rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: A palace has  $32$  rooms and  $40$  corridors. As shown in the figure below, each room is represented by a dot, and each corridor is represented by a segment connecting the rooms. Put  $n$  robots in these rooms, such that there is at most one robot in each room. Each robot is assigned to a corridor connected to its room. And the following condition is satisfied:  $\bullet$  Let all robots move along the assigned corridors at the same time. They will arrive at the rooms at another ends of the corridors at the same time. During this process, any two robots will not meet each other. And each robot will arrive at a different room in the end. 

Assume that the maximum value of the positive integer  $n$  we can take as above is  $N$ . How many ways to put  $N$  robots in the palace and assign the corridors to them satisfying the above condition? 

Remark: Any two robot are not distinguished from each other. 
Answer: \binom{32}{16}
Problem: Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Kat will win?
Answer: \frac{1}{7}
Problem: How many positive integers a less than  $100$  such that  $4a^2 + 3a + 5$  is divisible by  $6$ .
Answer: 32
Problem: Seyed has 998 white coins, a red coin, and an unusual coin with one red side and one white side. He can not see the color of the coins instead he has a scanner which checks if all of the coin sides touching the scanner glass are white. Is there any algorithm to find the red coin by using the scanner at most 17 times?
Answer:  f(1000) \leq 17
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: In the future, each country in the world produces its Olympic athletes via cloning and strict training
programs. Therefore, in the finals of the 200 m free, there are two indistinguishable athletes from each
of the four countries. How many ways are there to arrange them into eight lanes?
Answer: 2520
Problem: Let  $S$  be a real number. It is known that however we choose several numbers from the interval  $(0, 1]$  with sum equal to  $S$ , these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5. 

Find the greatest possible value of  $S$ . 
Answer: 5.5
Problem: Find all rational numbers  $k$  such that  $0 \le k \le \frac{1}{2}$  and  $\cos k \pi$  is rational.
Answer:  k = 0, \frac{1}{2}, \frac{1}{3}
Problem: Find all odd prime numbers  $p$  for which there exists a natural number  $g$  for which the sets \[A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}\] and \[B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}\] are equal.
Answer:  p = 3
Problem: The diagonals  $AC$  and  $CE$  of the regular hexagon  $ABCDEF$  are divided by inner points  $M$  and  $N$  respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine  $r$  if  $B,M$  and  $N$  are collinear.
Answer: r = \frac{\sqrt{3}}{3}
Problem: Consider  a sphere  $S$  of radius   $R$  tangent to a plane  $\pi$ . Eight other spheres of the same radius  $r$  are tangent to  $\pi$  and tangent to  $S$ , they form a "collar" in which each is tangent to its two neighbors. Express  $r$  in terms of  $R$ .
Answer:  r = R(2 - \\sqrt{2})
Problem: Suppose that  $n$  is a positive integer and that  $a$  is the integer equal to  $\frac{10^{2n}-1}{3\left(10^n+1\right)}.$ If the sum of the digits of  $a$  is 567, what is the value of  $n$ ?
Answer: 189
Problem: Define the sequences  $ a_n, b_n, c_n$  as follows.  $ a_0 \equal{} k, b_0 \equal{} 4, c_0 \equal{} 1$ .
 If  $ a_n$  is even then  $ a_{n \plus{} 1} \equal{} \frac {a_n}{2}$ ,  $ b_{n \plus{} 1} \equal{} 2b_n$ ,  $ c_{n \plus{} 1} \equal{} c_n$ . 
If  $ a_n$  is odd, then  $ a_{n \plus{} 1} \equal{} a_n \minus{} \frac {b_n}{2} \minus{} c_n$ ,  $ b_{n \plus{} 1} \equal{} b_n$ ,  $ c_{n \plus{} 1} \equal{} b_n \plus{} c_n$ . 
Find the number of positive integers  $ k < 1995$  such that some  $ a_n \equal{} 0$ .
Answer: u(2u + 1)
Problem:  Find all functions  $f : Z^+ \to Z^+$  such that  $n^3 - n^2 \le  f(n) \cdot (f(f(n)))^2 \le n^3 + n^2$  for every  $n$  in positive integers
Answer:  f(n) \in \{n-1, n, n+1\}
Problem: The area and the perimeter of the triangle with sides  $10,8,6$  are equal. Find all the triangles with integral sides whose area and perimeter are equal.
Answer: \{6, 25, 29\}, \{7, 15, 20\}, \{9, 10, 17\}, \{5, 12, 13\}, \{6, 8, 10\}
Problem: Adam places down cards one at a time from a standard 52 card deck (without replacement) in a pile. Each time he places a card, he gets points equal to the number of cards in a row immediately before his current card that are all the same suit as the current card. For instance, if there are currently two hearts on the top of the pile (and the third card in the pile is not hearts), then placing a heart would be worth 2 points, and placing a card of any other suit would be worth 0 points. What is the expected number of points Adam will have after placing all 52 cards?
Answer: \frac{624}{41}
Problem: Let  $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$ , $I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$ Find  $\lim_{n\to\infty} I_n.$ *2009 Tokyo Institute of Technology, Master Course in Mathematics*
Answer: 0
Problem: Alice and Bob play a game together as a team on a  $100 \times 100$  board with all unit squares initially white. Alice sets up the game by coloring exactly  $k$  of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least  $10$  red squares and color all of the remaining squares in it red. What is the
smallest  $k$  such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?

