rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Find all functions  $f: \mathbb{R} \rightarrow \mathbb{R}$  such that  $$ f(x^2 + y) \ge (\frac{1}{x} + 1)f(y) $$  holds for all  $x \in \mathbb{R} \setminus \{0\}$  and all  $y \in \mathbb{R}$ .

Answer:  f \equiv 0
Problem: For the all  $(m,n,k)$  positive integer triples such that  $|m^k-n!| \le n$  find the maximum value of  $\frac{n}{m}$
Answer: 2
Problem: Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is  $18$ . How many passcodes satisfy these conditions?
Answer: 36
Problem: Calvin and Hobbes play a game. First, Hobbes picks a family  $F$  of subsets of  $\{1, 2, . . . , 2020\}$ , known to both players. Then, Calvin and Hobbes take turns choosing a number from  $\{1, 2, . . . , 2020\}$  which is not already chosen, with Calvin going first, until all numbers are taken (i.e., each player has  $1010$  numbers). Calvin wins if he has chosen all the elements of some member of  $F$ , otherwise Hobbes wins. What is the largest possible size of a family  $F$  that Hobbes could pick while still having a winning strategy?

Answer: 4^{1010} - 3^{1010}
Problem: Let  $ F_n$  be a set of all possible connected figures, that consist of  $ n$  unit cells. For each element  $ f_n$  of this set, let  $ S(f_n)$  be the area of that minimal rectangle that covers  $ f_n$  and each side of the rectangle is parallel to the corresponding side of the cell. Find  $ max(S(f_n))$ ,where  $ f_n\in F_n$ ?

Remark: Two cells are called connected if they have a common edge.
Answer: n
Problem: Let  $a, b, c$  be the roots of the equation  $x^3-9x^2+11x-1 = 0$ , and define  $s =\sqrt{a}+\sqrt{b}+\sqrt{c}$ . 
Compute  $s^4 -18s^2 - 8s$  .
Answer: -37
Problem: Determine all ordered pairs of real numbers  $(a,b)$  such that the line  $y=ax+b$  intersects the curve  $y=\ln(1+x^2)$  in exactly one point.
Answer:  (0, 0)
Problem: Let  $\alpha$  be a root of  $x^6-x-1$ , and call two polynomials  $p$  and  $q$  with integer coefficients  $\textit{equivalent}$  if  $p(\alpha)\equiv q(\alpha)\pmod3$ .  It is known that every such polynomial is equivalent to exactly one of  $0,1,x,x^2,\ldots,x^{727}$ .  Find the largest integer  $n<728$  for which there exists a polynomial  $p$  such that  $p^3-p-x^n$  is equivalent to  $0$ .
Answer: 727
Problem: Represent the number  $ 2008$  as a sum of natural number such that the addition of the reciprocals of the summands yield 1.
Answer: 2008 = 8 + 10 \times 40 + 30 \times 40
Problem: Two types of pieces, bishops and rooks, are to be placed on a  $10\times 10$  chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop  $B$  is said to *attack* a piece  $P$  if  $B$  and  $P$  are on the same diagonal and there are no pieces between  $B$  and  $P$  on that diagonal; a rook  $R$  is said to attack a piece  $P$  if  $R$  and  $P$  are on the same row or column and there are no pieces between  $R$  and  $P$  on that row or column.
A piece  $P$  is *chocolate* if no other piece  $Q$  attacks  $P$ .
What is the maximum number of chocolate pieces there may be, after placing some pieces on the chessboard?
Answer: 50
Problem: Find the area of the region in the coordinate plane satisfying the three conditions $\star$    $x \le 2y$  $\star$   $y \le 2x$  $\star$   $x + y \le 60.$ 
Answer: 600
Problem: If  $p$  is a prime number, what is the product of elements like  $g$  such that  $1\le g\le p^2$  and  $g$  is a primitive root modulo  $p$  but it's not a primitive root modulo  $p^2$ , modulo  $p^2$ ?( $\frac{100}{6}$  points)
Answer: 1
Problem: Brice is eating bowls of rice. He takes a random amount of time  $t_1 \in (0,1)$  minutes to consume his first bowl, and every bowl thereafter takes  $t_n = t_{n-1} + r_n$  minutes, where  $t_{n-1}$  is the time it took him to eat his previous bowl and  $r_n \in (0,1)$  is chosen uniformly and randomly. The probability that it takes Brice at least 12 minutes to eat 5 bowls of rice can be expressed as simplified fraction  $\tfrac{m}{n}$ . Compute  $m+n$ .
Answer: 0 + 1 = 1
Problem: A deck of  $8056$  cards has  $2014$  ranks numbered  $1$ – $2014$ . Each rank has four suits - hearts, diamonds, clubs, and spades. Each card has a rank and a suit, and no two cards have the same rank and the same suit. How many subsets of the set of cards in this deck have cards from an odd number of distinct ranks?
Answer: \frac{16^{2014} - 14^{2014}}{2}
Problem: Let  $\mathbb X$  be the set of all bijective functions from the set  $S=\{1,2,\cdots, n\}$  to itself. For each  $f\in \mathbb X,$  define
\[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \  f^{(12)}(j)=j, 0, \ \ \ & \text{otherwise}\end{aligned}\right.\]
Determine  $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$ (Here  $f^{(k)}(x)=f(f^{(k-1)}(x))$  for all  $k\geq 2.$ )
Answer: n!
Problem: Each row of a  $24 \times 8$  table contains some permutation of the numbers  $1, 2, \cdots , 8.$  In each column the numbers are multiplied. What is the minimum possible sum of all the products?
Answer: 8 \cdot (8!)^3
Problem: Let  $S=\{1, 2, 3, \dots 2021\}$  and  $f:S \to S$  be a function such that  $f^{(n)}(n)=n$  for each  $n \in S$ .
Find all possible values for  $f(2021)$ .
(Here,  $f^{(n)}(n) = \underbrace{f(f(f\dots f(}_{n \text{ times} }n)))\dots))$ .)
Answer:  \{43, 86, 129, 172, 215, 258, 301, 344, 387, 430, 473, 516, 559, 602, 645, 688, 731, 774, 817, 860, 903, 946, 989, 1032, 1075, 1118, 1161, 1204, 1247, 1290, 1333, 1376, 1419, 1462, 1505, 1548, 1591, 1634, 1677, 1720, 1763, 1806, 1849, 1892, 1935, 1978, 2021\}
Problem: Define the sequence  $a_1,a_2,a_3,\ldots$  by  $a_n=\sum_{k=1}^n\sin(k)$ , where  $k$  represents radian measure. Find the index of the  $100$ th term for which  $a_n<0$ .
Answer: 628
Problem: If  $ c$  is a positive integer and  $ p$  is an odd prime, what is the smallest residue (in absolute value) of \[  \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\] 

