rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle  $C$  with radius  $30$ . Let  $K$  be the area of the region inside circle  $C$  and outside of the six circles in the ring. Find  $\lfloor K \rfloor$ .
Answer: 942
Problem: Let  $\omega \in \mathbb{C}$ , and  $\left |  \omega  \right | = 1$ . Find the maximum length of  $z = \left( \omega + 2 \right) ^3 \left( \omega - 3 \right)^2$ .
Answer: 108
Problem: For a positive integer  $ n$ , let  $ \theta(n)$  denote the number of integers  $ 0 \leq x < 2010$  such that  $ x^2 \minus{} n$  is divisible by  $ 2010$ . Determine the remainder when  $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$  is divided by  $ 2010$ .
Answer: 0
Problem: Let  $n$  be a positive integer. There are  $n$  purple and  $n$  white cows queuing in a line in some order. Tim wishes to sort the cows by colour, such that all purple cows are at the front of the line. At each step, he is only allowed to swap two adjacent groups of equally many consecutive cows. What is the minimal number of steps Tim needs to be able to fulfill his wish, regardless of the initial alignment of the cows?
Answer:  n
Problem: What is the largest possible number of edges in a graph on  $2n$  nodes, if there exists exactly one way to split its nodes into  $n$  pairs so that the nodes from each pair are connected by an edge?
Answer: n^2
Problem: Consider all permutations of natural numbers from  $1$  to  $100$ . A permutation is called  $\emph{double}$  when it has the following property: If you write this permutation twice in a row, then delete  $100$  numbers from them you get the remaining numbers  $1, 2, 3, \ldots , 100$  in order. How many  $\emph{double}$  permutations are there?
Answer: \binom{100}{50}
Problem: Suppose  $ P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0$  be a real polynomial of degree  $ n > 2$  with  $ a_n \equal{} 1$ ,  $ a_{n\minus{}1} \equal{} \minus{}n$ ,  $ a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2}$  such that all the roots of  $ P$  are real. Determine the coefficients  $ a_i$ .
Answer:  a_i = (-1)^{n-i} \binom{n}{i}
Problem: Compute the maximum integer value of  $k$  such that  $2^k$  divides  $3^{2n+3}+40n-27$  for any positive integer  $n$ .
Answer: 6
Problem: Farmer Boso has a busy farm with lots of animals. He tends to  $5b$  cows,  $5a +7$  chickens, and  $b^{a-5}$  insects. Note that each insect has  $6$  legs. The number of cows is equal to the number of insects. The total number of legs present amongst his animals can be expressed as  $\overline{LLL }+1$ , where  $L$  stands for a digit. Find  $L$ .
Answer: 3
Problem: Two points are located  $10$  units apart, and a circle is drawn with radius  $ r$  centered at one of the points. A tangent line to the circle is drawn from the other point. What value of  $ r$  maximizes the area of the triangle formed by the two points and the point of tangency?

Answer: 25
Problem: In trapezoid  $ABCD$ ,  $AD \parallel BC$  and  $\angle ABC + \angle CDA = 270^{\circ}$ .  Compute  $AB^2$  given that  $AB \cdot \tan(\angle BCD) = 20$  and  $CD = 13$ .
Answer: 260
Problem: Let  $ABC$  be an equilateral triangle. Let  $P$  and  $S$  be points on  $AB$  and  $AC$ , respectively, and let  $Q$  and  $R$  be points on  $BC$  such that  $PQRS$  is a rectangle. If  $PQ = \sqrt3 PS$  and the area of  $PQRS$  is  $28\sqrt3$ , what is the length of  $PC$ ?
Answer: 2\sqrt{7}
Problem: On circle there are points  $A$ ,  $B$  and  $C$  such that they divide circle in ratio  $3:5:7$ . Find angles of triangle  $ABC$ 
Answer: 36^
\circ, 60^
\circ, 84^
\circ
Problem: For positive integer  $n$  let  $a_n$  be the integer consisting of  $n$  digits of  $9$  followed by the digits  $488$ . For example,  $a_3 = 999,488$  and  $a_7 = 9,999,999,488$ . Find the value of  $n$  so that an is divisible by the highest power of  $2$ .
Answer: 6
Problem: Find all surjective functions  $ f: \mathbb{N} \to \mathbb{N}$  such that for every  $ m,n \in \mathbb{N}$  and every prime  $ p,$  the number  $ f(m + n)$  is divisible by  $ p$  if and only if  $ f(m) + f(n)$  is divisible by  $ p$ .

*Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran*
Answer: f(x) = x
Problem: Let  $f: \mathbb{R} \rightarrow \mathbb{R}$  be a function such that   $$ \displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,} $$   for all  $x \in  \mathbb{R}.$  Find all possible values of  $f(2).$ 
Answer: 2
Problem: Find the smallest positive integer  $n \geq 10$  such that  $n + 6$  is a prime and  $9n + 7$  is a perfect square.
Answer: 53
Problem: A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio.
Answer: \frac{1 + \sqrt{5}}{2}
Problem: Given that  $ x \plus{} \sin y \equal{} 2008$  and  $ x \plus{} 2008 \cos y \equal{} 2007$ , where  $ 0 \leq y \leq \pi/2$ , find the value of  $ x \plus{} y$ .
Answer: 2007 + \frac{\pi}{2}
Problem: Let  $S$  be a subset of  $\{0,1,2,\dots ,9\}$ . Suppose there is a positive integer  $N$  such that for any integer  $n>N$ , one can find positive integers  $a,b$  so that  $n=a+b$  and all the digits in the decimal representations of  $a,b$  (expressed without leading zeros) are in  $S$ . Find the smallest possible value of  $|S|$ .
Answer: 5
Problem: For each positive integer  $n$  denote:
\[n!=1\cdot 2\cdot 3\dots n\]
Find all positive integers  $n$  for which  $1!+2!+3!+\cdots+n!$  is a perfect square.
Answer:  n = 1 and n = 3
Problem: Find all positive integers  $p$ ,  $q$ ,  $r$  such that  $p$  and  $q$  are prime numbers and   $\frac{1}{p+1}+\frac{1}{q+1}-\frac{1}{(p+1)(q+1)} = \frac{1}{r}.$ 
Answer: \{p, q\} = \{2, 3\} \text{ or } \{3, 2\}
Problem: Find the smallest  $n \in \mathbb{N}$  such that if any 5 vertices of a regular  $n$ -gon are colored red, there exists a line of symmetry  $l$  of the  $n$ -gon such that every red point is reflected across  $l$  to a non-red point.
Answer: 14
Problem: After a cyclist has gone  $ \frac{2}{3}$  of his route, he gets a flat tire. Finishing on foot, he spends twice as long walking as he did riding. How many times as fast does he ride as walk?
Answer: 4
Problem: Let  $a_n\ (n\geq 1)$  be the value for which  $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$  is maximal. Find  $\lim_{n\to\infty} \ln a_n.$ 
Answer: -\ln 2
Problem: Let  $S$  is a finite set with  $n$  elements. We divided  $AS$  to  $m$  disjoint parts such that if  $A$ ,  $B$ ,  $A \cup B$  are in the same part, then  $A=B.$  Find the minimum value of  $m$ .
Answer: n+1
Problem: Consider the following modified algorithm for binary search, which we will call  $\textit{weighted binary search}$ :

\begin{tabular}{l}
01: \textbf{FUNCTION} SEARCH( $L$ , value) 
02: $\qquad$   hi  $\leftarrow$   $\operatorname{len}(L) - 1$  
03: $\qquad$   lo  $\leftarrow$  0 
04: $\qquad$   \textbf{WHILE} hi  $\geq$  lo 
05: $\qquad\qquad$  guess  $\leftarrow$   $\lfloor w \cdot\text{lo} + (1-w) \cdot \text{hi}\rfloor$  
06: $\qquad\qquad$  mid  $\leftarrow$   $L[\text{guess}]$  
07: $\qquad\qquad$  \textbf{IF} mid  $> \text{value}$  
08:  $\qquad\qquad\qquad$  hi  $\leftarrow$   $\text{guess} - 1$  
09:  $\qquad\qquad$  \textbf{ELSE IF} mid  $< \text{value}$  
10:  $\qquad\qquad\qquad$  lo  $\leftarrow$   $\text{guess} + 1$  
11:  $\qquad\qquad$  \textbf{ELSE} 
12:  $\qquad\qquad\qquad$  \textbf{RETURN} guess 
13: $\qquad$  \textbf{RETURN} -1 (not found) 
\end{tabular}

Assume  $L$  is a list of the integers  $\{1,2,\ldots,100\}$ , in that order. Further assume that accessing the  $k$ th index of  $L$  costs  $k+1$  tokens (e.g.  $L[0]$  costs  $1$  token). Let  $S$  be the set of all  $w\in[\tfrac12,1)$  which minimize the average cost when  $\texttt{value}$  is an integer selected at random in the range  $[1,50]$ . Given that  $S=\left(x,\tfrac {74}{99}\right]$ , determine  $x$ .
