top,  $20$  seconds to burn down the second centimeter, and  $10k$  seconds to burn down the  $k$ -th centimeter.  Suppose it takes  $T$  seconds for the candle to burn down completely.  Then  $\tfrac{T}{2}$  seconds after it is lit, the candle's height in centimeters will be  $h$ .  Find  $10h$ .
Answer: 350
Problem: In the coordinate plane, find the area of the region bounded by the curve  $ C: y\equal{}\frac{x\plus{}1}{x^2\plus{}1}$  and the line  $ L: y\equal{}1$ .
Answer: 1 - \frac{\pi}{4} - \frac{\ln(2)}{2}
Problem: The numbers 1447, 1005, and 1231 have something in common: each is a four-digit number beginning with 1 that has exactly two identical digits.  How many such numbers are there?
Answer: 432
Problem: Let be given an integer  $n\ge 2$  and a positive real number  $p$ . Find the maximum of
\[\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1},\]
where  $x_i$  are non-negative real numbers with sum  $p$ .
Answer: \frac{p^2}{4}
Problem: Find  the largest constant  $k$  such that the inequality  $$ a^2+b^2+c^2-ab-bc-ca \ge k \left|\frac{a^3-b^3}{a+b}+\frac{b^3-c^3}{b+c}+\frac{c^3-a^3}{c+a}\right| $$  holds for any for non negative real numbers  $a,b,c$  with   $(a+b)(b+c)(c+a)>0$ .
Answer: k = \frac{1}{2}
Problem: Let  $P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$ . What is the largest integer that is a divisor of  $P(n)$  for all positive even integers  $n$ ?
Answer: 15
Problem: Given a set of points in space, a *jump* consists of taking two points,  $P$  and  $Q,$  and replacing  $P$  with the reflection of  $P$  over  $Q$ . Find the smallest number  $n$  such that for any set of  $n$  lattice points in  $10$ -dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
Answer:  1025
Problem: Let  $x$  and  $y$  be real numbers satisfying the equation  $x^2-4x+y^2+3=0$ . If the maximum and minimum values of  $x^2+y^2$  are  $M$  and  $m$  respectively, compute the numerical value of  $M-m$ .
Answer: 8
Problem: Let  $A,B,C,...,Z$  be  $26$  nonzero real numbers. Suppose that  $T=TNYWR$ . Compute the smallest possible value of \[
	  \left\lceil A^2+B^2+\cdots+Z^2\right\rceil .
\] (The notation  $\left\lceil x\right\rceil$  denotes the least integer  $n$  such that  $n\geq x$ .)
Answer: 26
Problem: Find the number of pairs  $(n, q)$ , where  $n$  is a positive integer and  $q$  a non-integer rational number with  $0 < q < 2000$ , that satisfy  $\{q^2\}=\left\{\frac{n!}{2000}\right\}$ 
Answer: 1
Problem: Find the number of ordered pairs of integers  $(x, y)$  such that   $$ \frac{x^2}{y}- \frac{y^2}{x}= 3 \left( 2+ \frac{1}{xy}\right) $$  
Answer: 0
Problem: An isosceles triangle  $ ABC$  with  $ AB \equal{} AC$  is given on the plane. A variable circle  $ (O)$  with center  $ O$  on the line  $ BC$  passes through  $ A$  and does not touch either of the lines  $ AB$  and  $ AC$ . Let  $ M$  and  $ N$  be the second points of intersection of  $ (O)$  with lines  $ AB$  and  $ AC$ , respectively. Find the locus of the orthocenter of triangle  $ AMN$ .
Answer: BC
Problem: Positive real numbers  $a$  and  $b$  verify  $a^5+b^5=a^3+b^3$ . Find the greatest possible value of the expression  $E=a^2-ab+b^2$ .
Answer: 1
Problem:  $\textbf{Problem C.1}$ There are two piles of coins, each containing  $2010$  pieces. Two players  $A$  and  $B$  play a game taking turns ( $A$  plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?
Answer:  B
Problem: A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity  $v_1$ . The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity  $v_2$ . The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time  $v_3$ . The value of  $\frac{v_1}{v_3}+\frac{v_3}{v_1}$  can be expressed as a positive number  $\frac{m\sqrt{r}}{n}$ , where  $m$  and  $n$  are relatively prime, and  $r$  is not divisible by the square of any prime. Find  $m+n+r$ . If the number is rational, let  $r=1$ .
