rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Square  $S_1$  is inscribed inside circle  $C_1$ , which is inscribed inside square  $S_2$ , which is inscribed inside circle  $C_2$ , which is inscribed inside square  $S_3$ , which is inscribed inside circle  $C_3$ , which is inscribed inside square  $S_4$ .

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Let  $a$  be the side length of  $S_4$ , and let  $b$  be the side length of  $S_1$ . What is  $\tfrac{a}{b}$ ?
Answer: 2\sqrt{2}
Problem: Find all pairs of  $a$ ,  $b$  of positive integers satisfying the equation  $2a^2 = 3b^3$ .
Answer: (a, b) = (18k^3, 6k^2) \ \text{where} \ k \in \mathbb{N}
Problem: Except for the first two terms, each term of the sequence  $1000, x, 1000-x,\ldots$  is obtained by subtracting the preceding term from the one before that.  The last term of the sequence is the first negative term encounted.  What positive integer  $x$  produces a sequence of maximum length?
Answer: 618
Problem: Find all functions  $f:\mathbb{Z}_{>0}\mapsto\mathbb{Z}_{>0}$  such that   $$ xf(x)+(f(y))^2+2xf(y) $$   
is perfect square for all positive integers  $x,y$ . 

**This problem was proposed by me for the BMO 2017 and it was shortlisted. We then used it in our TST. 
Answer:  f(x) = x
Problem: What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?
Answer: 215
Problem: Let  $k>1$  be an integer. The sequence  $a_1,a_2, \cdots$  is defined as: $a_1=1, a_2=k$  and for all  $n>1$  we have:  $a_{n+1}-(k+1)a_n+a_{n-1}=0$ Find all positive integers  $n$  such that  $a_n$  is a power of  $k$ .
Answer:  n = 1  and  n = 2
Problem: On a typical morning Aiden gets out of bed, goes through his morning preparation, rides the bus, and walks from the bus stop to work arriving at work 120 minutes after getting out of bed. One morning Aiden got out of bed late, so he rushed through his morning preparation getting onto the bus in half the usual time, the bus ride took 25 percent longer than usual, and he ran from the bus stop to work in half the usual time it takes him to walk arriving at work 96 minutes after he got out of bed. The next morning Aiden got out of bed extra early, leisurely went through his morning preparation taking 25 percent longer than usual to get onto the bus, his bus ride took 25 percent less time than usual, and he walked slowly from the bus stop to work taking 25 percent longer than usual. How many minutes after Aiden got out of bed did he arrive at work that day?

Answer: 126
Problem: Consider  $7$ -gons inscribed in a circle such that all sides of the  $7$ -gon are of different length. Determine the maximal number of  $120^\circ$  angles in this kind of a  $7$ -gon.
Answer: 2
Problem: Anthony writes the  $(n+1)^2$  distinct positive integer divisors of  $10^n$ , each once, on a whiteboard. On a move, he may choose any two distinct numbers  $a$  and  $b$  on the board, erase them both, and write  $\gcd(a, b)$  twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made.
Answer: n^2 + n
Problem: Find all pairs of integers  $a,b$  for which there exists a polynomial  $P(x) \in \mathbb{Z}[X]$  such that product  $(x^2+ax+b)\cdot P(x)$  is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0  \] where each of  $c_0,c_1,\ldots,c_{n-1}$  is equal to  $1$  or  $-1$ .
Answer:  (a, b) \in \{ (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1) \}
Problem: Let  $T_1$  and  $T_2$  be the points of tangency of the excircles of a triangle  $ABC$  with its sides  $BC$  and  $AC$  respectively. It is known that the reflection of the incenter of  $ABC$  across the midpoint of  $AB$  lies on the circumcircle of triangle  $CT_1T_2$ . Find  $\angle BCA$ .
Answer: 90^
Problem: One can place a stone on each of the squares of a  $1999\times 1999$  board. Find the minimum number of stones that must be placed so that, for any blank square on the board, the total number of stones placed in the corresponding row and column is at least  $1999$ .
Answer: 19980001
Problem: In an exhibition there are  $100$  paintings each of which is made with exactly  $k$  colors. Find the minimum possible value of  $k$  if any  $20$  paintings have a common color but there is no color that is used in all paintings.
Answer: 21
Problem: A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime integers, find  $m+n$ .
Answer: 79
Problem: The sequence  $\{a_i\}_{i \ge 1}$  is defined by  $a_1 = 1$  and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all  $n \ge 2$ . Compute the eighth perfect square in the sequence.
Answer: 64
Problem: The altitudes of the triangle  ${ABC}$  meet in the point  ${H}$ . You know that  ${AB = CH}$ . Determine the value of the angle  $\widehat{BCA}$ . 