J. Suranyi
Answer: 0
Problem: Let  $x\geq 5, y\geq 6, z\geq 7$  such that  $x^2+y^2+z^2\geq 125$ . Find the minimum value of  $x+y+z$ .
Answer: 19
Problem:  A pair of positive integers  $(m,n)$  is called ***'steakmaker'*** if they maintain the equation 1 + 2 $^m$  = n $^2$ . For which values of m and n, the pair  $(m,n)$  are steakmaker, find the sum of  $mn$ 
Answer: 9
Problem: Let  $a$  and  $b$  be relatively prime positive integers such that  $a/b$  is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for  $1\leq i\leq 2007$ ,  $x_i$  is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of  $a+b$ .
Answer: 2008
Problem: A massless string is wrapped around a frictionless pulley of mass  $M$ . The string is pulled down with a force of 50 N, so that the pulley rotates due to the pull. Consider a point  $P$  on the rim of the pulley, which is a solid cylinder. The point has a constant linear (tangential) acceleration component equal to the acceleration of gravity on Earth, which is where this experiment is being held. What is the weight of the cylindrical pulley, in Newtons? 

*(Proposed by Ahaan Rungta)*

<details><summary>Note</summary>This problem was not fully correct. Within friction, the pulley cannot rotate. So we responded: 
<blockquote>Excellent observation! This is very true. To submit, I'd say just submit as if it were rotating and ignore friction. In some effects such as these, I'm pretty sure it turns out that friction doesn't change the answer much anyway, but, yes, just submit as if it were rotating and you are just ignoring friction. </blockquote>So do this problem imagining that the pulley does rotate somehow.</details>
Answer: 100 \, \text{N}
Problem: Upon cutting a certain rectangle in half, you obtain two rectangles that are scaled down versions of the original. What is the ratio of the longer side length to the shorter side length?
Answer: \sqrt{2}
Problem: Alex the Kat has written  $61$  problems for a math contest, and there are a total of  $187$  problems submitted. How many more problems does he need to write (and submit) before he has written half of the total problems?
Answer: 65
Problem: Let  $ T$  be the set of all positive integer divisors of  $ 2004^{100}$ . What is the largest possible number of elements of a subset  $ S$  of  $ T$  such that no element in  $ S$  divides any other element in  $ S$ ?
Answer: 101^2
Problem: Evaluate the product  $$ \prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}. $$
Answer: 1
Problem: An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network *feasible* if it satisfies the following conditions:



- All connections operate in both directions
- If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.


Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there?
Answer: 512
Problem: In how many ways can six marbles be placed in the squares of a  $6$ -by- $6$  grid such that no two marbles lie in the same row or column?
Answer: 720
Problem:  $p$  is a prime number such that its remainder divided by 8 is 3. Find all pairs of rational numbers  $(x,y)$  that satisfy the following equation.  $$ p^2 x^4-6px^2+1=y^2 $$  
Answer: (0, \pm 1)
Problem: Find all functions  $f:\mathbb{R}\to\mathbb{R}$  for which 
\[ x(f(x+1)-f(x)) = f(x), \] 
for all  $x\in\mathbb{R}$  and 
\[ | f(x) - f(y) | \leq |x-y| , \] 
for all  $x,y\in\mathbb{R}$ .
Answer: f(x) = cx where |c| \leq 1
Problem: Find the greatest possible distance between any two points inside or along the perimeter of an equilateral triangle with side length  $2$ .
Answer: 2
Problem: Find all integers  $n \leq 3$  such that there is a set  $S_n$  formed by  $n$  points of the plane that satisfy the following two conditions:
    Any three points are not collinear.
    No point is found inside the circle whose diameter has ends at any two points of  $S_n$ .**NOTE:** The points on the circumference are not considered to be inside the circle.

Answer: n = 1, 2, 3
Problem: Find all positive integers  $n\geqslant 2$  such that there exists a permutation  $a_1$ ,  $a_2$ ,  $a_3$ , \ldots,  $a_{2n}$  of the numbers   $1, 2, 3, \ldots, 2n$  satisfying   $$ a_1\cdot a_2 + a_3\cdot a_4 + \ldots + a_{2n-3} \cdot  a_{2n-2} = a_{2n-1} \cdot a_{2n}. $$  
Answer: n = 2
Problem: Determine the smallest  $n$  such that  $n \equiv (a - 1)$  mod  $a$  for all  $a \in \{2,3,..., 10\}$ .
Answer: 2519
Problem: Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of  $\$  1, \ $ 2,\$  5 $, and $ \ $ 10$  bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly  $\$ 1{,}000{,}000 $, and whoever makes the last move wins the $ \ $1{,}000{,}000$  in the pile (if the pile is worth more than  $\$ 1{,}000{,}000$ after a move, then the person who made the last move loses instead, and the other person wins the amount of cash in the pile). Assuming optimal play, how many dollars will the winning player gain?
Answer: 1,000,000
Problem: Tim is participating in the following three math contests.  On each contest his score is the number of correct answers.
     $\bullet$  The Local Area Inspirational Math Exam consists of 15 problems.
     $\bullet$  The Further Away Regional Math League has 10 problems.
     $\bullet$  The Distance-Optimized Math Open has 50 problems.
For every positive integer  $n$ , Tim knows the answer to the  $n$ th problems on each contest (which are pairwise distinct), if they exist; however, these answers have been randomly permuted so that he does not know which answer corresponds to which contest.  Unaware of the shuffling, he competes with his modified answers.  Compute the expected value of the sum of his scores on all three contests.
Answer: 25
Problem: We only know that the password of a safe consists of  $7$  different digits. The safe will open if we enter  $7$  different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than  $7$  attempts?

