rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Numbers  $0$  and  $2013$  are written at two opposite vertices of a cube. Some real numbers are to be written at the remaining  $6$  vertices of the cube. On each edge of the cube the difference between the numbers at its endpoints is written. When is the sum of squares of the numbers written on the edges minimal?
Answer: \frac{3 \cdot 2013^2}{2}
Problem: A set of  $8$  problems was prepared for an examination. Each student was given  $3$  of them. No two students received more than one common problem. What is the largest possible number of students?
Answer: 8
Problem: Solve the following system of equations for real  $x,y$  and  $z$ :
\begin{eqnarray*}
x &=& \sqrt{2y+3}
y &=& \sqrt{2z+3}
z &=& \sqrt{2x+3}.
\end{eqnarray*}
Answer: x = y = z = 3
Problem: Two players,  $A$  and  $B$ , play a game on a board which is a rhombus of side  $n$  and angles of  $60^{\circ}$  and  $120^{\circ}$ , divided into  $2n^2$  equilateral triangles, as shown in the diagram for  $n=4$ . $A$  uses a red token and  $B$  uses a blue token, which are initially placed in cells containing opposite corners of the board (the  $60^{\circ}$  ones). In turns, players move their token to a neighboring cell (sharing a side with the previous one). To win the game, a player must either place his token on the cell containing the other player's token, or get to the opposite corner to the one where he started.
If  $A$  starts the game, determine which player has a winning strategy.
Answer: Player A (Red) has a winning strategy.
Problem: In how many ways can the numbers from  $2$  to  $2022$  be arranged so that the first number is a multiple of  $1$ , the second number is a multiple of  $2$ , the third number is a multiple of  $3$ , and so on untile the last number is a multiple of  $2021$ ?
Answer: 13
Problem: Bill is buying cans of soup. Cans come in  $2$  shapes. Can  $A$  is a rectangular prism shaped can with dimensions  $20\times16\times10$ , and can  $B$  is a cylinder shaped can with radius  $10$  and height  $10$ . Let  $\alpha$  be the volume of the larger can, and  $\beta$  be the volume of the smaller can. What is  $\alpha-\beta$ ?
Answer: 3200 - 1000\pi \approx 58.41
Problem: Let  $ a_1\equal{}a_2\equal{}1$  and \[ a_{n\plus{}2}\equal{}\frac{n(n\plus{}1)a_{n\plus{}1}\plus{}n^2a_n\plus{}5}{n\plus{}2}\minus{}2\]for each  $ n\in\mathbb N$ . Find all  $ n$  such that  $ a_n\in\mathbb N$ .
Answer:  n = 1, 2
Problem: Diagonals of trapezium   $ABCD$  are mutually perpendicular and the midline of the trapezium is  $5$ . Find the length of the segment that connects the midpoints of the bases of the trapezium.
Answer: 5
Problem: Let set  $A=\{1,2,\ldots,n\} ,$  and  $X,Y$  be two subsets (not necessarily distinct) of  $A.$  Define that  $\textup{max} X$  and  $\textup{min} Y$  represent the greatest element of  $X$  and the least element of  $Y,$  respectively. Determine the number of two-tuples  $(X,Y)$  which satisfies  $\textup{max} X>\textup{min} Y.$ 
Answer: 2^{2n} - (n+1)2^n
Problem: The real numbers  $x, y, z$ , distinct in pairs satisfy   $$ \begin{cases} x^2=2 + y  y^2=2 + z  z^2=2 + x.\end{cases} $$   
Find the possible values of  $x^2 + y^2 + z^2$ .
Answer: 5, 6, 9
Problem: Sasha has  $10$  cards with numbers  $1, 2, 4, 8,\ldots, 512$ . He writes the number  $0$  on the board and invites Dima to play a game. Dima tells the integer  $0 < p < 10, p$  can vary from round to round. Sasha chooses  $p$  cards before which he puts a “ $+$ ” sign, and before the other cards he puts a “ $-$ " sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves.