Answer: 45^{\circ}
Problem: A subset  $B$  of  $\{1, 2, \dots, 2017\}$  is said to have property  $T$  if any three elements of  $B$  are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property  $T$  may contain.
Answer: 1009
Problem: Find all functions  $f:\mathbb{R}\to\mathbb{R}$  which satisfy the following conditions:  $f(x+1)=f(x)+1$  and  $f(x^2)=f(x)^2.$ *Based on a problem of Romanian Masters of Mathematics*
Answer: f(x) = x
Problem: Let  $a$  and  $b$  be positive integers that satisfy  $ab-7a-11b+13=0$ . What is the minimum possible value of  $a+b$ ?
Answer: 34
Problem: A wheel is rolled without slipping through  $15$  laps on a circular race course with radius  $7$ . The wheel is perfectly circular and has radius  $5$ . After the three laps, how many revolutions around its axis has the wheel been turned through?
Answer: 21
Problem: At what smallest  $n$  is there a convex  $n$ -gon for which the sines of all angles are equal and the lengths of all sides are different?
Answer: 5
Problem: Let  $M, \alpha, \beta \in \mathbb{R} $  with  $M > 0$  and  $\alpha, \beta \in (0,1)$ . If  $R>1$  is a real number, we say that a sequence of positive real numbers  $\{ C_n \}_{n\geq 0}$  is  $R$ -*inoceronte* if  $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$  for all  $n \geq 1$ . Determine the smallest real  $R>1$  for which exists a  $R$ -*inoceronte* sequence  $ \{ C_n \}_{n\geq 0}$  such that  $\sum_{n=1}^{\infty} \beta ^n C_n^{\alpha}$  diverges.
Answer:  R = \beta^{-\frac{1}{\alpha}}
Problem: Find all positive integers    $x , y$  which are roots of the equation 

                                                   $2 x^y-y= 2005$ 
<u>Babis</u>
Answer: (x, y) = (1003, 1)
Problem: Let  $ABC$  be a triangle. For every point  $M$  belonging to segment  $BC$  we denote by  $B'$  and  $C'$  the orthogonal projections of  $M$  on the straight lines  $AC$  and  $BC$ . Find points  $M$  for which the length of segment  $B'C'$  is a minimum.
Answer:  M
Problem: Let  $G$  be the centroid of triangle  $ABC$ . Find the biggest  $\alpha$  such that there exists a triangle for which there are at least three angles among  $\angle GAB, \angle GAC, \angle GBA, \angle GBC, \angle GCA, \angle GCB$  which are  $\geq \alpha$ .
Answer: \alpha_0 = \arcsin\left(\frac{\sqrt{3}}{3}\right)
Problem: The integers  $ 1,2,...,n$  are placed in order so that each value is either strictly bigger than all the preceding values or is strictly smaller than all preceding values. In how many ways can this be done?
Answer: 2^{n-1}
Problem: Three of the  $16$  squares from a  $4 \times 4$  grid of squares are selected at random. The probability that at least one corner square of the grid is selected is  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ . $  \begin{tabular}{ | l | c | c | r| }
    \hline
     &  &  &  \hline
     &  &  &  \hline
        &  &  &  \hline
  &  &  & 
    \hline
  \end{tabular}
$ 
Answer: 45
Problem: A sequence  $ (S_n), n \geq 1$  of sets of natural numbers with  $ S_1 = \{1\}, S_2 = \{2\}$  and
\[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}.
\]
Determine  $ S_{1024}.$ 
Answer: 1024
Problem: A rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has  $20$  regular triangular faces,  $30$  square faces, and  $12$  regular pentagonal faces, as shown below. How many rotational symmetries does a rhombicosidodecahedron have?
Answer: 60
Problem: Let  $H$  be a regular hexagon of side length  $x$ . Call a hexagon in the same plane a "distortion" of  $H$  if
and only if it can be obtained from  $H$  by translating each vertex of  $H$  by a distance strictly less than  $1$ . Determine the smallest value of  $x$  for which every distortion of  $H$  is necessarily convex.
Answer: 2
Problem: Consider two circles of radius one, and let  $O$  and  $O'$  denote their centers.  Point  $M$  is selected on either circle.  If  $OO' = 2014$ , what is the largest possible area of triangle  $OMO'$ ?
Answer: 1007
Problem: We draw two lines  $(\ell_1) , (\ell_2)$  through the orthocenter  $H$  of the triangle  $ABC$  such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.
Answer:  \triangle ABC
Problem: Each two of  $n$  students, who attended an activity, have different ages. It is given that each student shook hands with at least one student, who did not shake hands with anyone younger than the other. Find all possible values of  $n$ .
Answer: n \text{ even}
Problem: Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?

