rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Let  $p$  be a prime number such that  $\frac{28^p-1}{2p^2+2p+1}$  is an integer. Find all possible values of number of divisors of  $2p^2+2p+1$ .
Answer: 2
Problem: Suppose that  $ABCD$  is a rectangle with sides of length  $12$  and  $18$ . Let  $S$  be the region of points contained in  $ABCD$  which are closer to the center of the rectangle than to any of its vertices. Find the area of  $S$ .
Answer: 54
Problem: Determine the integers  $n\geqslant 2$  for which the equation  $x^2-\hat{3}\cdot x+\hat{5}=\hat{0}$  has a unique solution in  $(\mathbb{Z}_n,+,\cdot).$ 
Answer:  n = 11
Problem: Let  $f$  be a function such that  $f(0)=1$ ,  $f^\prime (0)=2$ , and \[f^{\prime\prime}(t)=4f^\prime(t)-3f(t)+1\] for all  $t$ .  Compute the  $4$ th derivative of  $f$ , evaluated at  $0$ .
Answer: 54
Problem: Let  $\mathcal M$  be the set of  $n\times n$  matrices with integer entries. Find all  $A\in\mathcal M$  such that  $\det(A+B)+\det(B)$  is even for all  $B\in\mathcal M$ .
Answer:  A \equiv 0 \pmod{2}
Problem: In  $ xy$  plane, find the minimum volume of the solid by rotating the region boubded by the parabola  $ y \equal{} x^2 \plus{} ax \plus{} b$  passing through the point  $ (1,\ \minus{} 1)$  and the  $ x$  axis about the  $ x$  axis
Answer: \frac{16\pi}{15}
Problem: Find the least positive integer  $k$  such that for any  $a \in [0, 1]$  and any positive integer  $n,$ \[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]
Answer:  k = 4
Problem: The chord  $[CD]$  is parallel to the diameter  $[AB]$  of a circle with center  $O$ . The tangent line at  $A$  meet  $BC$  and  $BD$  at  $E$  and  $F$ . If  $|AB|=10$ , calculate  $|AE|\cdot |AF|$ .
Answer: 100
Problem: The beaver is chess piece that move to  $2$  cells by horizontal or vertical.  Every cell of  $100 \times 100$  chessboard colored in some color,such that we can not get from one cell to another with same color with one move of beaver or knight. What minimal color do we need?
Answer: 4
Problem: Find all triples  $\left(x,\ y,\ z\right)$  of integers satisfying the following system of equations:
 $x^3-4x^2-16x+60=y$ ; $y^3-4y^2-16y+60=z$ ; $z^3-4z^2-16z+60=x$ .
Answer: (3, 3, 3), (-4, -4, -4), (5, 5, 5)
Problem: A college math class has  $N$  teaching assistants. It takes the teaching assistants  $5$  hours to grade homework assignments. One day, another teaching assistant joins them in grading and all homework assignments take only  $4$  hours to grade. Assuming everyone did the same amount of work, compute the number of hours it would take for  $1$  teaching assistant to grade all the homework assignments.
Answer: 20
Problem: Find the least positive integer  $n$  such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \]
Answer: 1
Problem: Find the pairs of real numbers  $(a,b)$  such that the biggest of the numbers  $x=b^2-\frac{a-1}{2}$  and  $y=a^2+\frac{b+1}{2}$  is less than or equal to  $\frac{7}{16}$ 
Answer: (a, b) = \left(\frac{1}{4}, -\frac{1}{4}\right)
Problem: Kevin the Koala eats  $1$  leaf on the first day of its life,  $3$  leaves on the second,  $5$  on the third, and in general eats  $2n-1$  leaves on the  $n$ th day. What is the smallest positive integer  $n>1$  such that the total number of leaves Kevin has eaten his entire  $n$ -day life is a perfect sixth power?
Answer: 8
Problem: Find all real numbers  $a$  for which there exists a function  $f$  defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality  $$ f(f(x))=x^2f(x)+ax^2 $$  for all real  $x$ .
Answer: a = 0
Problem: The *cross* of a convex  $n$ -gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a  $3 \times 4$  rectangle is  $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$ .

