rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Let  $ f ( x , y ) = ( x + y ) / 2 , g ( x , y ) = \sqrt { x y } , h ( x , y ) = 2 x y / ( x + y ) $ , and let   $$  S = \{ ( a , b ) \in \mathrm { N } \times \mathrm { N } | a \neq b \text { and } f( a , b ) , g ( a , b ) , h ( a , b ) \in \mathrm { N } \}  $$   where  $\mathbb{N}$  denotes the positive integers. Find the minimum of  $f$  over  $S$ .
Answer: 5
Problem: Let
		\[
		S = \sum_{i = 1}^{2012} i!.
		\]
		The tens and units digits of  $S$  (in decimal notation) are  $a$  and  $b$ , respectively. Compute  $10a + b$ .
Answer: 13
Problem: Given a real number  $\alpha$ , a function  $f$  is defined on pairs of nonnegative integers by $f(0,0) = 1, f(m,0) = f(0,m) = 0$  for  $m > 0$ , $f(m,n) = \alpha f(m,n-1)+(1- \alpha)f(m -1,n-1)$  for  $m,n > 0$ .
Find the values of  $\alpha$  such that  $| f(m,n)| < 1989$  holds for any integers  $m,n \ge 0$ .
Answer: 0 < \alpha < 1
Problem:  $2$  darts are thrown randomly at a circular board with center  $O$ , such that each dart has an equal probability of hitting any point on the board. The points at which they land are marked  $A$  and  $B$ . What is the probability that  $\angle AOB$  is acute?
Answer: \frac{1}{2}
Problem: Find the smallest positive integer, such that the sum and the product of it and  $10$  are both square numbers. 
Answer: 90
Problem: Find the smallest integer  $n > 3$  such that, for each partition of  $\{3, 4,..., n\}$  in two sets, at least one of these sets contains three (not necessarily distinct) numbers  $ a,  b,  c$  for which  $ab = c$ .
Answer: 243
Problem: Find  $x$  so that the arithmetic mean of  $x, 3x, 1000$ , and  $3000$  is  $2018$ .
Answer: 1018
Problem: Right triangle  $ABC$  has right angle at  $C$  and  $\angle BAC=\theta$ ; the point  $D$  is chosen on  $AB$  so that  $|AC|=|AD|=1$ ; the point  $E$  is chosen on  $BC$  so that  $\angle CDE=\theta$ .  The perpendicular to  $BC$  at  $E$  meets  $AB$  at  $F$ .  Evaluate  $\lim_{\theta\to 0}|EF|$ .
Answer: 0
Problem: Let  $ABCD$  be a square with unit sides. Which interior point  $P$  will the expression  $\sqrt2 \cdot AP + BP + CP$  have a minimum value, and what is this minimum?
Answer: 3
Problem: Let  $a$  and  $b$  be complex numbers satisfying the two equations
\begin{align*}
a^3 - 3ab^2 & = 36 
b^3 - 3ba^2 & = 28i.
\end{align*}
Let  $M$  be the maximum possible magnitude of  $a$ . Find all  $a$  such that  $|a| = M$ .
Answer:  a = 3, \quad a = 3\omega, \quad a = 3\omega^2
Problem: The point  $D$  at the side  $AB$  of triangle  $ABC$  is given. Construct points  $E,F$  at sides  $BC, AC$  respectively such that the midpoints of  $DE$  and  $DF$  are collinear with  $B$  and the midpoints of  $DE$  and  $EF$  are collinear with  $C.$ 
Answer:  E and F as described above.
Problem: A country has  $ 1998$  airports connected by some direct flights. For any three airports, some two are not connected by a direct flight. What is the maximum number of direct flights that can be offered?
Answer: 998001
Problem: The base  $4$  repeating decimal  $0.\overline{12}_4$  can be expressed in the form  $\frac{a}{b}$  in base 10, where  $a$  and  $b$  are relatively prime positive integers. Compute the sum of  $a$  and  $b$ .
Answer: 7
Problem: Rectangle  $ABCD$  has side lengths  $AB = 6\sqrt3$  and  $BC = 8\sqrt3$ . The probability that a randomly chosen point inside the rectangle is closer to the diagonal  $\overline{AC}$  than to the outside of the rectangle is  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .
Answer: 17
Problem: For each positive natural number  $n$  let  $d (n)$  be the number of its divisors including  $1$  and  $n$ .
For which positive natural numbers  $n$ , for every divisor  $t$  of  $n$ , that  $d (t)$  is a divisor of  $d (n)$ ?
Answer:  n is a product of distinct prime numbers.
Problem: Let  $M$  and  $N$  be two palindrome numbers, each having  $9$  digits and the palindromes don't start with  $0$ . If  $N>M$  and between  $N$  and  $M$  there aren't any palindromes, find all values of  $N-M$ .
Answer: 100000011
Problem: In the triangle ABC, D is the foot of the altitude from A to BC, and M is the midpoint of the line
segment BC. The three angles ∠BAD, ∠DAM and ∠MAC are all equal. Find the angles of the
triangle ABC.