Answer: 8
Problem: What is the smallest positive integer  $n$  such that  $2^n - 1$  is a multiple of  $2015$ ?
Answer: 60
Problem: The natural number  $n>1$  is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers?
Answer: 4
Problem: Find all numbers  $x \in \mathbb Z$  for which the number
\[x^4 + x^3 + x^2 + x + 1\]
is a perfect square.
Answer: -1, 0, 3
Problem: If  $ a\equal{}2b\plus{}c$ ,  $ b\equal{}2c\plus{}d$ ,  $ 2c\equal{}d\plus{}a\minus{}1$ ,  $ d\equal{}a\minus{}c$ , what is  $ b$ ?
Answer: b = \frac{2}{9}
Problem: The pattern in the figure below continues inward infinitely. The base of the biggest triangle is 1. All triangles are equilateral. Find the shaded area.

[asy]
defaultpen(linewidth(0.8));
pen blu = rgb(0,112,191);
real r=sqrt(3);
fill((8,0)--(0,8r)--(-8,0)--cycle, blu);
fill(origin--(4,4r)--(-4,4r)--cycle, white);
fill((2,2r)--(0,4r)--(-2,2r)--cycle, blu);
fill((0,2r)--(1,3r)--(-1,3r)--cycle, white);[/asy]
Answer: \frac{\sqrt{3}}{4}
Problem: Find all real  $x$  in the interval  $[0, 2\pi)$  such that
\[27 \cdot 3^{3 \sin x} = 9^{\cos^2 x}.\]
Answer: x \in \left\{ \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6} \right\}
Problem: Let  $ABCD$  be a tetrahedron and  $O$  its incenter, and let the line  $OD$  be perpendicular to  $AD$ . Find the angle between the planes  $DOB$  and  $DOC.$ 
Answer: 90^
Problem: Yukihira is counting the minimum number of lines  $m$ , that can be drawn on the plane so that they intersect in exactly  $200$  distinct points.What is  $m$ ?
Answer: 21
Problem: Let  $ABCDEF$  be a regular hexagon with side length  $2$ . Calculate the area of  $ABDE$ .
Answer: 4\sqrt{3}
Problem: The number  $201212200619$  has a factor  $m$  such that  $6 \cdot 10^9 <m <6.5 \cdot 10^9$ . Find  $m$ .
Answer: 6490716149
Problem: Find the least positive integer  $N$  which is both a multiple of 19 and whose digits add to 23.
Answer: 779
Problem: Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number  $n$  greater than 10. Then, she multiplies all the digits of  $n$  and obtains an odd number. Find all possible values of the units digit of  $n$ . $\textit{Proposed by Pablo Serrano, Ecuador}$ 
Answer: 5
Problem: A number  $ x$  is uniformly chosen on the interval  $ [0,1]$ , and  $ y$  is uniformly randomly chosen on  $ [\minus{}1,1]$ . Find the probability that  $ x>y$ .
Answer: \frac{3}{4}
Problem: Solve the system  $\begin{cases} x^3 - y^3 = 26 
x^2y - xy^2 = 6  
\end{cases}$ <details><summary>other version</summary>solved below
Solve the system  $\begin{cases} x^3 - y^3 = 2b 
x^2y - xy^2 = b  
\end{cases}$</details>
Answer:  (x, y) = (c, c)
Problem: The graph of the equation  $y = 5x  + 24$  intersects the graph of the equation  $y = x^2$  at two points. The two points are a distance  $\sqrt{N}$  apart. Find  $N$ .
Answer: 3146
Problem: A baker uses  $6\tfrac{2}{3}$  cups of flour when she prepares  $\tfrac{5}{3}$  recipes of rolls. She will use  $9\tfrac{3}{4}$  cups of flour when she prepares  $\tfrac{m}{n}$  recipes of rolls where m and n are relatively prime positive integers. Find  $m + n.$ 
Answer: 55
Problem: Let  $n> 2$  be a positive integer. Given is a horizontal row of  $n$  cells where each cell is painted blue or red. We say that a block is a sequence of consecutive boxes of the same color.  Arepito the crab is initially standing at the leftmost cell. On each turn, he counts the number  $m$  of cells belonging to the largest block containing the square he is on, and does one of the following:

If the square he is on is blue and there are at least  $m$  squares to the right of him, Arepito moves  $m$  squares to the right;

If the square he is in is red and there are at least  $m$  squares to the left of him, Arepito moves  $m$  cells to the left; 

In any other case, he stays on the same square and does not move any further.  