Proposed by *Nikola Velov, Macedonia*
Answer: 100
Problem: King Midas spent  $\frac{100}{x}\%$  of his gold deposit yesterday. He is set to earn gold today. What percentage
of the amount of gold King Midas currently has would he need to earn today to end up with as much
gold as he started?
Answer: \frac{100}{x-1}\%
Problem: There are  $2^{2n+1}$  towns with  $2n+1$  companies and each two towns are connected with airlines from one of the companies. What’s the greatest number  $k$  with the following property:
We can close  $k$  of the companies and their airlines in such way that we can still reach each town from any other (connected graph).
Answer: k = n
Problem: Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?
Answer: \frac{6480}{2401}
Problem: Let  $ f, g: \mathbb {R} \to \mathbb {R} $  function such that  $ f (x + g (y)) = - x + y + 1 $  for each pair of real numbers  $ x $  e  $ y $ . What is the value of  $ g (x + f (y) $ ?
Answer: -x + y - 1
Problem: Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart.  A tall circular building 100 feet in diameter is centered midway between the paths.  At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart.  Let  $t$  be the amount of time, in seconds, before Jenny and Kenny can see each other again.  If  $t$  is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Answer: 163
Problem: A dog has three trainers:



- The first trainer gives him a treat right away.
- The second trainer makes him jump five times, then gives him a treat.
- The third trainer makes him jump three times, then gives him no treat.

The dog will keep picking trainers with equal probability until he gets a treat. (The dog's memory isn't so good, so he might pick the third trainer repeatedly!) What is the expected number of times the dog will jump before getting a treat?
Answer: 8
Problem: In a year that has  $365$  days, what is the maximum number of "Tuesday the  $13$ th" there can be?