Answer: \triangle ABC is a 90^\circ-60^\circ-30^\circ triangle.
Problem: Find all natural  $ x $  for which  $ 3x+1 $  and  $ 6x-2 $  are perfect squares, and the number  $ 6x^2-1 $  is prime.
Answer: 1
Problem: Let  $ABC$  be an isosceles triangle with  $AB = BC$ . Point  $E$  lies on the side  $AB$ , and  $ED$  is the perpendicular from  $E$  to  $BC$ . It is known that  $AE = DE$ . Find  $\angle DAC$ .
Answer: 45^
Problem: Positive numbers are written in the squares of a 10 × 10 table. Frogs sit in five squares and cover the numbers in these squares. Kostya found the sum of all visible numbers and got 10. Then each frog jumped to an adjacent square and Kostya’s sum changed to  $10^2$ . Then the frogs jumped again, and the sum changed to  $10^3$  and so on: every new sum was 10 times greater than the previous one. What maximum sum can Kostya obtain?

Answer: 10^6
Problem: For which real numbers  $c$  is there a straight line that intersects the curve
\[ y = x^4 + 9x^3 + cx^2 + 9x + 4\] 
in four distinct points?
Answer:  c \leq \frac{243}{8}
Problem: Let  $ABCDEF$  be a regular hexagon with side length 10 inscribed in a circle  $\omega$ .  $X$ ,  $Y$ , and  $Z$  are points on  $\omega$  such that  $X$  is on minor arc  $AB$ ,  $Y$  is on minor arc  $CD$ , and  $Z$  is on minor arc  $EF$ , where  $X$  may coincide with  $A$  or  $B$  (and similarly for  $Y$  and  $Z$ ). Compute the square of the smallest possible area of  $XYZ$ .
Answer: 7500
Problem: We define  $\lfloor x \rfloor$  as the largest integer less than or equal to  $x$ . What is
\[ \left\lfloor \frac{5^{2017015}}{5^{2015}+7} \right\rfloor \mod 1000?\]
Answer: 0
Problem: Let  $a_1,a_2,...,a_9$  be nine real numbers, not necessarily distinct, with average  $m$ . Let  $A$  denote the number of triples  $1 \le i < j < k \le 9$  for which  $a_i + a_j + a_k \ge 3m$ .  What is the minimum possible value of  $A$ ?
Answer: 28
Problem: Compute the square of the distance between the incenter (center of the inscribed circle) and circumcenter (center of the circumscribed circle) of a 30-60-90 right triangle with hypotenuse of length 2.
Answer: 2 - \sqrt{3}
Problem: Find all positive integers that can be written in the following way  $\frac{m^2 + 20mn + n^2}{m^3 + n^3}$ Also,  $m,n$  are relatively prime positive integers.
Answer: 1 \text{ and } 5
Problem: In the triangle  $ABC$  , let  $H$  be the foot of the altitude from  $A $ . Let  $E$  be the intersection point of the angle bisector from  $B$  with the side  $AC$ . Knowing that  $\angle BEA=45^o$ , determine  $\angle EHC$ .
Answer: 45^
Problem: Find the least positive integer  $n$ , such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
      - For  $i=0,1,\dots,2n$  it is  $2014 \leq a_i \leq 2015$ .
      - There is a real number  $\xi$  with  $P(\xi)=0$ .
Answer: 2014
Problem: Let  $P(x) = x^2-1$  be a polynomial, and let  $a$  be a positive real number satisfying  $$ P(P(P(a))) = 99. $$   The value of  $a^2$  can be written as  $m+\sqrt{n}$ , where  $m$  and  $n$  are positive integers, and  $n$  is not divisible by the square of any prime. Find  $m+n$ .
Answer: 12
Problem:  $\mathbb{N}$  is the set of positive integers. Determine all functions  $f:\mathbb{N}\to\mathbb{N}$  such that for every pair  $(m,n)\in\mathbb{N}^2$  we have that:
\[f(m)+f(n) \ | \ m+n .\]
Answer: f(n) = n \text{ for all } n \in \mathbb{N}
Problem: Consider a right circular cylinder with radius  $ r$  of the base, hight  $ h$ . Find the volume of the solid by revolving the cylinder about a diameter of the base.
Answer: \pi r^2 h
Problem: Five identical circles are placed in a line inside a larger one as shown. If the shown chord has length  $16,$  find the radius of the large circle.
Answer: 8
Problem: In a basketball tournament every two teams play two matches. As usual, the winner of a match gets  $2$  points, the loser gets  $0$ , and there are no draws. A single team wins the tournament with  $26$  points and exactly two teams share the last position with  $20$  points. How many teams participated in the tournament?
Answer: 12
Problem: A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.
Answer: 1536
Problem: While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments  $AB = 10$  and  $CD = 7$  on a plane. Across all possible configurations of  those two segments, determine the maximum possible area of quadrilateral  $ACBD$ .