Suppose  $S$  is a dodecagon ( $12$ -gon) inscribed in a unit circle. Find the greatest possible cross of  $S$ .
Answer: \frac{2\sqrt{66}}{11}
Problem: Maria and Joe are jogging towards each other on a long straight path. Joe is running at  $10$  mph and Maria at  $8$  mph. When they are  $3$  miles apart, a fly begins to fly back and forth between them at a constant rate of  $15$  mph, turning around instantaneously whenever it reachers one of the runners. How far, in miles, will the fly have traveled when Joe and Maria pass each other?
Answer: \frac{5}{2}
Problem: Consider the graph on  $1000$  vertices  $v_1, v_2, ...v_{1000}$  such that for all  $1 \le i < j  \le 1000$ ,  $v_i$  is connected to  $v_j$  if and only if  $i$  divides  $j$ . Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color.
Answer: 10
Problem: In the sequence  $00$ ,  $01$ ,  $02$ ,  $03$ ,  $\cdots$ ,  $99$  the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by  $1$  (for example,  $29$  can be followed by  $19$ ,  $39$ , or  $28$ , but not by  $30$  or  $20$ ). What is the maximal number of terms that could remain on their places?
Answer: 50
Problem: The nine horizontal and nine vertical lines on an  $8\times8$  checkerboard form  $r$  rectangles, of which  $s$  are squares.  The number  $s/r$  can be written in the form  $m/n,$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n.$ 
Answer: 125
Problem: Find the maximum value of  $\lambda ,$  such that for  $\forall x,y\in\mathbb R_+$  satisfying  $2x-y=2x^3+y^3,x^2+\lambda y^2\leqslant 1.$ 
Answer: \frac{\sqrt{5} + 1}{2}
Problem: Let  $f$  be a monic cubic polynomial such that the sum of the coefficients of  $f$  is  $5$  and such that the sum of the roots of  $f$  is  $1$ . Find the absolute value of the sum of the cubes of the roots of  $f$ .
Answer: 14
Problem: An equilateral triangle of side  $n$  has been divided into little equilateral triangles of side  $1$  in the usual way. We draw a path over the segments of this triangulation, in such a way that it visits exactly once each one of the  $\frac{(n+1)(n+2)}{2}$  vertices. What is the minimum number of times the path can change its direction?

The figure below shows a valid path on a triangular board of side  $4$ , with exactly  $9$  changes of direction.

[asy]
unitsize(30);
pair h = (1, 0);
pair v = dir(60);
pair d = dir(120);
for(int i = 0; i < 4; ++i)
{
	draw(i*v -- i*v + (4 - i)*h);
    draw(i*h -- i*h + (4 - i)*v);
    draw((i + 1)*h -- (i + 1)*h + (i + 1)*d);
}

draw(h + v -- v -- (0, 0) -- 2*h -- 2*h + v -- h + 2*v -- 2*v -- 4*v -- 3*h + v -- 3*h -- 4*h, linewidth(2));
draw(3*h -- 4*h, EndArrow);
fill(circle(h + v, 0.1));

[/asy]
Answer:  n
Problem: Find all polynomials  $ P(x)$  with real coefficients such that for every positive integer  $ n$  there exists a rational  $ r$  with  $ P(r)=n$ .
Answer:  P(x) \in \mathbb{Q}[x]
Problem: For what positive integers  $n$  and  $k$  do there exits integers  $a_1,a_2,\dots,a_n$  and  $b_1,b_2,\dots,b_k$  such that the products  $a_ib_j$  ( $1\le i\le n,1\le j\le k$ ) give pairwise different residues modulo  $nk$ ?
Answer: \gcd(n, k) = 1
Problem: Adam has RM2010 in his bank account. He donates RM10 to charity every day. His first donation is on Monday. On what day will he donate his last RM10?
Answer: Saturday
Problem: Define a sequence  $\{a_n\}$   by\[a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),\]  find  integer  $k$  such that    $a_{k}<1<a_{k+1}.$ (September 29, 2012, Hohhot)
Answer: k = 2012
Problem: \[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\]
where  $n$  is an positive integer. Find the largest  $k$  such that  $n^k$  divides  $[A]$ .