Answer:  x = \frac{1}{2}
Problem: A rectangle with sides  $a$  and  $b$  has an area of  $24$  and a diagonal of length  $11$ . Find the perimeter of this rectangle.
Answer: 26
Problem: Consider the sequence:  $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$ , for  $n>1$ , where  $\text{lcm} (a,b)$  means the least common multiple of  $a$  and  $b$ . Find the greatest common divisor of  $x_{1995}$  and  $x_{1996}$ .
Answer: 19
Problem: Let  $r$  be a real number such that  $\sqrt[3]{r} - \frac{1}{\sqrt[3]{r}}=2$ . Find  $r^3 - \frac{1}{r^3}$ .
Answer: 14
Problem: Given the polynomial sequence  $(p_{n}(x))$  satisfying  $p_{1}(x)=x^{2}-1$ ,  $p_{2}(x)=2x(x^{2}-1)$ , and  $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$ , for  $n\geq 2$ , let  $s_{n}$  be the sum of the absolute values of the coefficients of  $p_{n}(x)$ . For each  $n$ , find a non-negative integer  $k_{n}$  such that  $2^{-k_{n}}s_n$  is odd.
Answer:  k_n = 2v_2(n)
Problem: Find all triples of prime numbers  $(p, q, r)$  such that  $p^q + p^r$  is a perfect square.
Answer:  (p, q, r) \in \{ (2, 5, 2), (2, 2, 5), (2, 3, 3), (3, 3, 2), (2, 2n+1, 2n+1) \}
Problem: The roots of the polynomial  $f(x) = x^8 +x^7 -x^5 -x^4 -x^3 +x+ 1 $  are all roots of unity. We say that a real number  $r \in [0, 1)$  is nice if  $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$  is a root of the polynomial  $f$  and if  $e^{2i \pi r}$   has positive imaginary part. Let  $S$  be the sum of the values of nice real numbers  $r$ . If  $S =\frac{p}{q}$   for relatively prime positive integers  $p, q$ , find  $p + q$ .
Answer: 3
Problem: Find the smallest real constant  $p$  for which the inequality holds  $\sqrt{ab}- \frac{2ab}{a + b} \le p \left( \frac{a + b}{2} -\sqrt{ab}\right)$  with any positive real numbers  $a, b$ .
Answer:  p = 1
Problem:  $\Phi$  is the union of all triangles that are symmetric of the triangle  $ABC$  wrt a point  $O$ , as point  $O$  moves along the triangle's sides. If the  area of the triangle is  $E$ , find the area of  $\Phi$ .
Answer: 2E
Problem: Find maximum of the expression  $(a -b^2)(b - a^2)$ , where  $0 \le  a,b  \le 1$ .
Answer: \frac{1}{16}
Problem: Find the positive integer  $n$  such that  $\frac12 \cdot \frac34 + \frac56 \cdot \frac78 +   \frac{9}{10}\cdot \frac{11}{12 }= \frac{n}{1200}$  .
Answer: 2315
Problem: When submitting problems, Steven the troll likes to submit silly names rather than his own. On day  $1$ , he gives no
name at all. Every day after that, he alternately adds  $2$  words and  $4$  words to his name. For example, on day  $4$  he
submits an  $8\text{-word}$  name. On day  $n$  he submits the  $44\text{-word name}$  “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute  $n$ .
Answer: 16
Problem: A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is  $60$  centimetres and its base has radius  $30$  centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.
Answer: 108000 \, \text{cm}^3
Problem: Find the smallest positive integer  $a$  such that  $x^4+a^2$  is not prime for any integer  $x$ .
Answer: 8
Problem: Positive integers  $x, y, z$  satisfy  $(x + yi)^2 - 46i = z$ . What is  $x + y + z$ ?
Answer: 552
Problem: Given a complete bipartite graph on  $n,n$  vertices (call this  $K_{n,n}$ ), we colour all its edges with  $2$  colours , red and blue . What is the least value of  $n$  such that for any colouring of the edges of the graph , there will exist at least one monochromatic  $4$  cycle ?