Answer: 35
Problem: Consider a function  $f:\mathbb{R} \to \mathbb{R}$ . Given that  $f(0)=0$ .  $\lim_{x\to0} \frac{f(h)}{h}=7$ , and  $f(x+y)=f(x)+f(y)+3xy$  for all  $x,y\in{\mathbb{R}}$ . Compute  $f(7)$ 
Answer: 122.5
Problem: Let the sequence  $(a_n)_{n \in \mathbb{N}}$ , where  $\mathbb{N}$  denote the set of natural numbers, is given with  $a_1=2$  and  $a_{n+1}$   $=$   $a_n^2$   $-$   $a_n+1$ . Find the minimum real number  $L$ , such that for every  $k$   $\in$   $\mathbb{N}$ \begin{align*} \sum_{i=1}^k \frac{1}{a_i} < L \end{align*}
Answer: 1
Problem: A trapezoid has bases with lengths equal to  $5$  and  $15$  and legs with lengths equal to  $13$  and  $13.$  Determine the area of the trapezoid.
Answer: 120
Problem:  $A,B,C$  and  $D$  are points on the parabola  $y = x^2$  such that  $AB$  and  $CD$  intersect on the  $y$ -axis. Determine the  $x$ -coordinate of  $D$  in terms of the  $x$ -coordinates of  $A,B$  and  $C$ , which are  $a, b$  and  $c$  respectively.
Answer: \frac{ab}{c}
Problem: Determine the prime numbers  $p$  for which the number  $a = 7^p - p - 16$  is a perfect square.