*(5 points for Juniors and 4 points for Seniors)*
Answer: 6
Problem: Regular octagon  $A_1A_2A_3A_4A_5A_6A_7A_8$  is inscribed in a circle of area  $1$ .  Point  $P$  lies inside the circle so that the region bounded by  $\overline{PA_1}$ ,  $\overline{PA_2}$ , and the minor arc  $\widehat{A_1A_2}$  of the circle has area  $\tfrac17$ , while the region bounded by  $\overline{PA_3}$ ,  $\overline{PA_4}$ , and the minor arc  $\widehat{A_3A_4}$  of the circle has area  $\tfrac 19$ .  There is a positive integer  $n$  such that the area of the region bounded by  $\overline{PA_6}$ ,  $\overline{PA_7}$ , and the minor arc  $\widehat{A_6A_7}$  is equal to  $\tfrac18 - \tfrac{\sqrt 2}n$ .  Find  $n$ .
Answer: 504
Problem: Find all functions  $f:[0,1] \to \mathbb{R}$  such that the inequality \[(x-y)^2\leq|f(x) -f(y)|\leq|x-y|\] is satisfied for all  $x,y\in [0,1]$ 
Answer: f(x) = \pm x + C
Problem: Find all real numbers a such that all solutions to the quadratic equation  $ x^2 \minus{} ax \plus{} a \equal{} 0$  are integers.
Answer:  a = 0, 4
Problem: Let  $a$  be a positive real number,  $n$  a positive integer, and define the *power tower*  $a\uparrow n$  recursively with  $a\uparrow 1=a$ , and  $a\uparrow(i+1)=a^{a\uparrow i}$  for  $i=1,2,3,\ldots$ .  For example, we have  $4\uparrow 3=4^{(4^4)}=4^{256}$ , a number which has  $155$  digits.  For each positive integer  $k$ , let  $x_k$  denote the unique positive real number solution of the equation  $x\uparrow k=10\uparrow (k+1)$ .  Which is larger:  $x_{42}$  or  $x_{43}$ ?
Answer:  x_{42}
Problem: Thirty-four countries participated in a jury session of the IMO, each represented by the leader and the deputy leader of the team. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with his deputy. After the meeting, the leader of the Illyrian team asked every other participant the number of people they had shaken hands with, and all the answers she got were different. How many people did the deputy leader of the Illyrian team greet ?
Answer: 33
Problem: 1. Let  $f(n) = n^2+6n+11$  be a function defined on positive integers. Find the sum of the first three prime values  $f(n)$  takes on.
Answer: 753
Problem: Compute the smallest positive integer that is  $3$  more than a multiple of  $5$ , and twice a multiple of  $6$ .
Answer: 48
Problem: Let  $[n] = \{1, 2, 3, ... ,n\}$  and for any set  $S$ , let $ P(S)$  be the set of non-empty subsets of  $S$ . What is the last digit of  $|P(P([2013]))|$ ?
Answer: 7
Problem: For any positive integer  $n$ , let  $a_n=\sum_{k=1}^{\infty}[\frac{n+2^{k-1}}{2^k}]$ , where  $[x]$  is the largest integer that is equal or less than  $x$ . Determine the value of  $a_{2015}$ .
Answer: 2015
Problem: Solve in integers the system of equations:    $$ x^2-y^2=z $$     $$ 3xy+(x-y)z=z^2 $$  
Answer:  (2, 1, 3), (1, 2, -3), (1, 0, 1), (0, 1, -1), (0, 0, 0)
Problem: How many positive integers less than  $2019$  are divisible by either  $18$  or  $21$ , but not both?
Answer: 176
Problem: Let the sequence  $\{a_i\}^\infty_{i=0}$  be defined by  $a_0 =\frac12$  and  $a_n = 1 + (a_{n-1} - 1)^2$ . Find the product   $$ \prod_{i=0}^\infty a_i=a_0a_1a_2\ldots $$  
Answer: \frac{2}{3}
Problem: The sum of the areas of all triangles whose vertices are also vertices of a  $1\times 1 \times 1$  cube is  $m+\sqrt{n}+\sqrt{p}$ , where  $m$ ,  $n$ , and  $p$  are integers.  Find  $m+n+p$ .
Answer: 348
Problem: Determine the positive integers expressible in the form  $\frac{x^2+y}{xy+1}$ , for at least  $2$  pairs  $(x,y)$  of positive integers
Answer: 1
Problem: Find all functions  $f: \mathbb{R}^{+} \rightarrow \mathbb{R}$  and  $g: \mathbb{R}^{+} \rightarrow \mathbb{R}$  such that  $$ f(x^2 + y^2) = g(xy) $$  holds for all  $x, y \in \mathbb{R}^{+}$ .