Answer: 1
Problem: In two weeks three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks. In four weeks, two cows eat all the grass on two hectares of land, together with all the grass that regrows there during the four weeks.
 How many cows will eat all the grass on six hectares of land in six weeks, together  with all the grass that regrows there over the six weeks?

 (Assume:  $\bullet$  the quantity of grass on each hectare is the same when the cows begin to  graze, $\bullet$  the rate of growth of the grass is uniform during the time of grazing, $\bullet$  the cows eat the same amount of grass each week.)
Answer: 3
Problem: For natural numbers  $x$  and  $y$ , let  $(x,y)$  denote the greatest common divisor of  $x$  and  $y$ . How many pairs of natural numbers  $x$  and  $y$  with  $x \le y$  satisfy the equation  $xy = x + y + (x, y)$ ?
Answer: 3
Problem: Let  $p$  be an odd prime number. For positive integer  $k$  satisfying  $1\le k\le p-1$ , the number of divisors of  $kp+1$  between  $k$  and  $p$  exclusive is  $a_k$ . Find the value of  $a_1+a_2+\ldots + a_{p-1}$ .
Answer: p-2
Problem: Let  $n$  be a fixed positive integer. How many ways are there to write  $n$  as a sum of positive integers,
\[n = a_1 + a_2 + \cdots a_k\]
with  $k$  an arbitrary positive integer and  $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$ ? For example, with  $n = 4$ , there are four ways:  $4$ ,  $2 + 2$ ,  $1 + 1 + 2$ ,  $1 + 1 + 1 + 1$ .
Answer: n
Problem: Let  $\alpha < 0 < \beta$  and consider the polynomial  $f(x) = x(x-\alpha)(x-\beta)$ . Let  $S$  be the set of real numbers  $s$  such that  $f(x) - s$  has three different real roots. For  $s\in S$ , let  $p(x)$  the product of the smallest and largest root of  $f(x)-s$ . Determine the smallest possible value that  $p(s)$  for   $s\in S$ .
Answer:  -\frac{1}{4} (\beta - \alpha)^2
Problem: Given infinite sequences  $a_1,a_2,a_3,\cdots$  and  $b_1,b_2,b_3,\cdots$  of real numbers satisfying  $\displaystyle a_{n+1}+b_{n+1}=\frac{a_n+b_n}{2}$  and  $\displaystyle a_{n+1}b_{n+1}=\sqrt{a_nb_n}$  for all  $n\geq1$ . Suppose  $b_{2016}=1$  and  $a_1>0$ . Find all possible values of  $a_1$ 
Answer: 2^{2015}
Problem: Let  $n=2^{31}3^{19}.$   How many positive integer divisors of  $n^2$  are less than  $n$  but do not divide  $n$ ?
Answer: 589
Problem: Find all positive integers  $k$  for which number  $3^k+5^k$  is a power of some integer with exponent greater than  $1$ .
Answer:  k = 1
Problem: In each cell of  $5 \times 5$  table there is one number from  $1$  to  $5$  such that every number occurs exactly once in every row and in every column. Number in one column is *good positioned* if following holds:
 - In every row, every number which is left from *good positoned* number is smaller than him, and every number which is right to him is greater than him, or vice versa.
 -  In every column, every number which is above from *good positoned* number is smaller than him, and every number which is below to him is greater than him, or vice versa.
What is maximal number of good positioned numbers that can occur in this table?
Answer: 5
Problem: To celebrate the  $20$ th LMT, the LHSMath Team bakes a cake. Each of the  $n$  bakers places  $20$  candles on the cake. When they count, they realize that there are  $(n -1)!$  total candles on the cake. Find  $n$ .
Answer:  n = 6
Problem: William is thinking of an integer between 1 and 50, inclusive. Victor can choose a positive integer  $m$  and ask William: "does  $m$  divide your number?", to which William must answer truthfully. Victor continues asking these questions until he determines William's number. What is the minimum number of questions that Victor needs to guarantee this?
Answer: 15
Problem: Malcolm writes a positive integer on a piece of paper. Malcolm doubles this integer and subtracts 1, writing this second result on the same piece of paper. Malcolm then doubles the second integer and adds 1, writing this third integer on the paper. If all of the numbers Malcolm writes down are prime, determine all possible values for the first integer.