Answer: \frac{\pi \sqrt{3}}{6}
Problem: The two diagonals of a quadrilateral have lengths  $12$  and  $9$ , and the two diagonals are perpendicular to each other. Find the area of the quadrilateral.
Answer: 54
Problem: Let  $\phi(n)$  be the number of positive integers less than  $n$  that are relatively prime to  $n$ , where  $n$  is a positive integer. Find all pairs of positive integers  $(m,n)$  such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]
Answer:  (2, 2)
Problem: Find a real function  $f : [0,\infty)\to \mathbb R$  such that  $f(2x+1) = 3f(x)+5$ , for all  $x$  in  $[0,\infty)$ .
Answer: f(x) = -\frac{5}{2}
Problem:  Parallelogram  $ABCD$  is given such that  $\angle ABC$  equals  $30^o$  . Let  $X$  be the foot of the perpendicular from  $A$  onto  $BC$ , and  $Y$  the foot of the perpendicular from  $C$  to  $AB$ . If  $AX = 20$  and  $CY = 22$ , find the area of the parallelogram.

Answer: 880
Problem: Let  $f(x)=(x-a)^3$ . If the sum of all  $x$  satisfying  $f(x)=f(x-a)$  is  $42$ , find  $a$ .
Answer: 14
Problem:  Find all functions  $f,g: \mathbb{R} \to \mathbb{R}$  such that satisfies  $$ f(x^2-g(y))=g(x)^2-y $$  for all  $x,y \in \mathbb{R}$ 
Answer: f(x) = g(x) = x \ \ \forall x \in \mathbb{R}
Problem: Let  $A$  and  $B$  be two sets such that  $A \cup B$  is the set of the positive integers, and  $A \cap B$  is the empty set. It is known that if two positive integers have a prime larger than  $2013$  as their difference, then one of them is in  $A$  and the other is in  $B$ . Find all the possibilities for the sets  $A$  and  $B$ .
Answer:  A contains all even numbers and B contains all odd numbers.
Problem: How many integers divide either  $2018$  or  $2019$ ? Note:  $673$  and  $1009$  are both prime.
Answer: 7
Problem: Find the area of the region bounded by a function  $y=-x^4+16x^3-78x^2+50x-2$  and the tangent line which is tangent to the curve at exactly two distinct points.

