rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Find the smallest possible value of a real number  $c$  such that for any  $2012$ -degree monic polynomial
\[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\]
with real coefficients,  we can obtain a new polynomial  $Q(x)$  by multiplying some of its coefficients by  $-1$  such that every root  $z$  of  $Q(x)$  satisfies the inequality
\[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]
Answer:  c = \cot\left(\frac{\pi}{2(2011)}\right)
Problem: Let  $N$  be the number of integer sequences  $a_1, a_2, \dots, a_{2^{16}-1}$  satisfying \[0 \le a_{2k + 1} \le a_k \le a_{2k + 2} \le 1\] for all  $1 \le k \le 2^{15}-1$ . Find the number of positive integer divisors of  $N$ .
Answer: 65537
Problem: We say that an integer  $x\in\{1,\dots,102\}$  is  $\textit{square-ish}$  if there exists some integer  $n$  such that  $x\equiv n^2+n\pmod{103}$ . Compute the product of all  $\textit{square-ish}$  integers modulo  $103$ .
Answer: 0
Problem: A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get  $-1$  mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)
Answer: 501
Problem: The sequence  $\{a_n\}_{n\ge 1}$  of real numbers is defined as follows:  $$ a_1=1, \quad \text{and}\quad a_{n+1}=\frac{1}{2\lfloor a_n \rfloor -a_n+1} \quad \text{for all} \quad n\ge 1 $$  Find  $a_{2024}$ .
Answer: \frac{1}{2}
Problem: There are  $2n-1$  lamps in a row. In the beginning only the middle one is on (the  $n$ -th one), and the other ones are off. Allowed move is to take two non-adjacent lamps which are turned off such that all lamps between them are turned on and switch all of their states from on to off and vice versa. What is the maximal number of moves until the process terminates?
Answer: n-1
Problem: Sofia and Viktor are playing the following game on a  $2022 \times 2022$  board:
 - Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table;
 - Then Viktor without seeing the table, chooses a positive integer  $n$ ;
 - After that Viktor looks at the table covered with dominoes, chooses and fixes  $n$  of them;
 - Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before.
If she achieves that, she wins, otherwise Viktor wins. What is the minimum number  $n$  for which Viktor can always win, no matter the starting covering of dominoes.
Answer: 1011^2
Problem: Let  $n $  be a positive integer. Let  $f $  be a function from nonnegative integers to themselves. Let  $f (0,i)=f (i,0)=0$ ,  $f (1, 1)=n $ , and  $ f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}] $  for positive integers  $i, j$  such that  $i*j>1$ . Find the number of pairs  $(i,j) $  such that  $f (i, j) $  is an odd number.(  $[x]$  is the floor function).
Answer: n
Problem: Find all pairs of positive integers  $(a,b)$  such that  $a^2 + b^2$  divides both  $a^3 + 1$  and  $b^3 + 1$ .
Answer: (1, 1)
Problem: A convex polyhedron has  $m$  triangular faces (there can be faces of other kind too). From each vertex there are exactly 4 edges. Find the least possible value of  $m$ .
Answer: 8
Problem: Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green given that it has no red?

[i]Proposed by Matthew Weiss
Answer: \frac{2}{5}
Problem: Find the smallest positive integer  $ n$  such that  $ 107n$  has the same last two digits as  $ n$ .
Answer: 50
Problem: Right triangle  $ABC$  is inscribed in circle  $W$ .  $\angle{CAB}=65$  degrees, and  $\angle{CBA}=25$  degrees. The median from  $C$  to  $AB$  intersects  $W$  and line  $D$ . Line  $l_1$  is drawn tangent to  $W$  at  $A$ . Line  $l_2$  is drawn tangent to  $W$  at  $D$ . The lines  $l_1$  and  $l_2$  intersect at  $P$ Determine  $\angle{APD}$ 
Answer: 50^\circ
Problem: Find all  $t\in \mathbb Z$  such that: exists a function  $f:\mathbb Z^+\to \mathbb Z$  such that: $f(1997)=1998$  $\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$ 
Answer:  t = -1
Problem: Determine the smallest integer  $j$  such that it is possible to fill the fields of the table  $10\times 10$  with numbers from  $1$  to  $100$  so that every  $10$  consecutive numbers lie in some of the  $j\times j$  squares of the table.

