rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: A rhombus  $\mathcal R$  has short diagonal of length  $1$  and long diagonal of length  $2023$ . Let  $\mathcal R'$  be the rotation of  $\mathcal R$  by  $90^\circ$  about its center. If  $\mathcal U$  is the set of all points contained in either  $\mathcal R$  or  $\mathcal R'$  (or both; this is known as the *union* of  $\mathcal R$  and  $\mathcal R'$ ) and  $\mathcal I$  is the set of all points contained in both  $\mathcal R$  and  $\mathcal R'$  (this is known as the *intersection* of  $\mathcal R$  and  $\mathcal R'$ , compute the ratio of the area of  $\mathcal I$  to the area of  $\mathcal U$ .
Answer: \frac{1}{2023}
Problem: Determine the integers  $x$  such that  $2^x + x^2 + 25$  is the cube of a prime number
Answer:  x = 6
Problem: Let  $T = \{9^k : k \ \text{is an integer}, 0 \le k \le 4000\}$ .  Given that  $9^{4000}$  has 3817 digits and that its first (leftmost) digit is 9, how many elements of  $T$  have 9 as their leftmost digit?
Answer: 184
Problem: Each rational number is painted either white or red. Call such a coloring of the rationals *sanferminera* if for any distinct rationals numbers  $x$  and  $y$  satisfying one of the following three conditions: [list=1][*] $xy=1$ ,
[*] $x+y=0$ ,
[*] $x+y=1$ ,[/list]we have  $x$  and  $y$  painted different colors. How many sanferminera colorings are there?
Answer: 2
Problem: If  $m, n$ , and  $p$  are three different natural numbers, each between  $2$  and  $9$ , what then are all the possible integer value(s) of the expression  $\frac{m+n+p}{m+n}$ ?
Answer: 1
Problem: Consider the number obtained by writing the numbers  $1,2,\ldots,1990$  one after another. In this number every digit on an even position is omitted; in the so obtained number, every digit on an odd position is omitted; then in the new number every digit on an even position is omitted, and so on. What will be the last remaining digit?
Answer: 9
Problem: Five bricklayers working together finish a job in  $3$  hours. Working alone, each bricklayer takes at most  $36$  hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone?
Answer: 270
Problem: Suppose that  $n\ge3$  is a natural number. Find the maximum value  $k$  such that there are real numbers  $a_1,a_2,...,a_n \in [0,1)$  (not necessarily distinct) that for every natural number like  $j \le k$  , sum of some  $a_i$ -s is  $j$ .
Answer:  k = n-2
Problem: Let  $(x_n)$  be a sequence of positive integers defined as follows:  $x_1$  is a fixed six-digit number and for any  $n \geq 1$ ,  $x_{n+1}$  is a prime divisor of  $x_n + 1$ . Find  $x_{19} + x_{20}$ .
Answer: 5
Problem: Find all functions  $f:\mathbb{Z}^{+}\to\mathbb{Z}^{+}$  such that   $$ m^{2}+f(n)^{2}+(m-f(n))^{2}\geq f(m)^{2}+n^{2} $$   for all pairs of positive integers  $(m,n)$ .
Answer: f(n) = n
Problem: A line through the origin passes through the curve whose equation is  $5y=2x^2-9x+10$  at two points whose  $x-$ coordinates add up to  $77.$   Find the slope of the line.
Answer: 29
Problem: Let  $ k$  be a positive integer. Suppose that the integers  $ 1,2,3,\dots,3k \plus{} 1$  are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by  $ 3$  ? Your answer should be in closed form, but may include factorials.
Answer: \frac{(k+1)! k!}{(2k)! (3k + 1)}
Problem: Suppose we list the decimal representations of the positive even numbers from left to right. Determine the  $2015^{th}$  digit in the list.
Answer: 8
Problem: If  $a$  and  $b$  are positive integers such that  $a^2-b^4= 2009$ , find  $a+b$ .
Answer: 47
Problem: Let  $a$  be real numbers. Given a matrix  $A=\left(\begin{array}{cc} a & 1-a  1-a & a \end{array}\right).$  Let  $\left(\begin{array}{c} x_0  y_0 \end{array}\right)=\left(\begin{array}{c} 1  0 \end{array}\right),$  and the sequences  $\{x_n\},\ \{y_n\}$  are defined by the following equation.