Answer: 1023
Problem: Suppose that each of  $n$  people knows exactly  one piece of information and all  $n$  pieces are  different.  Every time person  $A$  phones person  $B$ ,  $A$  tells  $B$  everything he knows, while tells  $A$  nothing. What is the minimum of phone calls between pairs of people needed for everyone to know everything?
Answer: 2n - 2
Problem: A positive integer is called *sabroso*if when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example,  $143$  is sabroso, since  $143 + 341 =484 = 22^2$ . Find all two-digit sabroso numbers.
Answer: 29, 38, 47, 56, 65, 74, 83, 92
Problem: Let  $a$  and  $b$  be the roots of the polynomial  $x^2+2020x+c$ . Given that  $\frac{a}{b}+\frac{b}{a}=98$ , compute  $\sqrt c$ .
Answer: 202
Problem: Kevin is trying to solve an economics question which has six steps. At each step, he has a probability  $p$  of making a sign error. Let  $q$  be the probability that Kevin makes an even number of sign errors (thus answering the question correctly!). For how many values of  $0 \le p \le 1$  is it true that  $p+q=1$ ?
Answer: 2
Problem: Suppose you are given a fair coin and a sheet of paper with the polynomial  $x^m$  written on it. Now for each toss of the coin, if heads show up, you must erase the polynomial  $x^r$  (where  $r$  is going to change with time - initially it is  $m$ ) written on the paper and replace it with  $x^{r-1}$ . If tails show up, replace it with  $x^{r+1}$ . What is the expected value of the polynomial I get after  $m$  such tosses? (Note: this is a different concept from the most probable value)
Answer: x^m
Problem: From the set of integers  $ \{1,2,3,\ldots,2009\}$ , choose  $ k$  pairs  $ \{a_i,b_i\}$  with  $ a_i<b_i$  so that no two pairs have a common element. Suppose that all the sums  $ a_i\plus{}b_i$  are distinct and less than or equal to  $ 2009$ . Find the maximum possible value of  $ k$ .
Answer: 803
Problem: Let  $ 2n$  distinct points on a circle be given. Arrange them into disjoint pairs in an arbitrary way and join the couples by chords. Determine the probability that no two of these  $ n$  chords intersect. (All possible arrangement into pairs are supposed to have the same probability.)
Answer: \frac{2^n}{(n+1)!}
Problem: Let C be the number of ways to arrange the letters of the word CATALYSIS, T be the number of ways to arrange the letters of the word TRANSPORT, S be the number of ways to arrange the letters of the word STRUCTURE, and M be the number of ways to arrange the letters of the word MOTION. What is  $\frac{C - T + S}{M}$  ?
Answer: 126
Problem: The real-valued function  $f$  is defined for all positive integers. For any integers  $a>1, b>1$  with  $d=\gcd (a, b)$ , we have
\[f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right) \]
Determine all possible values of  $f(2001)$ .
Answer: f(2001) \in \{0, \frac{1}{2}\}
Problem: The natural number  $a_n$  is obtained by writing together and ordered, in decimal notation , all natural numbers between  $1$  and  $n$ . So we have for example that  $a_1 = 1$ , $a_2 = 12$ ,  $a_3 = 123$ ,  $. . .$  ,  $a_{11} = 1234567891011$ ,   $...$  . Find all values of  $n$  for which  $a_n$  is not divisible by  $3$ .
Answer:  n \equiv 1 \pmod{3}
Problem: Determine all pairs  $(p, q)$ ,  $p, q \in \mathbb {N}$ , such that $(p + 1)^{p - 1} + (p - 1)^{p + 1} = q^q$ .
Answer:  (p, q) = (1, 1); (2, 2)
Problem: Four positive integers  $a, b, c, d$  satisfy the condition:  $a < b < c < d$ . For what smallest possible value of  $d$  could the following condition be true: the arithmetic mean of numbers  $a, b, c$  is twice smaller than the arithmetic mean of numbers  $a, b, c, d$ ?