Czech Republic
Answer: 5
Problem: In the  $xy$ -plane, for  $a>1$  denote by  $S(a)$  the area of the figure bounded by the curve  $y=(a-x)\ln x$  and the  $x$ -axis.

Find the value of integer  $n$  for which  $\lim_{a\rightarrow \infty} \frac{S(a)}{a^n\ln a}$  is non-zero real number.
Answer:  n = 2
Problem: Max has a light bulb and a defective switch. The light bulb is initially off, and on the  $n$ th time the switch is flipped, the light bulb has a  $\tfrac 1{2(n+1)^2}$  chance of changing its state (i.e. on  $\to$  off or off  $\to$  on). If Max flips the switch 100 times, find the probability the light is on at the end.
Answer: \frac{25}{101}
Problem: For how many ordered triples  $ (a,b,c)$  of positive integers are the equations  $ abc\plus{}9 \equal{} ab\plus{}bc\plus{}ca$  and  $ a\plus{}b\plus{}c \equal{} 10$  satisfied?
Answer: 4
Problem: For a bijective function  $g : R \to R$ , we say that a function  $f : R \to R$  is its superinverse if it satisfies the following identity  $(f \circ g)(x) = g^{-1}(x)$ , where  $g^{-1}$  is the inverse of  $g$ . Given  $g(x) = x^3 + 9x^2 + 27x + 81$  and  $f$  is its superinverse, find  $|f(-289)|$ .
Answer: 10
Problem: Find all triples of natural numbers  $(a,b,c)$  such that  $a$ ,  $b$  and  $c$  are in geometric progression and  $a + b + c = 111$ .
Answer:  (1, 10, 100), (100, 10, 1), (37, 37, 37)
Problem: Find all odd postive integers less than  $2007$  such that the sum of all of its positive divisors is odd.
Answer: 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849
Problem: In each square of a  $4$  by  $4$  grid, you put either a  $+1$  or a  $-1$ . If any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative. What is the minimum number of  $+1$ 's needed to be placed to be able to satisfy the conditions
Answer: 10
Problem: For how many natural numbers  $n$  between  $1$  and  $2014$  (both inclusive) is  $\frac{8n}{9999-n}$  an integer?

Answer: 1
Problem: If  $p$ ,  $q$ , and  $r$  are nonzero integers satisfying \[p^2+q^2 = r^2,\] compute the smallest possible value of  $(p+q+r)^2$ .
Answer: 4
Problem: A point  $(x,y)$  in the first quadrant lies on a line with intercepts  $(a,0)$  and  $(0,b)$ , with  $a,b > 0$ . Rectangle  $M$  has vertices  $(0,0)$ ,  $(x,0)$ ,  $(x,y)$ , and  $(0,y)$ , while rectangle  $N$  has vertices  $(x,y)$ ,  $(x,b)$ ,  $(a,b)$ , and  $(a,y)$ . What is the ratio of the area of  $M$  to that of  $N$ ?
Answer: 1
Problem: Suppose that  $a$  and  $b$  are real numbers such that  $\sin(a)+\sin(b)=1$  and  $\cos(a)+\cos(b)=\frac{3}{2}$ . If the value of  $\cos(a-b)$  can be written as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , determine  $100m+n$ .
Answer: 508
Problem: Three couples sit for a photograph in  $2$  rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible?
Answer: 96
Problem: A triangle ABC with  $\angle A = 60^o$  is given. Points  $M$  and  $N$  on  $AB$  and  $AC$  respectively are such that the circumcenter of  $ABC$  bisects segment  $MN$ . Find the ratio  $AN:MB$ .


by E.Bakaev
Answer: 1
Problem:  Determine all sequences of equal ratios of the form 

     \[
     \frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8}
     \]

     which  simultaneously satisfy the following conditions:

      $\bullet$  The set  $\{ a_1, a_2, \ldots , a_8 \}$  represents all positive divisors of  $24$ .