Answer: 2 \text{ and } 3
Problem: Let  $AD$  and  $BC$  be the parallel sides of a trapezium  $ABCD$ . Let  $P$  and  $Q$  be the midpoints of the diagonals  $AC$  and  $BD$ . If  $AD = 16$  and  $BC = 20$ , what is the length of  $PQ$ ?
Answer: 2
Problem: Find the largest positive integer  $n$  such that the following is true:
There exists  $n$  distinct positive integers  $x_1,~x_2,\dots,x_n$  such that whatever the numbers  $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$  are, not all null, the number  $n^3$  do not divide  $\sum_{k=1}^n a_kx_k$ .
Answer: 9
Problem: Given is a chessboard 8x8. We have to place  $n$  black queens and  $n$  white queens, so that no two queens attack. Find the maximal possible  $n$ . 
(Two queens attack each other when they have different colors. The queens of the same color don't attack each other)
Answer: 8
Problem: Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!”
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, “Yes! Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn’t know my favorite number.”
Claire says, “Now I know your favorite number!” What is Cat’s favorite number?
Answer: 25
Problem: Find all integer sequences of the form  $ x_i, 1 \le i \le 1997$ , that satisfy  $ \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k$ .
Answer:  x_i = 0
Problem: Let  $P$  be a regular  $2006$ -gon. A diagonal is called *good* if its endpoints divide the boundary of  $P$  into two parts, each composed of an odd number of sides of  $P$ . The sides of  $P$  are also called *good*.
Suppose  $P$  has been dissected into triangles by  $2003$  diagonals, no two of which have a common point in the interior of  $P$ . Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.
Answer: 1003
Problem: Find all functions  $f:\mathbb{N}_0\to\mathbb{N}_0$  for which  $f(0)=0$  and
\[f(x^2-y^2)=f(x)f(y) \]
for all  $x,y\in\mathbb{N}_0$  with  $x>y$ .
Answer:  f(x) = 0
Problem: (**4**) Let  $ a$ ,  $ b$  be constants such that  $ \lim_{x\rightarrow1}\frac {(\ln(2 \minus{} x))^2}{x^2 \plus{} ax \plus{} b} \equal{} 1$ . Determine the pair  $ (a,b)$ .
Answer:  (a, b) = (-2, 1)
Problem: A *pucelana* sequence is an increasing sequence of  $16$  consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with  $3$ -digit numbers only?
Answer: 2
Problem:  $1989$  equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.
Answer: 4
Problem: Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37} b(a\plus{}d)\equiv b\pmod {37} c(a\plus{}d)\equiv c\pmod{37} bc\plus{}d^2\equiv d\pmod{37} ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]
Answer: 1
Problem: Let  $f(x)=\int_0^ x \frac{1}{1+t^2}dt.$  For  $-1\leq x<1$ , find  $\cos \left\{2f\left(\sqrt{\frac{1+x}{1-x}}\right)\right\}.$ *2011 Ritsumeikan University entrance exam/Science and Technology*
Answer: -x
Problem: Let  $A$  and  $B$  be points on a circle  $\mathcal{C}$  with center  $O$  such that  $\angle AOB = \dfrac {\pi}2$ . Circles  $\mathcal{C}_1$  and  $\mathcal{C}_2$  are internally tangent to  $\mathcal{C}$  at  $A$  and  $B$  respectively and are also externally tangent to one another. The circle  $\mathcal{C}_3$  lies in the interior of  $\angle AOB$  and it is tangent  externally to  $\mathcal{C}_1$ ,  $\mathcal{C}_2$  at  $P$  and  $R$  and internally tangent to  $\mathcal{C}$  at  $S$ . Evaluate the value of   $\angle PSR$ .
Answer: 45^	extcirc
Problem: Find all pairs of positive integers  $ a,b$  such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a  a^2+a+1 &\equiv  0 \pmod b . \end{align*}
Answer: (1, 1)
Problem: Eddy draws  $6$  cards from a standard  $52$ -card deck. What is the probability that four of the cards that he draws have the same value?