Lucian Petrescu
Answer: p = 3
Problem: For some  $k > 0$  the lines  $50x + ky = 1240$  and  $ky = 8x + 544$  intersect at right angles at the point  $(m,n)$ . Find  $m + n$ .
Answer: 44
Problem: A positive integer  $k$  is  $2014$ -ambiguous if the quadratics  $x^2+kx+2014$  and  $x^2+kx-2014$  both have two integer roots. Compute the number of integers which are  $2014$ -ambiguous.
Answer: 0
Problem: Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola  $ xy\equal{}1$  and both branches of the hyperbola  $ xy\equal{}\minus{}1.$  (A set  $ S$  in the plane is called *convex* if for any two points in  $ S$  the line segment connecting them is contained in  $ S.$ )
Answer: 4
Problem:  $ABCDEF GH$  is a regular octagon with  $10$  units side . The circle with center  $A$  and radius  $AC$  intersects  the circle  with center  $D$  and radius  $CD$  at point  $ I$ , different from  $C$ . What is the length of the segment  $IF$ ?
Answer: 10
Problem: Find the minimum value of  $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$ .
Answer: 1 - 2e^{-1}
Problem: Let  $a, b, c$  be three non-zero integers. It is known that the sums  $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$  and  $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$  are integers. Find these sums.
Answer: 3 \text{ or } -3
Problem: Find all functions  $f,g$ : $R \to R$  such that  $f(x+yg(x))=g(x)+xf(y)$  for  $x,y \in R$ .
Answer:  f(x) = x
Problem: Let  $\|x\|_*=(|x|+|x-1|-1)/2$ . Find all  $f:\mathbb{N}\to\mathbb{N}$  such that 
\[f^{(\|f(x)-x\|_*)}(x)=x, \quad\forall x\in\mathbb{N}.\]
Here  $f^{(0)}(x)=x$  and  $f^{(n)}(x)=f(f^{(n-1)}(x))$  for all  $n\in\mathbb{N}$ .
Answer:  f(x) \in \{x, x+1\} \ \ \forall x \in \mathbb{N}
Problem: Let  $ABC$  be an equilateral triangle with side length  $1$ . This triangle is rotated by some angle about its center to form triangle  $DEF.$  The intersection of  $ABC$  and  $DEF$  is an equilateral hexagon with an area that is  $\frac{4} {5}$  the area of  $ABC.$  The side length of this hexagon can be expressed in the form  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. What is  $m+n$ ?
Answer: 7
Problem: A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was  $7/9$ . How many players took part at the tournament?
Answer: 33
Problem: Place the numbers  $ 1, 2, 3, 4, 5, 6, 7, 8$  at the vertices of a cuboid such that the sum of any  $ 3$  numbers on a side is not less than  $ 10$ .  Find the smallest possible sum of the 4 numbers on a side.
Answer: 15
Problem: Find all positive integers  $N$  such that the following holds: There exist pairwise coprime positive integers  $a,b,c$  with  $$ \frac1a+\frac1b+\frac1c=\frac N{a+b+c}. $$  
Answer: 9, 10, 11
Problem: Yannick is playing a game with  $100$  rounds, starting with  $1$  coin. During each round, there is an  $n\%$  chance that he gains an extra coin, where  $n$  is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game?
Answer: \left(\frac{101}{100}\right)^{100}
Problem: Let  $f(x) = x^3 - 3x + b$  and  $g(x) = x^2 + bx -3$ , where  $b$  is a real number. What is the sum of all possible values of  $b$  for which the equations  $f(x)$  = 0 and  $g(x) = 0$  have a common root?
Answer: 0
Problem: Points  $K$  and  $L$  are taken on the sides  $BC$  and  $CD$  of a square  $ABCD$  so that  $\angle AKB = \angle AKL$ . Find  $\angle KAL$ .
Answer: 45^	extcirc}
Problem: A finite set of integers is called *bad* if its elements add up to  $2010$ . A finite set of integers is a *Benelux-set* if none of its subsets is bad. Determine the smallest positive integer  $n$  such that the set  $\{502, 503, 504, . . . , 2009\}$  can be partitioned into  $n$  Benelux-sets.
(A partition of a set  $S$  into  $n$  subsets is a collection of  $n$  pairwise disjoint subsets of  $S$ , the union of which equals  $S$ .)

*(2nd Benelux Mathematical Olympiad 2010, Problem 1)*
Answer:  n = 2
Problem: Darwin takes an  $11\times 11$  grid of lattice points and connects every pair of points that are 1 unit apart, creating a  $10\times 10$  grid of unit squares. If he never retraced any segment, what is the total length of all segments that he drew?

*Ray Li.*

<details><summary>Clarifications</summary>[list=1][*]The problem asks for the total length of all *unit* segments (with two lattice points in the grid as endpoints) he drew.[/list]</details>
Answer: 220
Problem: Given are  $n$   $(n > 2)$  points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops?
Answer: \binom{n}{2}
Problem: Is the number \[
\left( 1 + \frac12 \right) \left( 1 + \frac14 \right) \left( 1 + \frac16 \right)\cdots\left( 1 + \frac{1}{2018} \right)
\] greater than, less than, or equal to  $50$ ?
Answer: < 50
Problem: The mean of  $\frac12 , \frac34$  , and  $\frac56$  differs from the mean of  $\frac78$  and  $\frac{9}{10}$  by  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .
Answer: 859
Problem: For  $n\geq 2$  , an equilateral triangle is divided into  $n^2$  congruent smaller equilateral triangles. Detemine all ways in which real numbers can be assigned to the  $\frac{(n+1)(n+2)}{2}$  vertices so that three such numbers sum to zero whenever the three vertices form a triangle with edges parallel to the sides of the big triangle.
Answer: 0
Problem: The set  $M= \{1;2;3;\ldots ; 29;30\}$  is  divided in  $k$ subsets such that if  $a+b=n^2, (a,b \in M, a\neq b, n$  is an
integer number  $)$ , then  $a$  and  $b$  belong different subsets.
Determine the minimum value of  $k$ .
Answer: 3
Problem: Let  $ABC$  be a triangle such that  $AB=6,BC=5,AC=7.$  Let the tangents to the circumcircle of  $ABC$  at  $B$  and  $C$  meet at  $X.$  Let  $Z$  be a point on the circumcircle of  $ABC.$  Let  $Y$  be the foot of the perpendicular from  $X$  to  $CZ.$  Let  $K$  be the intersection of the circumcircle of  $BCY$  with line  $AB.$  Given that  $Y$  is on the interior of segment  $CZ$  and  $YZ=3CY,$  compute  $AK.$ 
Answer: \frac{147}{10}
Problem: Find all function  $f:\mathbb{N} \longrightarrow \mathbb{N}$  such that