Answer: 10
Problem:  $A$  and  $B$  are two opposite vertices of an  $n \times n$  board. Within each small square of the board, the diagonal parallel to  $AB$  is drawn, so that the board is divided in  $2n^{2}$  equal triangles. A coin moves from  $A$  to  $B$  along the grid, and for every segment of the grid that it visits, a seed is put in each triangle that contains the segment as a side. The path followed by the coin is such that no segment is visited more than once, and after the coins arrives at  $B$ , there are exactly two seeds in each of the  $2n^{2}$  triangles of the board. Determine all the values of  $n$  for which such scenario is possible.
Answer:  n = 2
Problem: In parallelogram  $ABCD$ , the longer side is twice the shorter side. Let  $XYZW$  be the quadrilateral formed by the internal bisectors of the angles of  $ABCD$ . If the area of  $XYZW$  is  $10$ , find the area of  $ABCD$ 
Answer: 40
Problem: In a triangle  $ABC$ ,  $M$  is the midpoint of  $BC$ . Given that  $\angle MAC=15^{\circ}$ , find the maximum possible value of  $\angle ABC$ .
Answer: 30^\circ
Problem:  $PQ$  is a diameter of a circle.  $PR$  and  $QS$  are chords with intersection at  $T$ . If  $\angle PTQ= \theta$ , determine the ratio of the area of  $\triangle QTP$  to the area of  $\triangle SRT$  (i.e. area of  $\triangle QTP$ /area of  $\triangle SRT$ ) in terms of trigonometric functions of  $\theta$ 
Answer: \frac{1}{\cos^2(\theta)}
Problem: For an integer  $n\ge 3$  and a permutation  $\sigma=(p_{1},p_{2},\cdots ,p_{n})$  of  $\{1,2,\cdots , n\}$ , we say  $p_{l}$  is a  $landmark$  point if  $2\le l\le n-1$  and  $(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0$ . For example , for  $n=7$ ,
the permutation  $(2,7,6,4,5,1,3)$  has four landmark points:  $p_{2}=7$ ,  $p_{4}=4$ ,  $p_{5}=5$  and  $p_{6}=1$ . For a given  $n\ge 3$  , let  $L(n)$  denote the number of permutations of  $\{1,2,\cdots ,n\}$  with exactly one landmark point. Find the maximum  $n\ge 3$  for which  $L(n)$  is a perfect square.
Answer: 3
Problem: The sides  $AD$  and  $BC$  of a convex quadrilateral  $ABCD$  are extended to meet at  $E$ .  Let  $H$  and  $G$  be the midpoints of  $BD$  and  $AC$ , respectively.  Find the ratio of the area of the triangle  $EHG$  to that of the quadrilateral  $ABCD$ .
Answer: \frac{1}{4}
Problem: Let triangle ABC with  $ AB=c$   $AC=b$   $BC=a$    $R$  circumradius,  $p$  half peremetr of  $ABC$ .
I f  $\frac{acosA+bcosB+ccosC}{asinA+bsinB+csinC}=\frac{p}{9R}$   then find all  value of  $cosA$ .
Answer: a = b = c
Problem: For which  $ n\in \mathbb{N}$  do there exist rational numbers  $ a,b$  which are not integers such that both  $ a \plus{} b$  and  $ a^n \plus{} b^n$  are integers?
Answer:  n \equiv 1 \pmod{2}
Problem: For any  $h = 2^{r}$  ( $r$  is a non-negative integer), find all  $k \in \mathbb{N}$  which satisfy the following condition: There exists an odd natural number  $m > 1$  and  $n \in \mathbb{N}$ , such that  $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$ .