Answer: \frac{3}{4165}
Problem: For the give functions in  $\mathbb{N}$ :**(a)** Euler's  $\phi$  function ( $\phi(n)$ - the number of natural numbers smaller than  $n$  and coprime with  $n$ );**(b)** the  $\sigma$  function such that the  $\sigma(n)$  is the sum of natural divisors of  $n$ .
solve the equation  $\phi(\sigma(2^x))=2^x$ .
Answer: 1
Problem: To fold a paper airplane, Austin starts with a square paper  $F OLD$  with side length  $2$ . First, he  folds corners  $L$  and  $D$  to the square’s center. Then, he folds corner  $F$  to corner  $O$ . What is the longest distance between two corners of the resulting figure?
Answer: \sqrt{5}
Problem: In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation  $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$ . Find the minimum of  square areas.
Answer: 4
Problem: Let  $x$  and  $y$  be real numbers such that 
\[
  2 < \frac{x - y}{x + y} < 5.
\]
If  $\frac{x}{y}$  is an integer, what is its value?
Answer: -2
Problem: Let  $a_1\in (0,1)$  and  $(a_n)_{n\ge 1}$  a sequence of real numbers defined by  $a_{n+1}=a_n(1-a_n^2),\ (\forall)n\ge 1$ . Evaluate  $\lim_{n\to \infty} a_n\sqrt{n}$ .
Answer: \frac{\sqrt{2}}{2}
Problem: Let  $[ {x} ]$  be the greatest integer less than or equal to  $x$ , and let  $\{x\}=x-[x]$ . 
Solve the equation:  $[x] \cdot \{x\} = 2005x$ 
Answer: x = 0 \text{ and } x = \frac{-1}{2006}
Problem: A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass  $a$  and  $b$  and entangles them; this process destroys the omon with mass  $a$ , preserves the one with mass  $b$ , and creates a new omon whose mass is  $\frac 12 (a+b)$ . The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than  $1000$ . What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer?
Answer: 9
Problem:  $ABCDEF$  is a regular hexagon. Let  $R$  be the overlap between  $\vartriangle ACE$  and  $\vartriangle BDF$ . What is the area of  $R$  divided by the area of  $ABCDEF$ ?
Answer: \frac{1}{3}
Problem: There are 2000 cities in Graphland; some of them are connected by roads. 
For every city the number of roads going from it is counted. It is known that there are exactly two equal numbers among all the numbers obtained. What can be these numbers?
Answer:  n or n-1
Problem: Let  $a_1,a_2,a_3,a_4$  be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let  $i,j,k\in\{1,2,3,4\}$  with  $i \neq j$ ,  $j\neq k$ , and  $k\neq i $ . Determine the maximum number of triples  $(i,j,k)$  for which   $$ ({\rm gcd}(a_i,a_j))^2|a_k. $$  
Answer: 8
Problem: China Mathematical Olympiad 2018 Q6
Given the positive integer  $n ,k$   $(n>k)$  and  $ a_1,a_2,\cdots ,a_n\in (k-1,k)$  ,if positive number  $x_1,x_2,\cdots ,x_n$  satisfying:For any set  $\mathbb{I} \subseteq  \{1,2,\cdots,n\}$  , $|\mathbb{I} |=k$ ,have  $\sum_{i\in \mathbb{I}  }x_i\le \sum_{i\in \mathbb{I}  }a_i$  , find the maximum value of  $x_1x_2\cdots  x_n.$ 
Answer:  a_1 a_2 \cdots a_n
Problem: Consider the points  $1,\frac12,\frac13,...$  on the real axis. Find the smallest value  $k \in \mathbb{N}_0$  for which all points above can be covered with 5 **closed** intervals of length  $\frac1k$ .
Answer:  k = 10
Problem: Given a quadratic function  $y$  with respect to  $x$ .   $y$  takes the values of  $-1, -2$  and  $1$  for the values of  $x=-1,\ 0,\ 1$  respectively.
Find the length of the line segment of the graph of the curve cut by the  $x$ -axis.