forall positive integers  $x$  and  $y$ ,  $\frac{f(x+y)-f(x)}{f(y)}$   is again a positive integer not exceeding  $2022^{2022}$ .
Answer: f(n) = n
Problem: Let  $G=10^{10^{100}}$  (a.k.a. a googolplex). Then \[\log_{\left(\log_{\left(\log_{10} G\right)} G\right)} G\] can be expressed in the form  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Determine the sum of the digits of  $m+n$ .
Answer: 18
Problem: Let  $ABCD$  be a tetrahedron. Let  $a$  be the length of  $AB$  and let  $S$  be the area of the projection of the tetrahedron onto a plane perpendicular to  $AB$ . Determine the volume of the tetrahedron in terms of  $a$  and  $S$ .
Answer: \frac{1}{3} S a
Problem: Determine the biggest possible value of  $ m$  for which the equation  $ 2005x \plus{} 2007y \equal{} m$  has unique solution in natural numbers.
Answer: 2 \cdot 2005 \cdot 2007
Problem: Let  $S$  denote the sum of the 2011th powers of the roots of the polynomial  $(x-2^0)(x-2^1) \cdots (x-2^{2010}) - 1$ . How many ones are in the binary expansion of  $S$ ?
Answer: 2011
Problem: Find all the pairs of positive numbers such that the last
digit of their sum is 3, their difference is a primer number and
their product is a perfect square.
Answer:  (9, 4)
Problem: 
	 For positive integers  $n$ , let  $S_n$  be the set of integers  $x$  such that  $n$  distinct lines, no three concurrent, can divide a plane into  $x$  regions (for example,  $S_2=\{3,4\}$ , because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum  $i$  such that  $S_i$  contains at least 4 elements?

Answer: 4
Problem: Find all pairs  $(p,q)$  of positive primes such that the equation  $3x^2 - px + q = 0$  has two distinct rational roots.
Answer: (p, q) = (5, 2), (7, 2)
Problem: In quadrilateral  $ABCD$ , we have  $AB = 5$ ,  $BC = 6$ ,  $CD = 5$ ,  $DA = 4$ , and  $\angle ABC = 90^\circ$ . Let  $AC$  and  $BD$  meet at  $E$ . Compute  $\dfrac{BE}{ED}$ .
Answer: \frac{5}{4}
Problem: Let  $a,b,c>0$  such that  $a \geq bc^2$  ,  $b \geq ca^2$  and  $c \geq ab^2$  . Find the maximum value that the expression :  $$ E=abc(a-bc^2)(b-ca^2)(c-ab^2) $$   can acheive.
Answer: 0
Problem: A binary string is a string consisting of only 0’s and 1’s (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0’s? Express your answer as a fraction.
Answer: \frac{55}{64}
Problem: Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?
Answer: 5
Problem: Deﬁne a number to be  $boring$  if all the digits of the number are the same. How many positive integers less than  $10000$  are both prime and boring?
Answer: 1
Problem: Determine all triples  $(x, y, z)$  of nonnegative real numbers that verify the following system of equations:  $$ x^2 - y = (z -1)^2  $$     $$ y^2 - z = (x -1)^2 $$    $$ z^2 - x = (y - 1)^2 $$  
Answer: (1, 1, 1)
Problem: For any positive integer  $k$ , denote the sum of digits of  $k$  in its decimal representation by  $S(k)$ . Find all polynomials  $P(x)$  with integer coefficients such that for any positive integer  $n \geq 2016$ , the integer  $P(n)$  is positive and   $$ S(P(n)) = P(S(n)). $$
Answer: P(x) = x
Problem: Radii of five concentric circles  $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$  form a geometric progression with common ratio  $q$  in this order. What is the maximal value of  $q$  for which it's possible to draw a broken line  $A_0A_1A_2A_3A_4$  consisting of four equal segments such that  $A_i$  lies on  $\omega_i$  for every  $i=\overline{0,4}$ ?