\[ \left(\begin{array}{c} x_n  y_n \end{array}\right)=A \left(\begin{array}{c} x_{n-1}  y_{n-1} \end{array}\right),\ \ n=1,2,\ \cdots \]

then find the necessray and sufficient condition for  $a$  such that the sequence  $\{x_n\}$  converges.
Answer: 0 < a < 1
Problem: For which values of N is it possible to write numbers from 1 to N in some order so that for any group of two or more consecutive numbers, the arithmetic mean of these numbers is not whole?
Answer:  N must be even.
Problem: Set  $M$  contains  $n \ge 2$  positive integers. It's known that for any two different  $a, b \in M$ ,  $a^2+1$  is divisible by  $b$ . What is the largest possible value of  $n$ ?
Answer: 2
Problem: Find all positive integers  $k$  for which the equation:   $$  \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n) $$   has no solution in integers positive  $(m,n)$  with  $m\neq n$ .

Answer: k = 2
Problem: Find all triplets  $(a, b, c)$  of positive integers, such that  $a+bc, b+ac, c+ab$  are primes and all divide  $(a^2+1)(b^2+1)(c^2+1)$ .
Answer: (1, 1, 1)
Problem: A mouse has a wheel of cheese which is cut into  $2018$  slices. The mouse also has a  $2019$ -sided die, with faces labeled  $0,1,2,\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on  $k$ , and the mouse has at least  $k$  slices of cheese remaining, then the mouse eats  $k$  slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?
Answer: 2019
Problem: Jay notices that there are  $n$  primes that form an arithmetic sequence with common difference  $12$ . What is the maximum possible value for  $n$ ?
Answer: 5
Problem: Given  $100$  quadratic polynomials  $f_1(x)=ax^2+bx+c_1, ... f_{100}(x)=ax^2+bx+c_{100}$ . One selected  $x_1, x_2... x_{100}$  - roots of  $f_1, f_2, ... f_{100}$  respectively.What is the value of sum  $f_2(x_1)+...+f_{100}(x_{99})+f_1(x_{100})?$ ---------
Also 9.1 in 3rd round of Russian National Olympiad
Answer: 0
Problem: In triangle  $\triangle ABC$ , the angle  $\angle BAC$  is less than  $90^o$ . The perpendiculars from  $C$  on  $AB$  and from  $B$  on  $AC$  intersect the circumcircle of  $\triangle ABC$  again at  $D$  and  $E$  respectively. If  $|DE| =|BC|$ , find the measure of the angle  $\angle BAC$ .
Answer: 60^	extcirc }
Problem: In parallelogram  $ABCD$ ,  $AB = 10$ , and  $AB = 2BC$ . Let  $M$  be the midpoint of  $CD$ , and suppose that  $BM = 2AM$ . Compute  $AM$ .
Answer: 2\sqrt{5}
Problem: For real numbers  $a,\ b$  with  $0\leq a\leq \pi,\ a<b$ , let  $I(a,\ b)=\int_{a}^{b} e^{-x} \sin x\ dx.$ Determine the value of  $a$  such that  $\lim_{b\rightarrow \infty} I(a,\ b)=0.$ 
Answer: \frac{3\pi}{4}
Problem:  $(x_{n})_{-\infty<n<\infty}$  is a sequence of real numbers which satisfies  $x_{n+1}=\frac{x_{n}^2+10}{7}$  for every  $n \in \mathbb{Z}$ . If there exist a real upperbound for this sequence, find all the values  $x_{0}$  can take.
Answer:  [2, 5]
Problem: For an integer  $n>3$  denote by  $n?$  the product of all primes less than  $n$ . Solve the equation  $n?=2n+16$ .
Answer:  n = 7
Problem: A train car held  $6000$  pounds of mud which was  $88$  percent water. Then the train car sat in the sun, and some of the water evaporated so that now the mud is only  $82$  percent water. How many pounds does the mud weigh now?
Answer: 4000
Problem: Let  $r$  be a fixed positive real number. It is known that for some positive integer  $n$  the following statement is true: for any positive real numbers  $a_1,...,a_n$  satisfying the equation  $a_1+...+a_n=r(\frac{1}{a_1}+...+\frac{1}{a_n})$  they also satisfy the equation  $\frac{1}{\sqrt{r}-a_1}+...+\frac{1}{\sqrt{r}-a_n}=\frac{1}{\sqrt{r}}$  ( $a_i\neq\sqrt{r}$ ). Find  $n$ .
Answer:  n = 2
Problem: Find the smallest positive integer  $n$  that satisfies the following:

We can color each positive integer with one of  $n$  colors such that the equation  $w + 6x = 2y + 3z$  has no solutions in positive integers with all of  $w, x, y$  and  $z$  having the same color. (Note that  $w, x, y$  and  $z$  need not be distinct.)
Answer: 4
Problem: Let  $f, g$  be bijections on  $\{1, 2, 3, \dots, 2016\}$ . Determine the value of   $$ \sum_{i=1}^{2016}\sum_{j=1}^{2016}[f(i)-g(j)]^{2559}. $$   
Answer: 0
Problem: Find all functions  $f:\mathbb{R}\to\mathbb{R}$  such that   $$ f(x+y)\leq f(x^2+y) $$   for all  $x,y$ .
Answer: f(x) = c
Problem: Determine all  $n$  positive integers such that exists an  $n\times n$  where we can write  $n$  times each of the numbers from  $1$  to  $n$  (one number in each cell), such that the  $n$  sums of numbers in each line leave  $n$  distinct remainders in the division by  $n$ , and the  $n$  sums of numbers in each column leave  $n$  distinct remainders in the division by  $n$ .
Answer:  n
Problem: Find all polynomials with integer coefficients  $P$  such that for all positive integers  $n$ , the sequence   $$ 0, P(0), P(P(0)), \cdots $$   is eventually constant modulo  $n$ .
Answer:  P(x) = c
Problem: Assume  $A=\{a_{1},a_{2},...,a_{12}\}$  is a set of positive integers such that for each positive integer  $n \leq 2500$   there is a subset  $S$  of  $A$  whose sum of elements is  $n$ . If  $a_{1}<a_{2}<...<a_{12}$  , what is the smallest possible value of  $a_{1}$ ?
Answer: 1
Problem: It is given isosceles triangle  $ABC$  ( $AB=AC$ ) such that  $\angle BAC=108^{\circ}$ . Angle bisector of angle  $\angle ABC$  intersects side  $AC$  in point  $D$ , and point  $E$  is on side  $BC$  such that  $BE=AE$ . If  $AE=m$ , find  $ED$ 
Answer: m
Problem: Let  $P(x)=x^{2020}+x+2$ , which has  $2020$  distinct roots. Let  $Q(x)$  be the monic polynomial of degree  $\binom{2020}{2}$  whose roots are the pairwise products of the roots of  $P(x)$ . Let  $\alpha$  satisfy  $P(\alpha)=4$ . Compute the sum of all possible values of  $Q(\alpha^2)^2$ .
Answer: 2020 \times 2^{2019}
Problem: A square with center  $F$  was constructed on the side  $AC$  of triangle  $ABC$  outside it. After this, everything was erased except  $F$  and the midpoints  $N,K$  of sides  $BC,AB$ .
Restore the triangle.
Answer:  ABC
Problem: A lucky number is a number whose digits are only  $4$  or  $7.$  What is the  $17$ th smallest lucky number?