Answer:  n = 5
Problem: Using each of the digits  $1,2,3,\ldots ,8,9$  exactly once,we form nine,not necassarily distinct,nine-digit numbers.Their sum ends in  $n$  zeroes,where  $n$  is a non-negative integer.Determine the maximum possible value of  $n$ .
Answer: 8
Problem: A father and his son are skating around a circular skating rink. From time to time, the father overtakes the son. After the son starts skating in the opposite direction, they begin to meet five times more often. What is the ratio of the skating speeds of the father and the son?

 (Tairova)
Answer: \frac{3}{2}
Problem: A football tournament is played between 5 teams, each two of which playing exactly one match. 5 points are awarded for a victory and 0 – for a loss. In case of a draw 1 point is awarded to both teams, if no goals are scored, and 2 – if they have scored any. In the final ranking the five teams had points that were 5 consecutive numbers. Determine the least number of goals that could be scored in the tournament.
Answer: 6
Problem: The sequence  ${a_{n}}$   $n\in \mathbb{N}$  is given in a recursive way with  $a_{1}=1$ , $a_{n}=\prod_{i=1}^{n-1} a_{i}+1$ , for all  $n\geq 2$ .
Determine the least number  $M$ , such that  $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$  for all  $m\in \mathbb{N}$ 
Answer: 2
Problem: Let  $A', B' , C'$  be the projections of a point  $M$  inside a triangle  $ABC$  onto the sides  $BC, CA, AB$ , respectively. Deﬁne  $p(M ) = \frac{MA'\cdot MB'\cdot MC'}{MA \cdot MB \cdot MC}$  . Find the position of point  $M$  that maximizes  $p(M )$ .
Answer:  M \equiv I
Problem: Find all prime numbers  $ p,q$  less than 2005 and such that  $ q|p^2 \plus{} 4$ ,  $ p|q^2 \plus{} 4$ .
Answer:  (2, 2)
Problem: Define the sequence  $(x_{n})$ :  $x_{1}=\frac{1}{3}$  and  $x_{n+1}=x_{n}^{2}+x_{n}$ . Find  $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$ , wehere  $[$   $]$  denotes the integer part.
Answer: 2
Problem: Let n be an odd positive integer.Let M be a set of  $n$  positive integers ,which are 2x2 different. A set  $T$   $\in$   $M$  is called "good" if the product of its elements is divisible by the sum of the elements in  $M$ , but is not divisible by the square of the same sum. Given that  $M$  is "good",how many "good" subsets of  $M$  can there be?
Answer: 2^{n-1}
Problem: Let  $A$ ,  $B$ ,  $C$ ,  $D$  be four points on a line in this order.  Suppose that  $AC = 25$ ,  $BD = 40$ , and  $AD = 57$ . Compute  $AB \cdot CD + AD \cdot BC$ .
Answer: 1000
Problem: Let  $ABC$  be a triangle in which ( ${BL}$ is the angle bisector of  ${\angle{ABC}}$   $\left( L\in AC \right)$ ,  ${AH}$  is an altitude of $\vartriangle ABC$   $\left( H\in BC \right)$  and  ${M}$ is the midpoint of the side  ${AB}$ . It is known  that the midpoints of the segments  ${BL}$  and  ${MH}$  coincides. Determine the internal angles of triangle  $\vartriangle ABC$ .
Answer: 60^	ext{\circ}
Problem: Compute the sum of all real numbers x which satisfy the following equation   $$ \frac {8^x - 19 \cdot 4^x}{16 - 25 \cdot 2^x}= 2 $$  
Answer: 5
Problem: Determine the real numbers  $a$  such that the equation  $a 3^x + 3^{-x} = 3$  has exactly one solution  $x.$ 
Answer: a \in (-\infty, 0] \cup \left\{ \frac{9}{4} \right\}
Problem: Let  $ABCDE$  be a convex pentagon such that  $AB=AE=CD=1$ ,  $\angle ABC=\angle DEA=90^\circ$  and  $BC+DE=1$ . Compute the area of the pentagon.
Answer: 1
Problem: It takes a steamer  $5$  days to go from Gorky to Astrakhan downstream the Volga river and  $7$  days upstream from Astrakhan to Gorky. How long will it take for a raft to float downstream from Gorky to Astrakhan?