Note: The months of April, June, September and November have  $30$  days each, February has  $28$  and all others have  $31$  days.
Answer: 3
Problem: A regular  $2008$ -gon is located in the Cartesian plane such that  $(x_1,y_1)=(p,0)$  and  $(x_{1005},y_{1005})=(p+2,0)$ , where   $p$  is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\]
are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2007}+y_{2007}i),T&=(y_2+x_2i)(y_4+x_4i)(y_6+x_6i)\cdots(y_{2008}+x_{2008}i).\end{align*} Find the minimum possible value of  $|S-T|$ .
Answer: 0
Problem: 15. Let  $a_{n}$  denote the number of ternary strings of length  $n$  so that there does not exist a  $k<n$  such that the first  $k$  digits of the string equals the last  $k$  digits. What is the largest integer  $m$  such that  $3^{m} \mid a_{2023}$  ?
Answer: 2022
Problem: Given real numbers  $x,y,z,t\in (0,\pi /2]$  such that   $$ \cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1. $$  What is the minimum possible value of   $$ \cot (x) +\cot (y) +\cot (z) +\cot (t)? $$  
Answer: 4
Problem: Let  $n\geq 2$  and  $1 \leq r \leq n$ . Consider the set  $S_r=(A \in M_n(\mathbb{Z}_2), rankA=r)$ . Compute the sum  $\sum_{X \in S_r}X$ 
Answer:  O_n
Problem: Determine all pairs of positive integers  $(x, y)$  such that:  $$ x^4 - 6x^2 + 1 = 7\cdot 2^y $$  
Answer: (x, y) = (3, 2)
Problem: Solve the system of equations:
 $
\begin{matrix} 
x^2 + x - 1 = y  
y^2 + y - 1 = z  
z^2 + z - 1 = x.
\end{matrix}
$ 
Answer:  x = y = z = 1  and  x = y = z = -1
Problem: The radius  $r$  of a circle with center at the origin is an odd integer. 
There is a point ( $p^m, q^n$ ) on the circle, with  $p,q$  prime numbers and  $m,n$  positive integers. 
Determine  $r$ .
Answer: 5
Problem: The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table?
Answer: 8
Problem: A positive integer  $k$  is called *powerful* if there are distinct positive integers  $p, q, r, s, t$  such that  $p^2$ ,  $q^3$ ,  $r^5$ ,  $s^7$ ,  $t^{11}$  all divide k. Find the smallest powerful integer. 
Answer: 2^{34}
Problem: On a circumference, points  $A$  and  $B$  are on opposite arcs of diameter  $CD$ . Line segments  $CE$  and  $DF$  are perpendicular to  $AB$  such that  $A-E-F-B$  (i.e.,  $A$ ,  $E$ ,  $F$  and  $B$  are collinear on this order). Knowing  $AE=1$ , find the length of  $BF$ .
Answer: 1
Problem: Let  $\triangle ABC$  have  $M$  as the midpoint of  $BC$  and let  $P$  and  $Q$  be the feet of the altitudes from  $M$  to  $AB$  and  $AC$  respectively. Find  $\angle BAC$  if  $[MPQ]=\frac{1}{4}[ABC]$  and  $P$  and  $Q$  lie on the segments  $AB$  and  $AC$ .
Answer: 90^\circ
Problem: On the ellipse  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$  find the point  $T=(x_0,y_0)$  such that the triangle bounded by the axes of the ellipse and the tangent at that point has the least area.
Answer: \left( \pm \frac{a}{\sqrt{2}}, \pm \frac{b}{\sqrt{2}} \right)
Problem: Source: 2017 Canadian Open Math Challenge, Problem B2
-----
There are twenty people in a room, with  $a$  men and  $b$  women. Each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman. The total number of handshakes is  $106$ . Determine the value of  $a \cdot b$ .
Answer: 84
Problem: Let  $S$  be the set of all ordered triples  $(p,q,r)$  of prime numbers for which at least one rational number  $x$  satisfies  $px^2+qx+r=0.$  Which primes appear in seven or more elements of  $S?$ 
Answer: 2
Problem: Since counting the numbers from 1 to 100 wasn't enough to stymie Gauss, his teacher devised another clever problem that he was sure would stump Gauss. Defining  $\zeta_{15} = e^{2\pi i/15}$  where  $i = \sqrt{-1}$ , the teacher wrote the 15 complex numbers  $\zeta_{15}^k$  for integer  $0 \le k < 15$  on the board. Then, he told Gauss:

On every turn, erase two random numbers  $a, b$ , chosen uniformly randomly, from the board and then write the term  $2ab - a - b + 1$  on the board instead. Repeat this until you have one number left. What is the expected value of the last number remaining on the board?
Answer: 0
Problem: Find all functions  $f : \mathbb{N} \rightarrow \mathbb{R}$  such that for all triples  $a,b,c$  of positive integers the following holds :  $$ f(ac)+f(bc)-f(c)f(ab) \ge 1 $$  Proposed by *Mojtaba Zare*
Answer: f \equiv 1
Problem: Find the largest possible value of  $a$  such that there exist real numbers  $b,c>1$  such that
        \[a^{\log_b c}\cdot b^{\log_c a}=2023.\]
Answer: \sqrt{2023}
Problem: The set of positive real numbers  $x$  that satisfy  $2 | x^2 - 9 | \le 9 | x | $  is an interval  $[m, M]$ . Find  $10m + M$ .
Answer: 21
Problem: On Day  $1$ , Alice starts with the number  $a_1=5$ . For all positive integers  $n>1$ , on Day  $n$ , Alice randomly selects a positive integer  $a_n$  between  $a_{n-1}$  and  $2a_{n-1}$ , inclusive. Given that the probability that all of  $a_2,a_3,\ldots,a_7$  are odd can be written as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $m+n$ .
Answer: 65
Problem: Find all the values that can take the last digit of a "perfect" even number. (The natural number  $n$  is called "perfect" if the sum of all its natural divisors is equal twice the number itself.For example: the number  $6$  is perfect ,because  $1+2+3+6=2\cdot6$ ).
Answer: 6 \text{ and } 8
Problem: For her daughter’s  $12\text{th}$  birthday, Ingrid decides to bake a dodecagon pie in celebration. Unfortunately, the store does not sell dodecagon shaped pie pans, so Ingrid bakes a circular pie first and then trims off the sides in a way such that she gets the largest regular dodecagon possible. If the original pie was  $8$  inches in diameter, the area of pie that she has to trim off can be represented in square inches as  $a\pi - b$  where  $a, b$  are integers. What is  $a + b$ ?
Answer: 64
Problem: Find the smallest positive integer  $n$  for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit  $0$  when written in base ten.
Answer: 4
Problem: Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is  $2005$ . What is the number that goes into the leftmost box?

[asy]
size(300);
label("999",(2.5,.5));
label("888",(7.5,.5));

draw((0,0)--(9,0));
draw((0,1)--(9,1));
for (int i=0; i<=9; ++i)
{
draw((i,0)--(i,1));
}
[/asy]
Answer: 118
Problem: In square  $ABCD$ ,  $\overline{AC}$  and  $\overline{BD}$  meet at point  $E$ .
Point  $F$  is on  $\overline{CD}$  and  $\angle CAF = \angle FAD$ .
If  $\overline{AF}$  meets  $\overline{ED}$  at point  $G$ , and if  $\overline{EG} = 24$  cm, then find the length of  $\overline{CF}$ .
Answer: 48
Problem: 
	 Let  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$  be integers selected from the set  $\{1,2,\dots,100\}$ , uniformly and at random with replacement. Set \[ M = a + 2b + 4c + 8d + 16e + 32f. \] What is the expected value of the remainder when  $M$  is divided by  $64$ ?
Answer: \frac{63}{2}
Problem: Find all functions  $f : \mathbb{N} \rightarrow \mathbb{N}$  such that  $f(2m+2n)=f(m)f(n)$  for all natural numbers  $m,n$ .
Answer: f(x) = 1 \text{ for all } x \in \mathbb{N}
Problem: Find all functions  $ f:  N  \rightarrow N $  such that: $\bullet$   $ f (m) = 1 \iff m = 1 $ ; $\bullet$  If  $ d = \gcd (m, n) $ , then  $ f (mn) = \frac {f (m) f (n)} {f (d)} $ ; and $\bullet$    $ \forall m \in N  $ , we have  $ f ^ {2012} (m) = m $ .