*Author: Ray Li*

<details><summary>Clarifications</summary>[list=1][*]Lucky numbers are positive.
[*]"only 4 or 7" includes combinations of 4 and 7, as well as only 4 and only 7. That is, 4 and 47 are both lucky numbers.[/list]</details>
Answer: 4474
Problem: There is a  $40\%$  chance of rain on Saturday and a  $30\%$  of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is  $\frac{a}{b}$ , where  $a$  and  $b$  are relatively prime positive integers. Find  $a+b$ .
Answer: 107
Problem: Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square then it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a  $2\times4$  rectangle instead of  $2\times3$  like in chess). In this movement, it doesn't land on the squares between the beginning square and the final square it lands on.

A trip of the length  $n$  of the knight is a sequence of  $n$  squares  $C1, C2, ..., Cn$  which are all distinct such that the knight starts at the  $C1$  square and for each  $i$  from  $1$  to  $n-1$  it can use the movement described before to go from the  $Ci$  square to the  $C(i+1)$ .

Determine the greatest  $N \in \mathbb{N}$  such that there exists a path of the knight with length  $N$  on a  $5\times5$  board.
Answer: 12
Problem: Find the smallest integer  $k\ge3$  with the property that it is possible to choose two of the number  $1,2,...,k$  in such a way that their product is equal to the sum of the remaining  $k-2$  numbers.
Answer: 10
Problem: Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with  $0<l\le 2$ . Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.
Answer:  T
Problem: Find the point in the closed unit disc  $D=\{ (x,y) | x^2+y^2\le 1 \}$  at which the function  $f(x,y)=x+y$  attains its maximum .
Answer: \left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right)
Problem: Find the smallest positive value of  $36^k - 5^m$ , where  $k$  and  $m$  are positive integers.
Answer: 11
Problem: For every integer  $r > 1$  find the smallest integer  $h(r) > 1$  having the following property: For any partition of the set  $\{1, 2, . . ., h(r)\}$  into  $r$  classes, there exist integers  $a \geq 0, 1 \leq x \leq y$  such that the numbers  $a + x, a + y, a + x + y$  are contained in the same class of the partition.
Answer:  h(r) = 2r
Problem: There are 2019 students in a school, and some of these students are members of different student clubs. Each student club has an advisory board consisting of 12 students who are members of that particular club. An {\em advisory meeting} (for a particular club) can be realized only when each participant is a member of that club, and moreover, each of the 12 students forming the advisory board are present among the participants. It is known that each subset of at least 12 students in this school  can realize an advisory meeting for exactly one student club. Determine all possible numbers of different student clubs with exactly 27 members. 

Answer: \binom{2003}{11}
Problem: Find the number ot 6-tuples  $(x_1, x_2,...,x_6)$ , where  $x_i=0,1 or 2$  and  $x_1+x_2+...+x_6$  is even
Answer: 365
Problem: Two radii OA and OB of a circle c with midpoint O are perpendicular. Another circle touches c in point Q and the radii in points C and D, respectively. Determine  $ \angle{AQC}$ .
Answer:  45^
\circ
Problem: Determine the last four digits of a perfect square of a natural number, knowing that  the last three of them are the same.
Answer: 0000 \text{ or } 1444
Problem: Determine the least real number  $k$  such that the inequality  $$ \left(\frac{2a}{a-b}\right)^2+\left(\frac{2b}{b-c}\right)^2+\left(\frac{2c}{c-a}\right)^2+k \geq 4\left(\frac{2a}{a-b}+\frac{2b}{b-c}+\frac{2c}{c-a}\right) $$  holds for all real numbers  $a,b,c$ .
Answer: k = 8
Problem: Find all positive integers   $k$  such that for the first  $k$  prime numbers  $2, 3, \ldots, p_k$  there exist positive integers  $a$  and  $n>1$ , such that  $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$ .
Answer:  k = 1
Problem:  $2016$  bugs are sitting in different places of  $1$ -meter stick. Each bug runs in one or another direction with constant and equal speed. If two bugs face each other, then both of them change direction but not speed. If bug reaches one of the ends of the stick, then it flies away. What is the greatest number of contacts, which can be reached by bugs?
Answer: 1008^2
Problem: Ethan Song and Bryan Guo are playing an unfair game of rock-paper-scissors. In any game, Ethan has a 2/5 chance to win, 2/5 chance to tie, and 1/5 chance to lose. How many games is Ethan expected to win before losing?
Answer: 2
Problem: At Mallard High School there are three intermural sports leagues: football, basketball, and baseball.  There are 427 students participating in these sports: 128 play on football teams, 291 play on basketball teams, and 318 play on baseball teams.  If exactly 36 students participate in all three of the sports, how many students participate in exactly two of the sports?
Answer: 274
Problem: In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square  $\sqrt{5}$  away as long as it isn't occupied.