      $\bullet$   The common value of the ratios is a natural number.
Answer: \left\{ \frac{2}{1} = \frac{4}{2} = \frac{6}{3} = \frac{8}{4} = 2 \right\}, \left\{ \frac{3}{1} = \frac{6}{2} = \frac{12}{4} = \frac{24}{8} = 3 \right\}
Problem: Find all quadruplets (x, y, z, w) of positive real numbers that satisfy the following system: $\begin{cases}
\frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1}
x+y+z+w= 48
\end{cases}$ 
Answer: x = y = z = w = 12
Problem: Find all pairs of positive integers (m, n) for which it is possible to
paint each unit square of an m*n chessboard either black or white
in such way that, for any unit square of the board, the number
of unit squares which are painted the same color as that square
and which have at least one common vertex with it (including the
square itself) is even.
Answer: (m, n) such that at least one of m or n is even.
Problem: Find all natural numbers  $a>1$ , with the property that every prime divisor of  $a^6-1$  divides also at least one of the numbers  $a^3-1$ ,  $a^2-1$ .
Answer:  a = 2
Problem: Circles of radius 3 and 6 are externally tangent to each other and are internally tangent to a circle of radius 9. The circle of radius 9 has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.
Answer: 72
Problem: Given that  $a,b,c$  are integers with  $abc = 60$ , and that complex number  $\omega \neq 1$  satisfies  $\omega^3=1$ , find the minimum possible value of  $|a + b\omega + c\omega^2|$ .


Answer: \sqrt{3}
Problem: Let  $d$  be the greatest common divisor of  $2^{30^{10}}-2$  and  $2^{30^{45}}-2$ .  Find the remainder when  $d$  is divided by  $2013$ .
Answer: 2012
Problem: Determine all possible values of  $m+n$ , where  $m$  and  $n$  are positive integers satisfying \[\operatorname{lcm}(m,n) - \gcd(m,n) = 103.\]
Answer: 21, 105, 309
Problem: Using only a compass and a ruler, reconstruct triangle  $ABC$  given the following three points: point  $M$  the intersection of its medians, point  $I$  is the center of its inscribed circle and the point  $Q_a$  is touch point of the inscribed circle to side  $BC$ .
Answer:  \triangle ABC
Problem: A *domino* is a  $2 \times 1$  or  $1 \times  2$  tile. Determine in how many ways exactly  $n^2$  dominoes can be placed without overlapping on a  $2n \times  2n$  chessboard so that every  $2 \times  2$  square contains at least two uncovered unit squares which lie in the same row or column.
Answer: \binom{2n}{n}^2
Problem: Let  $n_0$  be the product of the first  $25$  primes. Now, choose a random divisor  $n_1$  of  $n_0$ , where a choice  $n_1$  is taken with probability proportional to  $\phi(n_1)$ . ( $\phi(m)$  is the number of integers less than  $m$  which are relatively prime to  $m$ .) Given this  $n_1$ , we let  $n_2$  be a random divisor of  $n_1$ , again chosen with probability proportional to  $\phi(n_2)$ . Compute the probability that  $n_2\equiv0\pmod{2310}$ .
Answer: \frac{256}{5929}
Problem: All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length  $s$ . Find the volume of the tetrahedron.
Answer: \frac{s^3\sqrt{2}}{12}
Problem: Let  $P$  be a  $2019-$ gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of  $P$  a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.)
Answer: 2018
Problem: What is the area of the smallest possible square that can be drawn around a regular hexagon of side length  $2$  such that the hexagon is contained entirely within the square?
Answer: 8
Problem: Find all natural two digit numbers such that when you substract by seven times the sum of its digit 
from the number you get a prime number.
Answer: 10, 31, 52, 73, 94
Problem: Let  $A$  be the set of all sequences from 0’s or 1’s with length 4. What’s the minimal number of sequences that can be chosen, so that an arbitrary sequence from  $A$  differs at most in 1 position from one of the chosen?
Answer: 4
Problem: Let  $ABCDEF$  be a regular hexagon with sidelength  $6$ , and construct squares  $ABGH$ ,  $BCIJ$ ,  $CDKL$ ,  $DEMN$ ,  $EFOP$ , and  $FAQR$  outside the hexagon. Find the perimeter of dodecagon  $HGJILKNMPORQ$ .
Answer: 72
Problem: Find all pairs   $(m,n)$  of positive odd integers, such that  $n \mid 3m+1$  and  $m \mid n^2+3$ .
Answer: (m,n) = (1,1), (43,13), (49,37)
Problem: The sequence  $ (a_n)$  satisfies  $ a_1 \equal{} 1$  and  $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$  for  $ n \geq 1$ . Let  $ k$  be the least integer greater than  $ 1$  for which  $ a_k$  is an integer. Find  $ k$ .
Answer: 41
Problem: The polynomial  $f(x)=1-x+x^2-x^3+\cdots-x^{19}+x^{20}$  is written into the form  $g(y)=a_0+a_1y+a_2y^2+\cdots+a_{20}y^{20}$ , where  $y=x-4$ , then  $a_0+a_1+\cdots+a_{20}=$ ________.
Answer: \frac{1 + 5^{21}}{6}
Problem: Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after  $t$  minutes he is at position  $(t,\sin t)$ .
 
Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way that after he has been in the forest for  $m$  minutes, his position is  $(m,\cos t)$ . 

What is the greatest distance between Alex and Farmer Tim while they are walking in these paths?
Answer: 3\sqrt{3}
Problem: Let  $XOY$  be a triangle with  $\angle XOY = 90^o$ . Let  $M$  and  $N$  be the midpoints of legs  $OX$  and  $OY$ , respectively. Suppose that  $XN = 19$  and  $YM =22$ . What is  $XY$ ?
Answer: 26
Problem: Let  $A_1$  be a midmoint of  $BC$ , and  $G$  is a centroid of the non-isosceles triangle  $\triangle ABC$ .  $GBKL$  and  $GCMN$  are the squares lying on the left with respect to rays  $GB$  and  $GC$  respectively. Let  $A_2$  be a midpoint of a segment connecting the centers of the squares  $GBKL$  and  $GCMN$ . Circumcircle of triangle  $\triangle A_{1}A_{2}G$  intersects  $BC$  at points  $A_1$  and  $X$ . Find  $\frac{A_{1}X}{XH}$ , where  $H$  is a base of altitude  $AH$  of the triangle  $\triangle ABC$ .
Answer: \frac{A_1X}{XH} = \frac{1}{2}
Problem: Find the number of ways to partition a set of  $10$  elements,  $S = \{1, 2, 3, . . . , 10\}$  into two parts; that is, the number of unordered pairs  $\{P, Q\}$  such that  $P \cup Q = S$  and  $P \cap Q = \emptyset$ .
Answer: 511
Problem: The numbers  $1, 2\ldots11$  are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.

Answer: \frac{10}{33}
Problem: The **Collaptz function** is defined as   $$ C(n) = \begin{cases} 3n - 1 & n\textrm{~odd},  \frac{n}{2} & n\textrm{~even}.\end{cases} $$   
We obtain the **Collaptz sequence** of a number by repeatedly applying the Collaptz function to that number. For example, the Collaptz sequence of  $13$  begins with  $13, 38, 19, 56, 28, \cdots$  and so on. Find the sum of the three smallest positive integers  $n$  whose Collaptz sequences do not contain  $1,$  or in other words, do not **collaptzse**.
Answer: 21
Problem: Determine all functions  $f:\mathbb R\to\mathbb R$  that satisfy equation:   $$  f(x^3+y^3) =f(x^3) + 3x^2f(x)f(y) + 3f(x)f(y)^2 + y^6f(y)  $$   
for all reals  $x,y$  
Answer: f(x) = 0
Problem: Ten birds land on a  $10$ -meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any  $x$ -meter portion of the wire, there is an  $\tfrac{x}{10}$  probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its closest neighbor?
Answer: \frac{1}{10^{10}}
Problem: Find all pairs of prime numbers  $p\,,q$  for which:
\[p^2 \mid q^3 + 1 \,\,\, \text{and} \,\,\, q^2 \mid p^6-1\]
Answer: \{(3, 2), (2, 3)\}
Problem: Evaluate  $ \int_{ \minus{} 1}^ {a^2} \frac {1}{x^2 \plus{} a^2}\ dx\ (a > 0).$ 