Proposed by Kunihiko Chikaya
Answer: \frac{1296}{5}
Problem: Find all postitive integers n  such that  $$ \left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot  \left\lfloor \frac{n}{4} \right\rfloor=n^2 $$  where  $\lfloor x \rfloor$  represents the largest integer less than the real number  $x$ .
Answer: 24
Problem: Determine all prime numbers  $p$  such that  $p^2 - 6$  and  $p^2 + 6$  are both prime numbers.
Answer:  p = 5
Problem: The polynomials  $P_{n}(x)$  are defined by  $P_{0}(x)=0,P_{1}(x)=x$  and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number  $n\geq 1$ , find all real numbers  $x$  satisfying the equation  $P_{n}(x)=0$ .
Answer: x = 0
Problem: Let  $n \ge 3$  be an integer. What is the largest possible number of interior angles greater than  $180^\circ$  in an  $n$ -gon in the plane, given that the  $n$ -gon does not intersect itself and all its sides have the same length?
Answer:  n-3
Problem: Let  $x_{1}$  be a positive real number and for every integer  $n \geq 1$  let  $x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}$ . 
If  $x_{5} = 43$ , what is the sum of digits of the largest prime factors of  $x_{6}$ ?
Answer: 13
Problem: A cuboid has an integer volume. Three of the faces have different areas, namely  $7, 27$ , and  $L$ . What is the smallest possible integer value for  $L$ ?
Answer: 21
Problem: Find all monotonic functions  $f: \mathbb{R} \rightarrow \mathbb{R}$  satisfying
\[f(f(x) - y) + f(x+y) = 0,\]
for every real  $x, y$ .
(Note that monotonic means that function is not increasing or not decreasing)
Answer:  f(x) = 0  or  f(x) = -x
Problem: Find the number of ordered triples of positive integers  $(a, b, c)$ , where  $a, b,c$  is a strictly increasing arithmetic progression,  $a + b + c = 2019$ , and there is a triangle with side lengths  $a, b$ , and  $c$ .
Answer: 672
Problem: There is a standard deck of  $52$  cards without jokers. The deck consists of four suits(diamond, club, heart, spade) which include thirteen cards in each. For each suit, all thirteen cards are ranked from “ $2$ ” to “ $A$ ” (i.e.  $2, 3,\ldots , Q, K, A$ ). A pair of cards is called a “*straight flush*” if these two cards belong to the same suit and their ranks are adjacent. Additionally, " $A$ " and " $2$ " are considered to be adjacent (i.e. "A" is also considered as " $1$ "). For example, spade  $A$  and spade  $2$  form a “*straight flush*”; diamond  $10$  and diamond  $Q$  are not a “*straight flush*” pair. Determine how many ways of picking thirteen cards out of the deck such that all ranks are included but no “*straight flush*” exists in them.
Answer: 3^{13} - 3
Problem: For  $\{1, 2, 3, \dots, n\}$  and each of its nonempty subsets a unique **alternating sum** is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers.  (For example, the alternating sum for  $\{1, 2, 4, 6, 9\}$  is  $9 - 6 + 4 - 2 + 1 = 6$  and for  $\{5\}$  it is simply 5.)  Find the sum of all such alternating sums for  $n = 7$ .
Answer: 448
Problem: Find all polynomial  $P(x)$  with degree  $\leq n$ and non negative coefficients such that   $$ P(x)P(\frac{1}{x})\leq P(1)^2 $$   for all positive  $x$ . Here  $n$  is a natuaral number
Answer: P(x) = a_k x^k for some k \leq n
Problem: A tree has  $10$  pounds of apples at dawn. Every afternoon, a bird comes and eats  $x$  pounds of apples. Overnight, the amount of food on the tree increases by  $10\%$ . What is the maximum value of  $x$  such that the bird can sustain itself indefinitely on the tree without the tree running out of food?
Answer: \frac{10}{11}
Problem: Let  $P$  be a point on the graph of the function  $y=x+\frac{2}{x}(x>0)$ .  $PA,PB$  are perpendicular to line  $y=x$  and  $x=0$ , respectively, the feet of perpendicular being  $A$  and  $B$ . Find the value of  $\overrightarrow{PA}\cdot \overrightarrow{PB}$ .
Answer: 0
Problem: Let  $\Omega_1$  and  $\Omega_2$  be two circles in the plane.  Suppose the common external tangent to  $\Omega_1$  and  $\Omega_2$  has length  $2017$  while their common internal tangent has length  $2009$ .  Find the product of the radii of  $\Omega_1$  and  $\Omega_2$ .
Answer: 8052
Problem: Find all injective functions  $ f : \mathbb{N} \to \mathbb{N}$  which satisfy
\[ f(f(n)) \le\frac{n \plus{} f(n)}{2}\]
for each  $ n \in \mathbb{N}$ .
Answer:  f(n) = n
Problem: Madam Mim has a deck of  $52$  cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bottom do, in fact, face up. Mim repeats this move until all cards have their backs facing up again. In total, how many moves did Mim make  $?$ 
Answer: 52
Problem: A regular tetrahedron with  $5$ -inch edges weighs  $2.5$  pounds. What is the weight in pounds of a similarly constructed regular tetrahedron that has  $6$ -inch edges? Express your answer as a decimal rounded to the nearest hundredth.
Answer: 4.32
Problem: Find all natural numbers  $ x,y$  such that  $ y| (x^{2}+1)$  and  $ x^{2}| (y^{3}+1)$ .
Answer:  (x, y) = (1, 1)
Problem: Find all prime numbers  $ p $  for which  $ 1 + p\cdot 2^{p} $  is a perfect square.