For each  $n$ , determine the smallest integer  $k$  for which there is an initial coloring of the row with  $k$  blue cells, for which Arepito will reach the rightmost cell. 
Answer: \left\lceil \frac{n+1}{2} \right\rceil
Problem: Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square.
Answer: \frac{\sqrt{5}-1}{4}
Problem: On each OMA lottery ticket there is a  $9$ -digit number that only uses the digits  $1, 2$  and  $3$  (not necessarily all three). Each ticket has one of the three colors red, blue or green. It is known that if two banknotes do not match in any of the  $9$  figures, then they are of different colors. Bill  $122222222$  is red,  $222222222$  is green, what color is bill  $123123123$ ?
Answer: red
Problem: Three congruent circles of radius  $2$  are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let  $K$  be the area of the triangle whose vertices are the midpoints of those arcs. If  $K = \sqrt{a} - b$  for positive integers  $a, b$ , find  $100a+b$ .
Answer: 300
Problem: $99$  identical balls lie on a table.  $50$  balls are made of copper, and  $49$  balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to  $2$  balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today?
Answer: 98
Problem: Consider a regular polygon with  $2^n$  sides, for  $n \ge 2$ , inscribed in a circle of radius  $1$ . Denote the area of this polygon by  $A_n$ . Compute  $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$ 
Answer: \frac{2}{\pi}
Problem: Let  $a,b,c,d,e,f$  be real numbers such that the polynomial
\[ p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f \]
factorises into eight linear factors  $x-x_i$ , with  $x_i>0$  for  $i=1,2,\ldots,8$ . Determine all possible values of  $f$ .
Answer: \frac{1}{256}
Problem: Find  $$  \inf_{\substack{ n\ge 1  a_1,\ldots ,a_n >0  a_1+\cdots +a_n <\pi }} \left( \sum_{j=1}^n a_j\cos \left( a_1+a_2+\cdots +a_j \right)\right) .  $$  
Answer: -\pi
Problem: There are  $15$  people, including Petruk, Gareng, and Bagong, which will be partitioned into  $6$  groups, randomly, that consists of  $3, 3, 3, 2, 2$ , and  $2$  people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.
Answer: \frac{3}{455}
Problem: Let  $ABC$  be a triangle inscribed in the parabola  $y=x^2$  such that the line  $AB \parallel$  the axis  $Ox$ . Also point  $C$  is closer to the axis  $Ox$  than the line  $AB$ . Given that the length of the segment  $AB$  is 1 shorter than the length of the altitude  $CH$  (of the triangle  $ABC$ ). Determine the angle  $\angle{ACB}$  .
Answer: \frac{\pi}{4}
Problem: A  $23 \times 23$  square is tiled with  $1 \times 1, 2 \times 2$  and  $3 \times 3$  squares. What is the smallest possible number of  $1 \times 1$  squares?
Answer: 1
Problem: Determine the largest possible value of the expression  $ab+bc+ 2ac$  for non-negative real numbers  $a, b, c$  whose sum is  $1$ .
Answer: \frac{1}{2}
Problem: Deﬁne the determinant  $D_1$  =  $|1|$ , the determinant  $D_2$  =      $|1 1|$   $|1 3|$       
, and the determinant  $D_3=$         
|1 1 1|
|1 3 3|
|1 3 5|         
.
In general, for positive integer n, let the determinant  $D_n$  have 1s in every position of its ﬁrst row and ﬁrst
column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that  $D_n$   $\geq$  2015. 
Answer: 12
Problem: Let  $n$  be a positive integer. Each number  $1, 2, ..., 1000$  has been colored with one of  $n$  colours. Each two numbers , such that one is a divisor of second of them, are colored with different colours. Determine minimal number  $n$  for which it is possible.
Answer:  n = 10
Problem: We form a decimal code of  $21$  digits. the code may start with  $0$ . Determine the probability that the fragment  $0123456789$  appears in the code.