You may not use  $ \tan ^{ \minus{} 1} x$  or Complex Integral here.
Answer: \frac{\pi}{2a}
Problem:  $p$  parallel lines are drawn in the plane and  $q$  lines perpendicular to them are also drawn. How many rectangles are bounded by the lines?
Answer: \frac{p(p-1)q(q-1)}{4}
Problem: For every  $x \ge -\frac{1}{e}\,$ , there is a unique number  $W(x) \ge -1$  such that 
\[
  W(x) e^{W(x)} = x.
\]
The function  $W$  is called Lambert's  $W$  function.  Let  $y$  be the unique positive number such that
\[
  \frac{y}{\log_{2} y} = - \frac{3}{5} \, .   
\]
The value of  $y$  is of the form  $e^{-W(z \ln 2)}$  for some rational number  $z$ .  What is the value of  $z$ ?
Answer: \frac{5}{3}
Problem: For positive integer  $n$  we define  $f(n)$  as sum of all of its positive integer divisors (including  $1$  and  $n$ ). Find all positive integers  $c$  such that there exists strictly increasing infinite sequence of positive integers  $n_1, n_2,n_3,...$  such that for all  $i \in \mathbb{N}$  holds  $f(n_i)-n_i=c$ 
Answer: c = 1
Problem: Let  $n = 2188 = 3^7+1$  and let  $A_0^{(0)}, A_1^{(0)}, ..., A_{n-1}^{(0)}$  be the vertices of a regular  $n$ -gon (in that order) with center  $O$  . For  $i = 1, 2, \dots, 7$  and  $j=0,1,\dots,n-1$ , let  $A_j^{(i)}$  denote  the centroid of the triangle \[ \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)}. \] Here the subscripts are taken modulo  $n$ . If \[ \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \] for relatively prime positive integers  $p$  and  $q$ , find  $p+q$ .
Answer: 2188
Problem: Let  $n$  be an integer.  If the tens digit of  $n^2$  is 7, what is the units digit of  $n^2$ ?
Answer: 6
Problem: Find all functions  $f: R \to R$  such that  $f (xy) \le yf (x) + f (y)$ , for all  $x, y\in R$ .
Answer:  f(x) = 0
Problem: **Q11.** Let be given a sequense  $a_1=5, \; a_2=8$  and  $a_{n+1}=a_n+3a_{n-1}, \qquad n=1,2,3,...$  Calculate the greatest common divisor of  $a_{2011}$  and  $a_{2012}$ .
Answer: 1
Problem: Consider a triangle  $ABC$  with  $\angle ACB=120^\circ$ . Let  $A’, B’, C’$  be the points of intersection of the angular bisector through  $A$ ,  $B$  and  $C$  with the opposite side, respectively.
Determine  $\angle A’C’B’$ .
Answer: \angle A'C'B' = 90^\circ
Problem: From the identity  $$  \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2}  \log \sin x \, dx + \int_{0}^{\pi \slash 2}  \log \cos x \, dx +\int_{0}^{\pi \slash 2}  \log  2 \, dx,  $$  deduce the value of  $\int_{0}^{\pi \slash 2}  \log \sin x \, dx.$ 
Answer: -rac{	ext{π}}{2} 	ext{log} 2
Problem: The kid cut out of grid paper with the side of the cell  $1$  rectangle along the grid  lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle.
- My rectangle ... - kid sobbed. - There is something strange about this figure!
- Nonsense, do not mention it -  Carlson said - waving his hand carelessly. - Here you see, in this figure the perimeter is the same as the area of ​​the rectangle was, and the area is the same as was the perimeter!
What size square did Carlson cut out?