Answer:  k
Problem: In a hidden friend, suppose no one takes oneself. We say that the hidden friend has "marmalade" if
there are two people  $A$  and  $ B$  such that A took  $B$  and  $ B $  took  $A$ . For each positive integer n, let  $f (n)$  be the number of hidden friends with n people where there is no “marmalade”, i.e.  $f (n)$  is equal to the number of permutations  $\sigma$  of { $1, 2,. . . , n$ } such that:

* $\sigma (i) \neq i$  for all  $i=1,2,...,n$ * there are no  $ 1 \leq i <j \leq n $  such that  $ \sigma (i) = j$  and  $\sigma (j) = i. $ Determine the limit

  $\lim_{n \to + \infty} \frac{f(n)}{n!}$ 
Answer:  \exp\left(-\frac{3}{2}\right)
Problem: A trapezoid  $ABCD$  is bicentral. The vertex  $A$ , the incenter  $I$ , the circumcircle  $\omega$  and its center  $O$  are given and the trapezoid is erased. Restore it using only a ruler.
Answer: The trapezoid ABCD is restored.
Problem: Let  $ABCDE$  be a convex pentagon such that  $AB = BC = CD$  and  $\angle BDE =
\angle EAC = 30 ^{\circ}$ . Find the possible values of  $\angle BEC$ .
Answer: 60^
Problem: Let  $ABC$  be a triangle with  $\angle A=\tfrac{135}{2}^\circ$  and  $\overline{BC}=15$ . Square  $WXYZ$  is drawn inside  $ABC$  such that  $W$  is on  $AB$ ,  $X$  is on  $AC$ ,  $Z$  is on  $BC$ , and triangle  $ZBW$  is similar to triangle  $ABC$ , but  $WZ$  is not parallel to  $AC$ . Over all possible triangles  $ABC$ , find the maximum area of  $WXYZ$ .
Answer: \frac{225\sqrt{2}}{8}
Problem: Let  $f(x) = 1 + \frac{2}{x}$ . Put  $f_1(x) = f(x)$ ,  $f_2(x) = f(f_1(x))$ ,  $f_3(x) = f(f_2(x))$ ,  $... $ . Find the solutions to  $x = f_n(x)$  for  $n > 0$ .	
Answer:  x = 2 \quad \text{and} \quad x = -1
Problem: Find all triples  $(p,n,k)$  of positive integers, where  $p$  is a Fermat's Prime, satisfying \[p^n + n = (n+1)^k\].

*Observation: a Fermat's Prime is a prime number of the form  $2^{\alpha} + 1$ , for  $\alpha$  positive integer.*
Answer:  (3, 1, 2) and (5, 2, 3)
Problem: Let  $O$  be the circumcenter of triangle  $ABC$ , and let  $l$  be the line passing through the midpoint of segment  $BC$  which is also perpendicular to the bisector of angle  $ \angle BAC $ . Suppose that the midpoint of segment  $AO$  lies on  $l$ . Find  $ \angle BAC $ .
Answer: 60^	extcirc}
Problem: Find the maximum number of colors used in coloring integers  $n$  from  $49$  to  $94$  such that if  $a, b$  (not necessarily different) have the same color but  $c$  has a different color, then  $c$  does not divide  $a+b$ .
Answer: 2
Problem: The mean number of days per month in  $2020$  can be written as  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .
Answer: 63
Problem:  $\textbf{Problem 1.}$  ****There are less than  $400$  marbles.** If they are distributed among  $3$  childrens, there is one left over if they are distributed among  $7$  children, there are 2 left over. Finally if they are distributed among  $5$  children, there are none left over.

What is the largest number of the marbels you have?
Answer: 310
Problem: Find all functions  $f : R \to R$  which satisfy  $f(x) = max_{y\in R} (2xy - f(y))$  for all  $x \in  R$ .
Answer: f(x) = x^2 \quad \forall x \in \mathbb{R}
Problem: Suppose convex hexagon  $ \text{HEXAGN}$  has  $ 120^\circ$ -rotational symmetry about a point  $ P$ —that is, if you rotate it  $ 120^\circ$  about  $ P$ , it doesn't change. If  $ PX\equal{}1$ , find the area of triangle  $ \triangle{GHX}$ .
Answer: \frac{\sqrt{3}}{4}
Problem: Find all triples  $(a,b, c)$  of real numbers for which there exists a non-zero function   $f: R \to R$ , such that   $$ af(xy + f(z)) + bf(yz + f(x)) + cf(zx + f(y)) = 0 $$   for all real  $x, y, z$ .