 Clarification:  $f^n (a) = f (f^{n-1} (a))$ 
Answer:  f(m) = m
Problem: Big Welk writes the letters of the alphabet in order, and starts again at  $A$  each time he gets to  $Z.$  What is the  $4^3$ -rd letter that he writes down?
Answer: L
Problem: Find the smallest  $\lambda \in \mathbb{R}$  such that for all  $n \in \mathbb{N}_+$ , there exists  $x_1, x_2, \ldots, x_n$  satisfying  $n = x_1 x_2 \ldots x_{2023}$ , where  $x_i$  is either a prime or a positive integer not exceeding  $n^\lambda$  for all  $i \in \left\{ 1,2, \ldots, 2023 \right\}$ .
Answer: \frac{2}{2024}
Problem: Find all functions  $f : \mathbb{Z}\rightarrow \mathbb{Z}$  such that for all prime  $p$  the following condition holds:  $$ p \mid ab + bc + ca \iff p \mid f(a)f(b) + f(b)f(c) + f(c)f(a) $$
Answer:  f(x) = \pm x
Problem: Find all real numbers $ x$  and  $y$  such that   $$ x^2 + y^2 = 2 $$    $$ \frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2. $$  
Answer: (1, 1)
Problem: Let  $N$  be the positive integer with 1998 decimal digits, all of them 1; that is,
\[N=1111\cdots 11.\]
Find the thousandth digit after the decimal point of  $\sqrt N$ .
Answer: 3
Problem: Find all functions  $f:\mathbb{R}\to\mathbb{R}$  such that
                          $f(f(x)f(1-x))=f(x)$     and      $f(f(x))=1-f(x)$ ,
for all real  $x$ .
Answer: f(x) = \frac{1}{2}
Problem: Find the number of permutations of the letters  $ABCDE$  where the letters  $A$  and  $B$  are not adjacent and the letters  $C$  and  $D$  are not adjacent. For example, count the permutations  $ACBDE$  and  $DEBCA$  but not  $ABCED$  or  $EDCBA$ .
Answer: 48
Problem: A rectangular prism with dimensions  $20$  cm by  $1$  cm by  $7$  cm is made with blue  $1$  cm unit cubes. The outside of the rectangular prism is coated in gold paint. If a cube is chosen at random and rolled, what is the probability that the side facing up is painted gold?
Answer: \frac{9}{35}
Problem: A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer  $n \ge 3$ , determine the largest integer  $m$  such that no planar  $n$ -gon splits into less than  $m$  triangles.
Answer: \lceil \frac{n}{3} \rceil
Problem: Let  $ S \equal{} \{1,2,\ldots,2008\}$ . For any nonempty subset  $ A\in S$ , define  $ m(A)$  to be the median of  $ A$  (when  $ A$  has an even number of elements,  $ m(A)$  is the average of the middle two elements). Determine the average of  $ m(A)$ , when  $ A$  is taken over all nonempty subsets of  $ S$ .
Answer: \frac{2009}{2}
Problem:  $p > 3$   is a prime. Find all integers  $a$ ,  $b$ , such that  $a^2 + 3ab + 2p(a+b) + p^2 = 0$ .
Answer: (a, b) = (-p, 0)
Problem: Compute the remainder when the largest integer below  $\frac{3^{123}}{5}$  is divided by  $16$ .
Answer: 5
Problem: There is the sequence of numbers  $1, a_2, a_3, ...$  such that satisfies  $1 \cdot a_2  \cdot a_3 \cdot  ...  \cdot a_n = n^2$ , for every integer  $n> 2$ . Determine the value of  $a_3 + a_5$ .
Answer: \frac{61}{16}
Problem: We say that some positive integer  $m$  covers the number  $1998$ , if  $1,9,9,8$  appear in this order as digits of  $m$ . (For instance  $1998$  is covered by  $2\textbf{1}59\textbf{9}36\textbf{98}$  but not by  $213326798$ .) Let  $k(n)$  be the number of positive integers that cover  $1998$  and have exactly  $n$  digits ( $n\ge 5$ ), all different from  $0$ . What is the remainder of  $k(n)$  on division by  $8$ ?
Answer: 1
Problem: Which positive integers can be written in the form \[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\] for positive integers  $x$ ,  $y$ ,  $z$ ?
Answer: all positive evens
Problem: Let  $f$  be the quadratic function with leading coefficient  $1$  whose graph is tangent to that of the lines  $y=-5x+6$  and  $y=x-1$ .  The sum of the coefficients of  $f$  is  $\tfrac pq$ , where  $p$  and  $q$  are positive relatively prime integers.  Find  $100p + q$ .
Answer: 2509