1956 Tokyo Institute of Technology entrance exam
Answer: \frac{\sqrt{17}}{2}
Problem: Collinear points  $A$ ,  $B$ , and  $C$  are given in the Cartesian plane such that  $A= (a, 0)$  lies along the x-axis,  $B$  lies along the line  $y=x$ ,  $C$  lies along the line  $y=2x$ , and  $\frac{AB}{BC}=2$ . If  $D= (a, a)$ , and the circumcircle of triangle  $ADC$  intersects the line  $y=x$  again at  $E$ , and ray  $AE$  intersects  $y=2x$  at  $F$ , evaluate  $\frac{AE}{EF}$ .
Answer: \frac{\sqrt{2}}{2}
Problem: Consider a hexagon with vertices labeled  $M$ ,  $M$ ,  $A$ ,  $T$ ,  $H$ ,  $S$  in that order. Clayton starts at the  $M$  adjacent to  $M$  and  $A$ , and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability  $\frac{1}{2}$ , and writes down the corresponding letter. Clayton stops moving when the string he's written down contains the letters  $M, A, T$ , and  $H$  in that order, not necessarily consecutively (for example, one valid string might be  $MAMMSHTH$ .) What is the expected length of the string Clayton wrote?
Answer: 6
Problem: Let  $n$  be a positive integer. A child builds a wall along a line with  $n$  identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?
Answer: 2^{n-1}
Problem: The vertices of a regular  $2012$ -gon are labeled  $A_1,A_2,\ldots, A_{2012}$  in some order. It is known that if  $k+\ell$  and  $m+n$  leave the same remainder when divided by  $2012$ , then the chords  $A_kA_{\ell}$  and  $A_mA_n$  have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled  $A_1$  and  $A_4$ . How is the tenth vertex labeled?
Answer: A_{28}
Problem: A cube with side length  $100cm$  is filled with water and has a hole through which the water drains into a cylinder of radius  $100cm.$  If the water level in the cube is falling at a rate of  $1 \frac{cm}{s} ,$  how fast is the water level in the cylinder rising? 
Answer: \frac{1}{\pi}
Problem: A pet shop sells cats and two types of birds: ducks and parrots. In the shop,  $\tfrac{1}{12}$  of animals are ducks, and  $\tfrac{1}{4}$  of birds are ducks. Given that there are  $56$  cats in the pet shop, how many ducks are there in the pet shop?
Answer: 7
Problem: The non-negative numbers  $x,y,z$  satisfy the relation  $x + y+ z = 3$ . Find the smallest possible numerical value and the largest possible numerical value for the expression  $$ E(x,y, z) = \sqrt{x(y + 3)} + \sqrt{y(z + 3)} + \sqrt{z(x + 3)} . $$  
Answer: 3 \leq E(x, y, z) \leq 6
Problem: Let  $[n]$  denote the set of integers  $\left\{ 1, 2, \ldots, n \right\}$ . We randomly choose a function  $f:[n] \to [n]$ , out of the  $n^n$  possible functions. We also choose an integer  $a$  uniformly at random from  $[n]$ . Find the probability that there exist positive integers  $b, c \geq 1$  such that  $f^b(1) = a$  and  $f^c(a) = 1$ . ( $f^k(x)$  denotes the result of applying  $f$  to  $x$   $k$  times.)
Answer: \frac{1}{n}
Problem: A bowl contained  $10\%$  blue candies and  $25\%$  red candies.  A bag containing three quarters red candies and one quarter blue candies was added to the bowl.  Now the bowl is  $16\%$  blue candies.  What percentage of the candies in the bowl are now red?
Answer: 45\%
Problem: Find all functions  $f: R \to R$  and  $g:R \to R$  such that  $f(x-f(y))=xf(y)-yf(x)+g(x)$  for all real numbers  $x,y$ .

I.Voronovich
Answer:  f(x) = 0  and  g(x) = 0  or  f(x) = x  and  g(x) = 0
Problem: Find the number of pairs  $(a, b)$  of positive integers with the property that the greatest common divisor of  $a$  and  $ b$  is equal to  $1\cdot 2 \cdot 3\cdot ... \cdot50$ , and the least common multiple of  $a$  and  $ b$  is  $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$ .
Answer: 32768