<details><summary>thanks </summary>Thanks to the user Vlados021 for translating the problem.</details>
Answer: \frac{1 + \sqrt{5}}{2}
Problem: In a sphere  $S_0$  we radius  $r$  a cube  $K_0$  has been inscribed. Then in the cube  $K_0$  another sphere  $S_1$  has been inscribed and so on to infinity. Calculate the volume of all spheres created in this way.
Answer: 2 \pi r^3
Problem: Find all quadruples of real numbers  $(a,b,c,d)$  such that the equalities 
\[X^2 + a X + b = (X-a)(X-c) \text{ and } X^2 + c X + d = (X-b)(X-d)\]
hold for all real numbers  $X$ .
Answer: (-1, -2, 2, 0), (0, 0, 0, 0)
Problem: For a positive integer  $a$ , define a sequence of integers  $x_1,x_2,\ldots$  by letting  $x_1=a$  and  $x_{n+1}=2x_n+1$  for  $n\geq 1$ . Let  $y_n=2^{x_n}-1$ . Determine the largest possible  $k$  such that, for some positive integer  $a$ , the numbers  $y_1,\ldots,y_k$  are all prime.
Answer: k = 2
Problem: Let  $(a_n)$  be defined by  $a_1=a_2=1$  and  $a_n=a_{n-1}+a_{n-2}$  for  $n>2$ . Compute the sum  $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$ .

Answer: 2
Problem: In three cisterns of milk lies  $780$  litres of milk. When we pour off from first cistern quarter of milk, from second cistern fifth of milk and from third cistern  $\frac{3}{7}$  of milk, in all cisterns remain same amount of milk. How many milk is in cisterns?
Answer:  x = 240 , y = 225 , z = 315
Problem: The point  $M$  moves such that the sum of squares of the lengths from  $M$  to faces of a cube, is fixed. Find the locus of  $M.$ 
Answer: The locus of M is a sphere centered at \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right).
Problem: Dan the dog spots Cate the cat 50m away. At that instant, Cate begins running away from Dan at 6 m/s, and Dan begins running toward Cate at 8 m/s. Both of them accelerate instantaneously and run in straight lines. Compute the number of seconds it takes for Dan to reach Cate.
Answer: 25 \, \text{seconds}
Problem: Find all positive integers  $a,b,c$  satisfying  $(a,b)=(b,c)=(c,a)=1$  and \[ \begin{cases} a^2+b\mid b^2+c b^2+c\mid c^2+a \end{cases} \] and none of prime divisors of  $a^2+b$  are congruent to  $1$  modulo  $7$ 
Answer:  (1, 1, 1)
Problem: Find all quadrupels  $(a, b, c, d)$  of positive real numbers that satisfy the following two equations:
\begin{align*}
ab + cd &= 8,
abcd &= 8 + a + b + c + d.
\end{align*}

Answer: (2, 2, 2, 2)
Problem: Find all triples  $(p, q, r)$  of prime numbers for which  $4q - 1$  is a prime number and   $$ \frac{p + q}{p + r} = r - p $$  holds.
Answer:  (2, 3, 3)
Problem: Zadam Heng bets Saniel Dun that he can win in a free throw contest. Zadam shoots until he has made  $5$  shots. He wins if this takes him  $9$  or fewer attempts. The probability that Zadam makes any given attempt is  $\frac{1}{2}$ . What is the probability that Zadam Heng wins the bet?
Answer: \frac{1}{2}
Problem: You have a  $3 \times 2021$  chessboard from which one corner square has been removed. You also have a set of  $3031$  identical dominoes, each of which can cover two adjacent chessboard squares. Let  $m$  be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps.
What is the remainder when  $m$  is divided by  $19$ ?
Answer: 0
Problem: Find the least  $n\in N$  such that among any  $n$  rays in space sharing a common origin there exist two which form an acute angle.
Answer: 7
Problem: Put  $1,2,....,2018$  (2018 numbers) in a row randomly and call this number  $A$ . Find the remainder of  $A$  divided by  $3$ .
Answer: 0
Problem: Deﬁne the sequence of positive integers  $a_n$  recursively by  $a_1=7$  and  $a_n=7^{a_{n-1}}$  for all  $n\geq 2$ .  Determine the last two digits of  $a_{2007}$ .
Answer: 43