Answer:  f(x) = c \quad \text{and} \quad g(x) = c \text{ for some constant } c \in \mathbb{R}
Problem: A paper equilateral triangle of side length  $2$  on a table has vertices labeled  $A,B,C.$  Let  $M$  be the point on the sheet of paper halfway between  $A$  and  $C.$  Over time, point  $M$  is lifted upwards, folding the triangle along segment  $BM,$  while  $A,B,$  and  $C$  on the table. This continues until  $A$  and  $C$  touch. Find the maximum volume of tetrahedron  $ABCM$  at any time during this process.
Answer: 1
Problem: A river is bounded by the lines  $x=0$  and  $x=25$ , with a current of 2 units/s in the positive y-direction. At  $t=0$ , a mallard is at  $(0, 0)$ , and a wigeon is at  $(25, 0)$ . They start swimming with a constant speed such that they meet at  $(x,22)$ . The mallard has a speed of 4 units/s relative to the water, and the wigeon has a speed of 3 units/s relative to the water. Find the value of  $x$ .
Answer: \frac{100}{7}
Problem: Sally randomly chooses three different numbers from the set  $\left\{1,2,\ldots,14\right\}$ . What is the probability that the sum of her smallest number and her biggest number is at least  $15$ ?
Answer: \frac{19}{91}
Problem: A regular  $2018$ -gon is inscribed in a circle. The numbers  $1, 2, ..., 2018$  are arranged on the vertices of the  $2018$ -gon, with each vertex having one number on it, such that the sum of any  $2$  neighboring numbers ( $2$  numbers are neighboring if the vertices they are on lie on a side of the polygon) equals the sum of the  $2$  numbers that are on the antipodes of those  $2$  vertices (with respect to the given circle). Determine the number of different arrangements of the numbers. 
(Two arrangements are identical if you can get from one of them to the other by rotating around the center of the circle).
Answer: 2 \times 1008!
Problem: Twelve students participated in a theater festival consisting of  $n$  different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of  $n$ .
Answer:  n = 4
Problem: Let  $ABCDEF$  be a regular hexagon with side length  $2$ . A circle with radius  $3$  and center at  $A$  is drawn. Find the area inside quadrilateral  $BCDE$  but outside the circle.
Answer: 4\sqrt{3} - 9\pi
Problem: An integer   $n>1$  is given .  Find the smallest positive number  $m$  satisfying the following conditions： for any set  $\{a,b\}$   $\subset \{1,2,\cdots,2n-1\}$  ,there are non-negative integers    $ x, y$   ( not all zero)  such  that    $2n|ax+by$  and  $x+y\leq m.$ 
Answer: n
Problem: Determine all triples  $(x, y, z)$  of positive real numbers which satisfies the following system of equations
\[2x^3=2y(x^2+1)-(z^2+1), \] \[ 2y^4=3z(y^2+1)-2(x^2+1), \] \[ 2z^5=4x(z^2+1)-3(y^2+1).\]
Answer: (1, 1, 1)
Problem: Find  $A^2$ , where  $A$  is the sum of the absolute values of all roots of the following equation: \begin{eqnarray*}x &=& \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}}\end{eqnarray*}
Answer: 383
Problem: Let  $b$  be a positive integer. Find all  $2002$ −tuples  $(a_1, a_2,\ldots , a_{2002})$ , of natural numbers such that
\[\sum_{j=1}^{2002} a_j^{a_j}=2002b^b.\]
Answer:  (a_1, a_2, \ldots, a_{2002}) = (b, b, \ldots, b)
Problem: In a room there are  $144$  people. They are joined by  $n$  other people who are each carrying  $k$  coins. When these coins are shared among all  $n + 144$  people, each person has  $2$  of these coins. Find the minimum possible value of  $2n + k$ .
Answer: 50
Problem: Let  $\mathbb{N} = \{1, 2, 3, \ldots\}$  be the set of positive integers. Find all functions  $f$ , defined on  $\mathbb{N}$  and taking values in  $\mathbb{N}$ , such that  $(n-1)^2< f(n)f(f(n)) < n^2+n$  for every positive integer  $n$ .
Answer:  f(n) = n