Answer: a + b + c = 0
Problem: Given a circle  $K$ , find the locus of vertices  $A$  of parallelograms  $ABCD$  with diagonals  $AC \leq BD$ , such that  $BD$  is inside  $K$ .
Answer: K
Problem: Suppose that  $P(x)$  is a polynomial with the property that there exists another polynomial  $Q(x)$  to satisfy  $P(x)Q(x)=P(x^2)$ .  $P(x)$  and  $Q(x)$  may have complex coefficients. If  $P(x)$  is a quintic with distinct complex roots  $r_1,\dots,r_5$ , find all possible values of  $|r_1|+\dots+|r_5|$ .
Answer: \{4, 5\}
Problem: (**3**) Let  $ \ell$  be the line through  $ (0,0)$  and tangent to the curve  $ y \equal{} x^3 \plus{} x \plus{} 16$ . Find the slope of  $ \ell$ .
Answer: 13
Problem: In the questions below:  $G$  is a ﬁnite group;  $H \leq G$  a subgroup of  $G; |G : H |$  the index of  $H$  in  $G; |X |$  the number of elements of  $X \subseteq G; Z (G)$  the center of  $G; G'$  the commutator subgroup of  $G; N_{G}(H )$  the normalizer of  $H$  in  $G; C_{G}(H )$  the centralizer of  $H$  in  $G$ ; and  $S_{n}$  the  $n$ -th symmetric group.

Let  $H \leq G, |H | = 3.$  What can be said about  $|N_{G}(H ) : C_{G}(H )|$ ?
Answer: |N_G(H) : C_G(H)| \text{ must be either } 1 \text{ or } 2
Problem: 
	 Let  $ABC$  be a triangle such that  $AB = 13$ ,  $BC = 14$ ,  $CA = 15$  and let  $E$ ,  $F$  be the feet of the altitudes from  $B$  and  $C$ , respectively.
Let the circumcircle of triangle  $AEF$  be  $\omega$ .
We draw three lines, tangent to the circumcircle of triangle  $AEF$  at  $A$ ,  $E$ , and  $F$ .
Compute the area of the triangle these three lines determine.
Answer: 123.36
Problem: Given triangle  $ ABC$ . Point  $ O$  is the center of the excircle touching the side  $ BC$ . Point  $ O_1$  is the reflection of  $ O$  in  $ BC$ . Determine angle  $ A$  if  $ O_1$  lies on the circumcircle of  $ ABC$ .
Answer: 60^	extdegree
Problem: The game of backgammon has a "doubling" cube, which is like a standard 6-faced die except that its faces are inscribed with the numbers 2, 4, 8, 16, 32, and 64, respectively.  After rolling the doubling cube four times at random, we let  $a$  be the value of the first roll,  $b$  be the value of the second roll,  $c$  be the value of the third roll, and  $d$  be the value of the fourth roll.  What is the probability that  $\frac{a + b}{c + d}$  is the average of  $\frac{a}{c}$  and  $\frac{b}{d}$  ?
Answer: \frac{73}{648}
Problem: In trapezoid  $ABCD$ , the sides  $AD$  and  $BC$  are parallel, and  $AB = BC = BD$ . The height  $BK$  intersects the diagonal  $AC$  at  $M$ . Find  $\angle CDM$ .
Answer: \angle CDM = 90^\circ
Problem: Given the following system of equations:  $$ \begin{cases} R I +G +SP = 50   R I +T + M = 63  G +T +SP = 25   SP + M = 13    M +R I = 48  N = 1 \end{cases} $$   
Find the value of L that makes  $LMT +SPR I NG = 2023$  true. 


Answer: \frac{341}{40}
Problem:  $2021$  points are given on a circle. Each point is colored by one of the  $1,2, \cdots ,k$  colors. For all points and colors  $1\leq r \leq k$ , there exist an arc such that at least half of the points on it are colored with  $r$ . Find the maximum possible value of  $k$ .