One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability  $p$ . Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors.

The value of  $p$  that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a, b$ . Compute  $100a + b$ .
Answer: 102
Problem: The straight river is one and a half kilometers wide and has a current of  $8$  kilometers per hour. A boat capable of traveling  $10$  kilometers per hour in still water, sets out across the water. How many minutes will it take the boat to reach a point directly across from where it started?
Answer: 15
Problem: Find all functions  $f: R \to R$ , such that equality  $f (xf (y) - yf (x)) = f (xy) - xy$  holds for all  $x, y \in R$ .
Answer: f(x) = x \text{ or } f(x) = |x| \text{ for all } x \in \mathbb{R}
Problem: Let  $p_1 = 2, p_2 = 3, p_3 = 5 ...$  be the sequence of prime numbers. Find the least positive even integer  $n$  so that  $p_1 + p_2 + p_3 + ... + p_n$  is not prime.
Answer: 8
Problem: The functions  $ f(x) ,\ g(x)$  satify that  $ f(x) \equal{} \frac {x^3}{2} \plus{} 1 \minus{} x\int_0^x g(t)\ dt,\ g(x) \equal{} x \minus{} \int_0^1 f(t)\ dt$ .
Let  $ l_1,\ l_2$  be the tangent lines of the curve  $ y \equal{} f(x)$ , which pass through the point  $ (a,\ g(a))$  on the curve  $ y \equal{} g(x)$ .
Find the minimum area of the figure bounded by the tangent tlines  $ l_1,\ l_2$  and the curve  $ y \equal{} f(x)$  .
Answer: 0
Problem: Let  $ABCD$  be a convex quadrilateral such that the triangle  $ABD$  is equilateral and the triangle  $BCD$  is isosceles, with  $\angle C = 90^o$ . If  $E$  is the midpoint of the side  $AD$ , determine the measure of the angle  $\angle CED$ .
Answer: \angle CED = 45^
Problem: Let  $ f(x) = x^3 + ax + b $ , with  $ a \ne b $ , and suppose the tangent lines to the graph of  $f$  at  $x=a$  and  $x=b$  are parallel. Find  $f(1)$ .
Answer: 1
Problem: Let  $n$  be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2} x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for  $n$ -tuples  $(x_{1},x_{2},\ldots,x_{n})$  of nonnegative real numbers.
Answer:  x_{1} = x_{2} = 	imes = x_{n} = 1
Problem: Princeton has an endowment of  $5$  million dollars and wants to invest it into improving campus life. The university has three options: it can either invest in improving the dorms, campus parties or dining hall food quality. If they invest  $a$  million dollars in the dorms, the students will spend an additional  $5a$  hours per week studying. If the university invests  $b$  million dollars in better food, the students will spend an additional  $3b$  hours per week studying. Finally, if the  $c$  million dollars are invested in parties, students will be more relaxed and spend  $11c - c^2$  more hours per week studying. The university wants to invest its  $5$  million dollars so that the students get as many additional hours of studying as possible. What is the maximal amount that students get to study?
Answer: 34
Problem: Find a function  $f(n)$  on the positive integers with positive integer values such that  $f( f(n) ) = 1993 n^{1945}$  for all  $n$ .
Answer:  f(n)
Problem: In parallelogram  $ABCD$ ,  $AC=10$  and  $BD=28$ . The points  $K$  and  $L$  in the plane of  $ABCD$  move in such a way that  $AK=BD$  and  $BL=AC$ . Let  $M$  and  $N$  be the midpoints of  $CK$  and  $DL$ , respectively. What is the maximum walue of  $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$  ?
Answer: 2
Problem: The degree measures of the six interior angles of a convex hexagon form an arithmetic sequence (not necessarily in cyclic order).  The common difference of this arithmetic sequence can be any real number in the open interval  $(-D, D)$ .  Compute the greatest possible value of  $D$ .
Answer: 24
Problem: Consider the set  $S$  of permutations of  $1, 2, \dots, 2022$  such that for all numbers  $k$  in the
permutation, the number of numbers less than  $k$  that follow  $k$  is even.