Answer: 35
Problem: You are given a positive integer  $k$  and not necessarily distinct positive integers  $a_1, a_2 , a_3 , \ldots,
a_k$ . It turned out that for any coloring of all positive integers from  $1$  to  $2021$  in one of the  $k$  colors so that there are exactly  $a_1$  numbers of the first color,  $a_2$  numbers of the second color,  $\ldots$ , and  $a_k$  numbers of the  $k$ -th color, there is always a number  $x \in \{1, 2, \ldots, 2021\}$ , such that the total number of numbers colored in the same color as  $x$  is exactly  $x$ . What are the possible values of  $k$ ?
Answer: 2021
Problem: A natural number  $n$  is called *nice* if it enjoys the following properties:
• The expression is made ​​up of  $4$  decimal digits;
• the first and third digits of  $n$  are equal;
• the second and fourth digits of  $n$  are equal;
• the product of the digits of  $n$  divides  $n^2$ .
Determine all nice numbers.
Answer: 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 1212, 2424, 3636, 4848, 1515
Problem: Find all the quaterns  $(x,y,z,w)$  of real numbers (not necessarily distinct) that solve the following system of equations:  $$ x+y=z^2+w^2+6zw $$    $$ x+z=y^2+w^2+6yw $$    $$ x+w=y^2+z^2+6yz $$    $$ y+z=x^2+w^2+6xw $$    $$ y+w=x^2+z^2+6xz $$    $$ z+w=x^2+y^2+6xy $$  
Answer:  \{(0, 0, 0, 0), \left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right), \left(-\frac{1}{4}, -\frac{1}{4}, \frac{3}{4}, -\frac{1}{4}\right), \left(-\frac{1}{2}, -\frac{1}{2}, \frac{5}{2}, -\frac{1}{2}\right)\}
Problem: There are  $5$  accents in French, each applicable to only specific letters as follows:



- The cédille: ç
- The accent aigu: é
- The accent circonflexe: â, ê, î, ô, û
- The accent grave: à, è, ù
- The accent tréma: ë, ö, ü


Cédric needs to write down a phrase in French. He knows that there are  $3$  words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If  $n$  is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of  $n$ .
Answer: 4
Problem: Determine all positive integers  $n$  for which there exists an integer  $m$  such that  $2^{n}-1$  divides  $m^{2}+9$ .
Answer: n = 2^x \text{ for non-negative integral } x
Problem: Submit a positive integer  $N$  between 1 and 20, inclusive. 
    If  $C$  is the total number of teams that submit  $N$  for this question, your score will be  $\lfloor\frac{N}{C}\rfloor$ *Lightning 5.4*
Answer: \left\lfloor \frac{N}{C} \right\rfloor
Problem: A triangle  $ABC$  with area  $1$  is given. Anja and Bernd are playing the following game: Anja chooses a point  $X$  on side  $BC$ . Then Bernd chooses a point  $Y$  on side  $CA$  und at last Anja chooses a point  $Z$  on side  $AB$ . Also,  $X,Y$  and  $Z$  cannot be a vertex of triangle  $ABC$ . Anja wants to maximize the area of triangle  $XYZ$  and Bernd wants to minimize that area.
What is the area of triangle  $XYZ$  at the end of the game, if both play optimally?
Answer: \frac{1}{4}
Problem: A survey of participants was conducted at the Olympiad.  $50\%$  of the participants liked the first round,  $60\%$  of the participants liked the second round,  $70\%$  of the participants liked the opening of the Olympiad. It is known that each participant liked either one option or all three. Determine the percentage of participants who rated all three events positively.
Answer: 40\%
Problem: Find all  $(x, y, z, n) \in {\mathbb{N}}^4$  such that  $ x^3 +y^3 +z^3 =nx^2 y^2 z^2$ .
Answer: (1, 1, 1, 3), (1, 2, 3, 1)
Problem: For how many integers  $n$  between  $ 1$  and  $2021$  does the infinite nested expression   $$ \sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}} $$   give a rational number?
Answer: 44
Problem: Let  $ABC$  be a triangle with  $AB=13$ ,  $BC=14$ , and  $CA=15$ . Let  $\ell$  be a line passing through two sides of triangle  $ABC$ . Line  $\ell$  cuts triangle  $ABC$  into two figures, a triangle and a quadrilateral, that have equal perimeter. What is the maximum possible area of the triangle?
Answer: \frac{1323}{26}
Problem: Find all positive integers  $k$ , for which the product of some consecutive  $k$  positive integers ends with  $k$ .