Answer: k = 2
Problem: What is the sum of all primes  $p$  such that  $7^p - 6^p + 2$  is divisible by 43? 

Answer: 3
Problem: Determine all sequences of non-negative integers  $a_1, \ldots, a_{2016}$  all less than or equal to  $2016$  satisfying  $i+j\mid ia_i+ja_j$  for all  $i, j\in \{ 1,2,\ldots, 2016\}$ .
Answer:  a_1 = a_2 = \ldots = a_{2016}
Problem: How many ways are there to fill in a  $2\times 2$  square grid with the numbers  $1,2,3,$  and  $4$  such that the numbers in any two grid squares that share an edge have an absolute difference of at most  $2$ ?
Answer: 8
Problem: The sequence  $\{a_{n}\}_{n\geq 0}$  is defined by  $a_{0}=20,a_{1}=100,a_{n+2}=4a_{n+1}+5a_{n}+20(n=0,1,2,...)$ . Find the smallest positive integer  $h$  satisfying  $1998|a_{n+h}-a_{n}\forall n=0,1,2,...$ 
Answer: 6
Problem: If you pick a random  $3$ -digit number, what is the probability that its hundreds digit is triple the ones digit?
Answer: \frac{1}{30}
Problem: Find all pairs  $(m, n)$  of positive integers such that  $m^2 + 2n^2 = 3(m + 2n)$ 
Answer: (m, n) = (3, 3), (4, 2)
Problem: Light of a blue laser (wavelength  $\lambda=475 \, \text{nm}$ ) goes through a narrow slit which has width  $d$ . After the light emerges from the slit, it is visible on a screen that is  $ \text {2.013 m} $  away from the slit. The distance between the center of the screen and the first minimum band is  $ \text {765 mm} $ . Find the width of the slit  $d$ , in nanometers.
Answer: 1250 \, \text{nm}
Problem: A point  $(a,b)$  in the plane is called *sparkling* if it also lies on the line  $ax+by=1$ .  Find the maximum possible distance between two sparkling points.
Answer: 2
Problem: In a directed graph with  $2013$  vertices, there is exactly one edge between any two vertices and for every vertex there exists an edge outwards this vertex. We know that whatever the arrangement of the edges, from every vertex we can reach  $k$  vertices using at most two edges. Find the maximum value of  $k$ .
Answer: k = 2012
Problem: Triangle  $ABC$  has  $AB=10$ ,  $BC=17$ , and  $CA=21$ . Point  $P$  lies on the circle with diameter  $AB$ . What is the greatest possible area of  $APC$ ?
Answer:  \frac{189}{2}
Problem: Let  $D$  be the foot of the altitude from  $A$  in a triangle  $ABC$ . The angle bisector at  $C$  intersects  $AB$  at a point  $E$ . Given that  $\angle CEA=\frac\pi4$ , compute  $\angle EDB$ .
Answer:  \frac{\pi}{4}
Problem: Find all ordered pairs  $(a,b)$  of positive integers that satisfy  $a>b$   and the equation  $(a-b)^{ab}=a^bb^a$ .
Answer: (4, 2)
Problem: Find all natural numbers  $n$  for which each natural number written with  $~$   $n-1$   $~$  'ones' and one 'seven' is prime.
Answer:  n = 1 and n = 2
Problem: Point  $M$  lies inside an isosceles right  $\triangle ABC$  with hypotenuse  $AB$  such that  $MA=5$ ,  $MB=7$ ,  $MC=4\sqrt{2}$ . Find  $\angle AMC$ .
Answer: \frac{7\pi}{12}
Problem: Given set  $S = \{ xy\left( {x + y} \right)\; |\; x,y \in \mathbb{N}\}$ .Let  $a$  and  $n$  natural numbers such that  $a+2^k\in S$  for all  $k=1,2,3,...,n$ .Find the greatest value of  $n$ .