Answer: 2
Problem: Andrew writes down all of the prime numbers less than  $50$ . How many times does he write the digit  $2$ ?
Answer: 3
Problem: Find all pairs of non-negative integers  $m$  and  $n$  that satisfy   $$ 3 \cdot  2^m + 1 = n^2. $$  
Answer:  (m, n) = (0, 2), (3, 5), (4, 7)
Problem: Define the sequence  $x_1, x_2, ...$  inductively by  $x_1 = \sqrt{5}$  and  $x_{n+1} = x_n^2 - 2$  for each  $n \geq 1$ . Compute $\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$ .
Answer: 1
Problem: Find the maximum value of  $M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}$  ,  $x,y, z > 0$ 
Answer: 1
Problem: suppose that  $p$  is a prime number. find that smallest  $n$  such that there exists a non-abelian group  $G$  with  $|G|=p^n$ .

SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries.

the exam time was 5 hours.
Answer: n = 3
Problem: A circle and a point  $P$  higher than the circle lie in the same vertical plane. A particle moves along a straight line under gravity from  $P$  to a point  $Q$  on the circle. Given that the distance travelled from  $P$  in time  $t$  is equal to  $\dfrac{1}{2}gt^2 \sin{\alpha}$ , where  $\alpha$  is the angle of inclination of the line  $PQ$  to the horizontal, give a geometrical characterization of the point  $Q$  for which the time taken from  $P$  to  $Q$  is a minimum.
Answer:  Q
Problem: On  $xy$ plane the parabola  $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$  intersects with the line  $y=x$  at the point  $P$  that is different from the origin.
Assumed that the circle  $C$  is touched to  $K$  at  $P$  and  $y$  axis at the point  $Q.$ 
Let  $S_{1}$  be the area of the region surrounded by the line passing through two points  $P,\ Q$  and  $K,$  or  $S_{2}$  be the area of the region surrounded by the line which is passing through  $P$  and parallel to  $x$  axis and  $K.$  Find the value of  $\frac{S_{1}}{S_{2}}.$ 
Answer: 1
Problem: Arthur is driving to David’s house intending to arrive at a certain time. If he drives at 60 km/h, he will arrive 5 minutes late. If he drives at 90 km/h, he will arrive 5 minutes early. If he drives at  n  km/h, he will arrive exactly on time. What is the value of n?
Answer: 72
Problem: What is the smallest positive integer  $n$  such that there exists a choice of signs for which \[1^2\pm2^2\pm3^2\ldots\pm n^2=0\] is true?
Answer: 7
Problem: Find the number of ways in which three numbers can be selected from the set  $\{1,2,\cdots ,4n\}$ , such that the sum of the three selected numbers is divisible by  $4$ .
Answer: n^3 + 3n \binom{n}{2} + \binom{n}{3}
Problem: Let  $ m, n \geq 1$  be two coprime integers and let also  $ s$  an arbitrary integer. Determine the number of subsets  $ A$  of   $ \{1, 2, ..., m \plus{} n \minus{} 1\}$  such that  $ |A| \equal{} m$  and  $ \sum_{x \in A} x \equiv s \pmod{n}$ .
Answer: \frac{1}{n} \binom{m+n-1}{m}
Problem: Find all numbers between  $100$  and  $999$  which equal the sum of the cubes of their digits.
Answer: 153, 370, 371, 407
Problem: For a given rational number  $r$ , find all integers  $z$  such that \[2^z + 2 = r^2\mbox{.}\]

*(Swiss Mathematical Olympiad 2011, Final round, problem 7)*
Answer: (\pm 2, 1) \text{ and } \left(\pm \frac{3}{2}, -2\right)
Problem: Determine all positive integers  $k$  for which there exist a positive integer  $m$  and a set  $S$  of positive integers such that any integer  $n > m$  can be written as a sum of distinct elements of  $S$  in exactly  $k$  ways.
Answer:  k = 2^m
Problem: Five parcels of land are given. In each step, we divide one parcel into three or four smaller ones. Assume that, after several steps, the number of obtained parcels equals four times the number of steps made. How many steps were performed?
Answer: 3, 4, \text{ or } 5
Problem: An angle is considered as a point and two rays coming out of it.
Find the largest value on  $n$  such that it is possible to place  $n$   $60$  degree angles on the plane in a way that any pair of these angles have exactly  $4$  intersection points.
Answer:  n = 2