Answer: \frac{12 \cdot 10^{11} - 30}{10^{21}}
Problem: Given are the lines  $l_1,l_2,\ldots ,l_k$  in the plane, no two of which are parallel and no three of which are concurrent. For which  $k$  can one label the intersection points of these lines by  $1, 2,\ldots , k-1$  so that in each of the given lines all the labels appear exactly once?
Answer:  k such that 2 \mid k
Problem: Compute the positive difference between the two real solutions to the equation   $$ (x-1)(x-4)(x-2)(x-8)(x-5)(x-7)+48\sqrt 3 = 0. $$  
Answer: \sqrt{25 + 8\sqrt{3}}
Problem: How many alphabetic sequences (that is, sequences containing only letters from  $a\cdots z$ ) of length  $2013$  have letters in alphabetic order?
Answer: \binom{2038}{25}
Problem: (My problem. :D)
Call the number of times that the digits of a number change from increasing to decreasing, or vice versa, from the left to right while ignoring consecutive digits that are equal the *flux* of the number. For example, the flux of 123 is 0 (since the digits are always increasing from left to right) and the flux of 12333332 is 1, while the flux of 9182736450 is 8. What is the average value of the flux of the positive integers from 1 to 999, inclusive?
Answer: \frac{175}{333}
Problem: Find the value of   $$ \sum_{1\le a<b<c} \frac{1}{2^a3^b5^c} $$   (i.e. the sum of  $\frac{1}{2^a3^b5^c}$  over all triples of positive integers  $(a, b, c)$  satisfying  $a<b<c$ )
Answer: \frac{1}{406}
Problem: Find in explicit form all ordered pairs of positive integers  $(m, n)$  such that  $mn-1$  divides  $m^2 + n^2$ .
Answer: (a_1, b_1) = (1, 2), (1, 3), (2, 1), (3, 1) \text{ and the explicit formula is } 5a_n - b_n = a_{n+1}, b_n = a_{n-1}
Problem: What is the maximum possible value for the sum of the squares of the roots of  $x^4+ax^3+bx^2+cx+d$  where  $a$ ,  $b$ ,  $c$ , and  $d$  are  $2$ ,  $0$ ,  $1$ , and  $7$  in some order?
Answer: 49
Problem: In triangle  $ABC$ , points  $M$  and  $N$  are on segments  $AB$  and  $AC$  respectively such that  $AM = MC$  and  $AN = NB$ . Let  $P$  be the point such that  $PB$  and  $PC$  are tangent to the circumcircle of  $ABC$ . Given that the perimeters of  $PMN$  and  $BCNM$  are  $21$  and  $29$  respectively, and that  $PB = 5$ , compute the length of  $BC$ .
Answer: \frac{200}{21}
Problem: Suppose  $ A_1,\dots, A_6$  are six sets each with four elements and  $ B_1,\dots,B_n$  are  $ n$  sets each with two elements, Let  $ S \equal{} A_1 \cup A_2 \cup \cdots \cup A_6 \equal{} B_1 \cup \cdots \cup B_n$ . Given that each elements of  $ S$  belogs to exactly four of the  $ A$ 's and to exactly three of the  $ B$ 's, find  $ n$ .
Answer: 9
Problem: Find all prime numbers  $p$  such that there exists a unique  $a \in \mathbb{Z}_p$  for which  $a^3 - 3a + 1 = 0.$ 
Answer: p = 3
Problem: Students are in classroom with  $n$  rows. In each row there are  $m$  tables. It's given that  $m,n \geq 3$ . At each table there is exactly one student. We call neighbours of the student students sitting one place right, left to him, in front of him and behind him. Each student shook hands with his neighbours. In the end there were  $252$  handshakes. How many students were in the classroom?
Answer: 72
Problem:  $O$  and  $I$  are the circumcentre and incentre of   $\vartriangle ABC$  respectively. Suppose  $O$  lies in the interior of   $\vartriangle ABC$  and  $I$  lies on the circle passing through  $B, O$ , and  $C$ . What is the magnitude of  $\angle B AC$  in degrees?
Answer: 60^	ext{\circ}
Problem: Find the least number  $ c$  satisfyng the condition  $\sum_{i=1}^n {x_i}^2\leq cn$   
and all real numbers  $x_1,x_2,...,x_n$  are greater than or equal to  $-1$  such that  $\sum_{i=1}^n {x_i}^3=0$ 
Answer: 1