Answer: 3
Problem: At a national math contest, students are being housed in single rooms and double rooms; it is known that  $75\%$  of the students are housed in double rooms. What percentage of the rooms occupied are double rooms?
Answer: 60\%
Problem: Let  $M$  be a finite subset of the plane such that for any two different points  $A,B\in M$  there is a point  $C\in M$  such that  $ABC$  is equilateral. What is the maximal number of points in  $M?$ 
Answer:  3
Problem: Find all prime numbers  $p$ , for which the number  $p + 1$  is equal to the product of all the prime numbers which are smaller than  $p$ .
Answer: p = 5
Problem: Determine all pairs of polynomials  $f$  and  $g$  with real coefficients such that \[ x^2 \cdot g(x) = f(g(x)). \]
Answer:  (f, g)
Problem: A Yule log is shaped like a right cylinder with height  $10$  and diameter  $5$ . Freya cuts it parallel to its bases into  $9$  right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by  $a\pi$ . Compute  $a$ .
Answer: 100
Problem: Which of the following rationals is greater ? $\frac{1995^{1994} + 1}{1995^{1995} + 1}$  or  $\frac{1995^{1995} + 1}{ 1995^{1996} +1}$ 
Answer: \frac{1995^{1994} + 1}{1995^{1995} + 1} > \frac{1995^{1995} + 1}{1995^{1996} + 1}
Problem: Suppose  $f(x)$  is a rational function such that  $3f\left(\dfrac{1}{x}\right)+\dfrac{2f(x)}{x}=x^2$  for  $x\neq 0$ .  Find  $f(-2)$ .
Answer: \frac{13}{5}
Problem: Let  $I$  be the set of points  $(x,y)$  in the Cartesian plane such that   $$ x>\left(\frac{y^4}{9}+2015\right)^{1/4} $$   Let  $f(r)$  denote the area of the intersection of  $I$  and the disk  $x^2+y^2\le r^2$  of radius  $r>0$  centered at the origin  $(0,0)$ . Determine the minimum possible real number  $L$  such that  $f(r)<Lr^2$  for all  $r>0$ .
Answer: \frac{\pi}{3}
Problem: Find all the integer pairs  $ (x,y)$  such that:
                                  \[ x^2\minus{}y^4\equal{}2009\]
Answer: (\pm 45, \pm 2)
Problem: Magicman and his helper want to do some magic trick. They have special card desk. Back of all cards is common color and face is one of  $2017$  colors.
Magic trick: magicman go away from scene. Then viewers should put on the table  $n>1$  cards in the row face up. Helper looks at these cards, then he turn all cards face down, except one, without changing order in row. Then magicman returns on the scene, looks at cards, then show on the one  card, that lays face down and names it face color. 
What is minimal  $n$  such that magicman and his helper can has strategy to make magic trick successfully?
Answer:  n = 2018
Problem: Find the sum of the squares of the values  $x$  that satisfy  $\frac{1}{x} + \frac{2}{x+3}+\frac{3}{x+6} = 1$ .
Answer: 33
Problem: The dates of three Sundays of a month were even numbers. What day of the week was the  $20$ th of the month?
Answer: Thursday
Problem: Alexander has chosen a natural number  $N>1$  and has written down in a line,and in increasing order,all his positive divisors  $d_1<d_2<\ldots <d_s$  (where  $d_1=1$  and  $d_s=N$ ).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these  $s-1$  numbers (the greatest common divisors) is equal to  $N-2$ .Find all possible values of  $N$ .
Answer: 3
Problem: Consider a unit circle with center  $O$ . Let  $P$  be a point outside the circle such that the two line segments passing through  $P$  and tangent to the circle form an angle of  $60^\circ$ . Compute the length of  $OP$ .
Answer: 2
Problem: Car A is traveling at 20 miles per hour. Car B is 1 mile behind, following at 30 miles per hour. A fast fly can move at 40 miles per hour. The fly begins on the front bumper of car B, and flies back and forth between the two cars. How many miles will the fly travel before it is crushed in the collision?
Answer: 4