Problem: Let  $ABC$  be an acute triangle. Let $H$  and  $D$  be points on  $[AC]$  and  $[BC]$ , respectively, such that  $BH \perp AC$  and  $HD \perp BC$ . Let  $O_1$  be the circumcenter of  $\triangle ABH$ , and  $O_2$  be the circumcenter of  $\triangle BHD$ , and  $O_3$  be the circumcenter of  $\triangle HDC$ . Find the ratio of area of  $\triangle O_1O_2O_3$  and  $\triangle ABH$ .
Answer: \frac{1}{4}
Problem: For  $x\geq 0,$  find the minimum value of  $x$  for which  $\int_0^x 2^t(2^t-3)(x-t)\ dt$  is minimized.
Answer: 0
Problem: Let  $p$  be an odd prime. Find all positive integers  $n$  for which  $\sqrt{n^2-np}$  is a positive integer.
Answer:  n = \left( \frac{(p+1)^2}{4} \right)
Problem: Let  $A$  be a  $2n\times 2n$  matrix, with entries chosen independently at random. Every entry is chosen to be  $0$  or  $1,$  each with probability  $1/2.$  Find the expected value of  $\det(A-A^t)$  (as a function of  $n$ ), where  $A^t$  is the transpose of  $A.$ 
Answer: \frac{(2n-1)!!}{2^n}
Problem: We inscribe a circle  $\omega$  in equilateral triangle  $ABC$  with radius  $1$ . What is the area of the region inside the triangle but outside the circle?
Answer: 3\sqrt{3} - \pi
Problem: Find the real number  $a$  such that the integral   $$ \int_a^{a+8}e^{-x}e^{-x^2}dx $$   attain its maximum.
Answer: a = -4.5
Problem: Let  $\mathbb{N}$  denote the set of positive integers, and let  $S$  be a set. There exists a function  $f :\mathbb{N} \rightarrow S$  such that if  $x$  and  $y$  are a pair of positive integers with their difference being a prime number, then  $f(x) \neq f(y)$ . Determine the minimum number of elements in  $S$ .
Answer: 4
Problem: Let  $O$  and  $I$  be the circumcenter and incenter of triangle  $ABC$ . The perpendicular from  $I$  to  $OI$  meets  $AB$  and the external bisector of angle  $C$  at points  $X$  and  $Y$  respectively. In what ratio does  $I$  divide the segment  $XY$ ?
Answer: 1:2
Problem: A polynomial  $P(x)$  is a \emph{base- $n$  polynomial} if it is of the form  $a_dx^d+a_{d-1}x^{d-1}+\cdots + a_1x+a_0$ , where each  $a_i$  is an integer between  $0$  and  $n-1$  inclusive and  $a_d>0$ . Find the largest positive integer  $n$  such that for any real number  $c$ , there exists at most one base- $n$  polynomial  $P(x)$  for which  $P(\sqrt 2+\sqrt 3)=c$ .
Answer: 9
Problem: This year February  $29$  fell on a Sunday. In what year will February  $29$  next fall on a Sunday?
Answer: 2032
Problem: Find all polynomials  $P(X)$  of fourth degree with real coefficients that verify the properties:**a)**  $P(-x)=P(x), \forall x\in\mathbb{R};$ **b)**  $P(x)\geq0, \forall x\in\mathbb{R};$ **c)**  $P(0)=1;$ **d)**  $P(X)$  has exactly two local minimums  $x_1$  and  $x_2$  such that  $|x_1-x_2|=2.$ 
Answer:  P(x) = a_4 x^4 - 2a_4 x^2 + 1 where 0 < a_4 \leq 1
Problem:  $N$  is a  $50$ -digit number (in decimal representation). All digits except the  $26$ th digit (from the left) are  $1$ . If  $N$  is divisible by  $13$ , find its  $26$ -th digit.
Answer: 9
Problem: Determine all functions  $ f: \mathbb{R} \mapsto \mathbb{R}$  with  $ x,y \in \mathbb{R}$  such that

\[ f(x \minus{} f(y)) \equal{} f(x\plus{}y) \plus{} f(y)\]
Answer:  f(x) = 0 \text{ or } f(x) = -2x
Problem: Determine all prime numbers of the form  $1 + 2^p + 3^p +...+ p^p$  where  $p$  is a prime number.
Answer:  p = 2
Problem: Congruent circles  $\Gamma_1$  and  $\Gamma_2$  have radius  $2012,$  and the center of  $\Gamma_1$  lies on  $\Gamma_2.$  Suppose that  $\Gamma_1$  and  $\Gamma_2$   intersect at  $A$  and  $B$ . The line through  $A$  perpendicular to  $AB$  meets  $\Gamma_1$  and  $\Gamma_2$  again at  $C$  and  $D$ , respectively. Find the length of  $CD$ .
Answer: 4024
Problem: Let  $ n$  and  $ k$  be positive integers. Please, find an explicit formula for
\[ \sum y_1y_2 \dots y_k,\]
where the summation runs through all  $ k\minus{}$ tuples positive integers  $ (y_1,y_2,\dots,y_k)$  satisfying  $ y_1\plus{}y_2\plus{}\dots\plus{}y_k\equal{}n$ .
Answer: \binom{n-1}{k-1}