Answer: 1, 2, 4
Problem: Diagonals of a quadrilateral  $ABCD$  are equal and meet at point  $O$ . The perpendicular bisectors to segments  $AB$  and  $CD$  meet at point  $P$ , and the perpendicular bisectors to  $BC$  and  $AD$  meet at point  $Q$ . Find angle  $\angle POQ$ .

by A.Zaslavsky
Answer: 90^
Problem: A *palindromic table* is a  $3 \times 3$  array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below. 
\[ \begin{array}[h]{ccc}
  O & M & O 
  N & M & N 
  O & M & O
\end{array} \]
How many palindromic tables are there that use only the letters  $O$  and  $M$ ?  (The table may contain only a single letter.)
Answer: 16
Problem: Three points  $X, Y,Z$  are on a striaght line such that  $XY = 10$  and  $XZ = 3$ . What is the product of all possible values of  $YZ$ ?
Answer: 91
Problem: If there is only  $1$  complex solution to the equation  $8x^3 + 12x^2 + kx + 1 = 0$ , what is  $k$ ?
Answer: 6
Problem: Mary told John her score on the American High School Mathematics Examination (AHSME), which was over 80.  From this, John was able to determine the number of problems Mary solved correctly.  If Mary's score had been any lower, but still over 80, John could not have determined this.  What was Mary's score?  (Recall that the AHSME consists of 30 multiple-choice problems and that one's score,  $s$ , is computed by the formula  $s = 30 + 4c - w$ , where  $c$  is the number of correct and  $w$  is the number of wrong answers; students are not penalized for problems left unanswered.)
Answer: 119
Problem: Given a set  $A$  which contains  $n$  elements. For any two distinct subsets  $A_{1}$ ,  $A_{2}$  of the given set  $A$ , we fix the number of elements of  $A_1 \cap A_2$ . Find the sum of all the numbers obtained in the described way.
Answer: n \left( 2^{2n-3} - 2^{n-2} \right)
Problem: There are  $38$  people in the California Baseball League (CBL). The CBL cannot start playing games until people are split into teams of exactly  $9$  people (with each person in exactly one team). Moreover, there must be an even number of teams. What is the fewest number of people who must join the CBL such that the CBL can start playing games? The CBL may not revoke membership of the  $38$  people already in the CBL.
Answer: 16
Problem: The Yamaimo family is moving to a new house, so they’ve packed their belongings into boxes, which weigh  $100\text{ kg}$  in total. Mr. Yamaimo realizes that  $99\%$  of the weight of the boxes is due to books. Later, the family unpacks some of the books (and nothing else). Mr. Yamaimo notices that now only  $95\%$  of the weight of the boxes is due to books. How much do the boxes weigh now in kilograms?
Answer: 20 \text{ kg}
Problem: Victor has  $3$  piles of  $3$  cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the  $3$  piles.) In how many orders can Victor draw the cards?
Answer: 1680
Problem: Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than  $1$ , the remainder is  $1$ . How many possible values are there for Sally's four-digit number?
Answer: 3
Problem: A circle with center at the origin and radius  $5$  intersects the abscissa in points  $A$  and  $B$ . Let  $P$   a point lying on the line  $x = 11$ , and the point  $Q$  is the intersection point of  $AP$  with this circle. We know what is the  $Q$  point is the midpoint of the  $AP$ . Find the coordinates of the point  $P$ .
Answer:  (11, 8)
Problem: Solve for integers  $x,y,z$ : \[ \{ \begin{array}{ccc} x + y &=& 1 - z  x^3 + y^3 &=& 1 - z^2 . \end{array}  \]
Answer: (k, -k, 1), (0, 1, 0), (1, 0, 0), (0, -2, 3), (-2, 0, 3), (-2, -3, 6), (-3, -2, 6)
Problem: A *9-cube* is a nine-dimensional hypercube (and hence has  $2^9$  vertices, for example). How many five-dimensional faces does it have?

(An  $n$  dimensional hypercube is defined to have vertices at each of the points  $(a_1,a_2,\cdots ,a_n)$  with  $a_i\in \{0,1\}$  for  $1\le i\le n$ )
Answer: 2016
Problem: Bill’s age is one third larger than Tracy’s age. In  $30$  years Bill’s age will be one eighth larger than Tracy’s age. How many years old is Bill?
Answer: 24