Answer:  p = 2 } and { p = 3
Problem: Given  $k\ge 2$ , for which polynomials  $P\in \mathbb{Z}[X]$  does there exist a function  $h:\mathbb{N}\rightarrow\mathbb{N}$  with  $h^{(k)}(n)=P(n)$ ?
Answer:  P(x) = x + c
Problem: Let  $ABCD$  be a regular tetrahedron with side length  $1$ . Let  $EF GH$  be another regular tetrahedron such that the volume of  $EF GH$  is  $\tfrac{1}{8}\text{-th}$  the volume of  $ABCD$ . The height of  $EF GH$  (the minimum distance from any of the vertices to its opposing face) can be written as  $\sqrt{\tfrac{a}{b}}$ , where  $a$  and  $b$  are positive coprime integers. What is  $a + b$ ?

Answer: 7
Problem: Adi the Baller is shooting hoops, and makes a shot with probability  $p$ . He keeps shooting hoops until he misses. The value of  $p$  that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as  $\frac{1}{\sqrt[b]{a}}$  for a prime  $a$  and positive integer  $b$ . Find  $a+b$ .


Proposed by Minseok Eli Park (wolfpack)
Answer: 37
Problem: There are  $24$  different pencils,  $4$  different colors, and  $6$  pencils of each color. They were given to  $6$  children in such a way that each got  $4$  pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors.