For example, for  $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$ If  $|S| = (a!)^b$  where  $a, b \in \mathbb{N}$ , then find the product  $ab$ .
Answer: 2022
Problem: Let  $x_1,x_2,\cdots,x_n$  be postive real numbers such that  $x_1x_2\cdots x_n=1$  , $S=x^3_1+x^3_2+\cdots+x^3_n$ .Find the maximum of  $\frac{x_1}{S-x^3_1+x^2_1}+\frac{x_2}{S-x^3_2+x^2_2}+\cdots+\frac{x_n}{S-x^3_n+x^2_n}$ 
Answer: 1
Problem: Find all  $x$  and  $y$  which are rational multiples of  $\pi$  with  $0<x<y<\frac{\pi}{2}$  and  $\tan x+\tan y =2$ .
Answer: None
Problem: There are eight different symbols designed on  $n\geq 2$  different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any  $k$  symbols  $(1\leq k\leq 7)$  the number of shirts containing at least one of the  $k$  symbols is even. Determine the value of  $n$ .
Answer:  n = 8
Problem: Determine all integers  $n$  such that both of the numbers:  $$ |n^3 - 4n^2 + 3n - 35| \text{ and }  |n^2 + 4n + 8| $$  are both prime numbers.
Answer: n \in \{-3, -1, 5\}
Problem: Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!" 

Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"

Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"

Claire says, "Now I know your favorite number!" What is Cat's favorite number?
Answer: 13
Problem: The only prime factors of an integer  $n$  are 2 and 3. If the sum of the divisors of  $n$  (including itself) is  $1815$ , find  $n$ .
Answer: 648
Problem: The positive integers are colored with black and white such that:
- There exists a bijection from the black numbers to the white numbers,
- The sum of three black numbers is a black number, and
- The sum of three white numbers is a white number.

Find the number of possible colorings that satisfies the above conditions.
Answer: 2
Problem: In a wagon, every  $m \geq 3$  people have exactly one common friend. (When  $A$  is  $B$ 's friend,  $B$  is also  $A$ 's friend. No one was considered as his own friend.) Find the number of friends of the person who has the most friends.
Answer:  m-1
Problem: Note that  $12^2=144$  ends in two  $4$ s and  $38^2=1444$  end in three  $4$ s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end.
Answer: 3
Problem: Alexander and Louise are a pair of burglars. Every morning, Louise steals one third of Alexander's money, but feels remorse later in the afternoon and gives him half of all the money she has. If Louise has no money at the beginning and starts stealing on the first day, what is the least positive integer amount of money Alexander must have so that at the end of the 2012th day they both have an integer amount of money?
Answer: 4 \cdot 3^{2012}
Problem: Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial  $y = x^2 + ax + b$  are drawn in the plane. The numbers  $a$  and  $b$  are not known. How to draw a unit segment using only ruler and compass?
Answer:  1
Problem: Given that for reals  $a_1,\cdots, a_{2004},$  equation  $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$  has  $2006$  positive real solution, find the maximum possible value of  $a_1.$ 
Answer: -2006
Problem: Let  $n>4$  be a positive integer, which is divisible by  $4$ . We denote by  $A_n$  the sum of the odd positive divisors of  $n$ . We also denote  $B_n$  the sum of the even positive divisors of  $n$ , excluding the number  $n$  itself. Find the least possible value of the expression   $$ f(n)=B_n-2A_n, $$  for all possible values of  $n$ , as well as for which positive integers  $n$  this minimum value is attained.
Answer: 4