Problem: Find all functions  $f:\mathbb{N} \rightarrow \mathbb{N}$  such that for every prime number  $p$  and natural number  $x$ ,  $$ \{ x,f(x),\cdots f^{p-1}(x) \}  $$  is a complete residue system modulo  $p$ . With  $f^{k+1}(x)=f(f^k(x))$  for every natural number  $k$  and  $f^1(x)=f(x)$ .
Answer: f(x) = x + 1
Problem: Let  $ n \geq 3$  and consider a set  $ E$  of  $ 2n \minus{} 1$  distinct points on a circle. Suppose that exactly  $ k$  of these points are to be colored black.  Such a coloring is **good** if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly  $ n$  points from  $ E$ .  Find the smallest value of  $ k$  so that every such coloring of  $ k$  points of  $ E$  is good.
Answer: n
Problem: Rita the painter rolls a fair  $6\text{-sided die}$ that has  $3$  red sides,  $2$  yellow sides, and  $1$  blue side. Rita rolls the die twice and mixes the colors that the die rolled. What is the probability that she has mixed the color purple?
Answer: \frac{1}{6}
Problem: Let  $a=256$ . Find the unique real number  $x>a^2$  such that
\[\log_a \log_a \log_a x = \log_{a^2} \log_{a^2} \log_{a^2} x.\]
Answer: 2^{32}
Problem: Find the number of squares in the sequence given by  $ a_0\equal{}91$  and  $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$  for  $ n \ge 0.$ 
Answer: 0
Problem: Seven cards numbered  $1$  through  $7$  lay stacked in a pile in ascending order from top to bottom ( $1$  on top,  $7$  on bottom). A shuffle involves picking a random card *of the six not currently on top*, and putting it on top. The relative order of all the other cards remains unchanged. Find the probability that, after  $10$  shuffles,  $6$  is higher in the pile than  $3$ .
Answer: \frac{3^{10} - 2^{10}}{2 \cdot 3^{10}}
Problem: Let  $A$  and  $B$  be sets such that there are exactly  $144$  sets which are subsets of either  $A$  or  $B$ . Determine the number of elements  $A \cup B$  has.
Answer: 8
Problem: For a positive integer  $n$ , let  $d(n)$  be the number of all positive divisors of  $n$ . Find all positive integers  $n$  such that  $d(n)^3=4n$ .
Answer: 2, 128, 2000
Problem: For positive integer  $n$ , define  $S_n$  to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where  $a_1,a_2,\ldots,a_n$  are positive real numbers whose sum is 17. There is a unique positive integer  $n$  for which  $S_n$  is also an integer. Find this  $n$ .
Answer: 12
Problem: Determine all triangles such that the lengths of the three sides and its area are given by four consecutive natural numbers.
Answer: 3, 4, 5
Problem: Starting from 37, adding 5 before each previous term, forms the following sequence:
\[37,537,5537,55537,555537,...\]
How many prime numbers are there in this sequence?
Answer: 1
Problem: Let  $p_1, p_2, . . . , p_{30}$  be a permutation of the numbers  $1, 2, . . . , 30.$  For how many permutations does the equality  $\sum^{30}_{k=1}|p_k - k| = 450 $  hold?
Answer: (15!)^2
Problem: A  $\emph{planar}$  graph is a connected graph that can be drawn on a sphere without edge crossings. Such a drawing will divide the sphere into a number of faces. Let  $G$  be a planar graph with  $11$  vertices of degree  $2$ ,  $5$  vertices of degree  $3$ , and  $1$  vertex of degree  $7$ . Find the number of faces into which  $G$  divides the sphere.
Answer: 7
Problem: There are  $2n$  different numbers in a row. By one move we can interchange any two numbers or interchange any  $3$  numbers cyclically (choose  $a,b,c$  and place  $a$  instead of  $b$ ,  $b$  instead of  $c$ ,  $c$  instead of  $a$ ). What is the minimal number of moves that is always sufficient to arrange the numbers in increasing order ?
Answer: n
Problem: Let  $ABC$  be an acute-angled, not equilateral triangle, where vertex  $A$  lies on the perpendicular bisector of the segment  $HO$ , joining the orthocentre  $H$  to the circumcentre  $O$ . Determine all possible values for the measure of angle  $A$ .
Answer: 60^	ext{o}