P.S. for 10 grade gives same problem with  $40$  pencils,  $10$  of each color and  $10$  children.
Answer: 5
Problem: Let  $ABC$  be a triangle with orthocenter  $H$ ; suppose  $AB=13$ ,  $BC=14$ ,  $CA=15$ . Let  $G_A$  be the centroid of triangle  $HBC$ , and define  $G_B$ ,  $G_C$  similarly. Determine the area of triangle  $G_AG_BG_C$ .
Answer: 9.33\overline{3}
Problem: For a function satisfying  $ f'(x) > 0$  for  $ a\leq x\leq b$ , let  $ F(x) \equal{} \int_a^b |f(t) \minus{} f(x)|\ dt$ . For what value of  $ x$  is  $ F(x)$  is minimized?
Answer:  x = \frac{a + b}{2}
Problem: Find all integers n greater than or equal to  $3$  such that  $\sqrt{\frac{n^2 - 5}{n + 1}}$  is a rational number.
Answer: 3
Problem: The MathMatters competition consists of 10 players  $P_1$ ,  $P_2$ ,  $\dots$ ,  $P_{10}$  competing in a ladder-style tournament.  Player  $P_{10}$  plays a game with  $P_9$ : the loser is ranked 10th, while the winner plays  $P_8$ .  The loser of that game is ranked 9th, while the winner plays  $P_7$ .  They keep repeating this process until someone plays  $P_1$ : the loser of that final game is ranked 2nd, while the winner is ranked 1st.  How many different rankings of the players are possible?
Answer: 512
Problem: A  $3$  by  $3$  determinant has three entries equal to  $2$ , three entries equal to  $5$ , and three entries equal to  $8$ . Find the maximum possible value of the determinant.
Answer: 405
Problem: Solve the system of equations for positive real numbers:  $$ \frac{1}{xy}=\frac{x}{z}+ 1,\frac{1}{yz} = \frac{y}{x} + 1, \frac{1}{zx} =\frac{z}{y}+ 1 $$  
Answer: x = y = z = \frac{1}{\sqrt{2}}
Problem: Steve needed to address a letter to  $2743$  Becker Road. He remembered the digits of the address, but he forgot the correct order of the digits, so he wrote them down in random order. The probability that Steve got exactly two of the four digits in their correct positions is  $\tfrac m n$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .
Answer: 5
Problem: Find all second degree polynomial  $d(x)=x^{2}+ax+b$  with integer coefficients, so that there exists an integer coefficient polynomial  $p(x)$  and a non-zero integer coefficient polynomial  $q(x)$  that satisfy: \[\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.\]
Answer:  d(x) = x^2 + ax + b
Problem: In a forest there are  $5$  trees  $A, B, C, D, E$  that are in that order on a straight line. At the midpoint of  $AB$  there is a daisy, at the midpoint of  $BC$  there is a rose bush, at the midpoint of  $CD$  there is a jasmine, and at the midpoint of  $DE$  there is a carnation. The distance between  $A$  and  $E$  is  $28$  m; the distance between the daisy and the carnation is  $20$  m. Calculate the distance between the rose bush and the jasmine.
Answer: 6
Problem: Let  $f(x) = \sin \frac{x}{3}+ \cos \frac{3x}{10}$  for all real  $x$ . 
Find the least natural number  $n$  such that  $f(n\pi  + x)= f(x)$  for all real  $x$ .
Answer: 60
Problem: Let  $r_1, r_2, ..., r_{2021}$  be the not necessarily real and not necessarily distinct roots of  $x^{2022} + 2021x = 2022$ . Let  $S_i = r_i^{2021}+2022r_i$  for all  $1 \le i \le 2021$ . Find  $\left|\sum^{2021}_{i=1} S_i \right| = |S_1 +S_2 +...+S_{2021}|$ .
Answer: 2021 \cdot 2022
Problem: Let  $M=\{0,1,2,\dots,2022\}$  and let  $f:M\times M\to M$  such that for any  $a,b\in M$ ,
\[f(a,f(b,a))=b\]
and  $f(x,x)\neq x$  for each  $x\in M$ . How many possible functions  $f$  are there  $\pmod{1000}$ ?
Answer: 0
Problem:  Let  $n$  be a positive integer. Mariano divides a rectangle into  $n^2$  smaller rectangles by drawing  $n-1$  vertical lines and  $n-1$  horizontal lines, parallel to the sides of the larger rectangle. On every step, Emilio picks one of the smaller rectangles and Mariano tells him its area. Find the least positive integer  $k$  for which it is possible that Emilio can do  $k$  conveniently thought steps in such a way that with the received information, he can determine the area of each one of the  $n^2$  smaller rectangles.

Answer: 2n - 1
Problem: Calculate 

\[T = \sum \frac{1}{n_1! \cdot n_2! \cdot \cdots n_{1994}! \cdot (n_2 + 2 \cdot n_3 + 3 \cdot n_4 + \ldots + 1993 \cdot n_{1994})!}\]

where the sum is taken over all 1994-tuples of the numbers  $n_1, n_2, \ldots, n_{1994} \in \mathbb{N} \cup \{0\}$  satisfying  $n_1 + 2 \cdot n_2 + 3 \cdot n_3 + \ldots + 1994 \cdot n_{1994} = 1994.$ 
Answer: \frac{1}{1994!}
Problem: In triangle  $ABC$ ,  $\angle ACB=50^{\circ}$  and  $\angle CBA=70^{\circ}$ . Let  $D$  be a foot of perpendicular from point  $A$  to side  $BC$ ,  $O$  circumcenter of  $ABC$  and  $E$  antipode of  $A$  in circumcircle  $ABC$ . Find  $\angle DAE$ 
Answer: 30^\circ
Problem: Let  $a_0 = 1$  and define the sequence  $\{a_n\}$  by \[a_{n+1} = \frac{\sqrt{3}a_n - 1}{a_n + \sqrt{3}}.\] If  $a_{2017}$  can be expressed in the form  $a+b\sqrt{c}$  in simplest radical form, compute  $a+b+c$ .
Answer: 4
Problem: Let   $$ N = \sum^{512}_{i=0}i {512 \choose i}. $$   What is the greatest integer  $a$  such that  $2^a$  is a divisor of  $N$ ?
Answer: 520