rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: We consider an  $n \times n$  table, with  $n\ge1$ . Aya wishes to color  $k$  cells of this table so that that there is a unique way to place  $n$  tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of  $k$  for which Aya's wish is achievable?
Answer: \frac{n(n+1)}{2}
Problem: In ZS Chess, an Ivanight attacks like a knight, except that if the attacked square is out of range, it goes
through the edge and comes out from the other side of the board, and attacks that square instead. The
ZS chessboard is an  $8 \times 8$  board, where cells are coloured with  $n$  distinct colours, where  $n$  is a natural
number, such that a Ivanight placed on any square attacks  $ 8 $  squares that consist of all  $n$  colours, and
the colours appear equally many times in those  $ 8 $  squares. For which values of  $n$  does such a ZS chess
board exist?
Answer:  n = 1, 2, 4, 8
Problem: The cells of a  $100 \times 100$  table are colored white. In one move, it is allowed to select some  $99$  cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring?
Answer: 100
Problem: A rectangle is inscribed in a circle of area  $32\pi$  and the area of the rectangle is  $34$ . Find its perimeter.
Answer: 28
Problem: For positive integers  $x,y$ , find all pairs  $(x,y)$  such that  $x^2y + x$  is a multiple of  $xy^2 + 7$ .
Answer: (x, y) = (7, 1), (14, 1), (35, 1), (7, 2), (7k, 7)
Problem: Jennifer wants to do origami, and she has a square of side length  $ 1$ . However, she would prefer to use a regular octagon for her origami, so she decides to cut the four corners of the square to get a regular octagon. Once she does so, what will be the side length of the octagon Jennifer obtains?
Answer: 1 - \frac{\sqrt{2}}{2}
Problem: Given a triangle on the plane, construct inside the triangle the point  $P$  for which the centroid of the triangle formed by the three projections of  $P$  onto the sides of the triangle happens to be  $P$ .
Answer: The Lemoine point
Problem: Eric has  $2$  boxes of apples, with the first box containing red and yellow apples and the second box containing green apples. Eric observes that the red apples make up  $\tfrac{1}{2}$  of the apples in the first box. He then moves all of the red apples to the second box, and observes that the red apples now make up  $\tfrac{1}{3}$  of the apples in the second box. Suppose that Eric has  $28$  apples in total. How many red apples does Eric have?
Answer: 7
Problem: Let  $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$  For constant  $a$  such that  $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$ , evaluate  $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$ 
Answer: \frac{1}{2}
Problem: Let  $ABC$  be an acute-angled scalene triangle and  $T$  be a point inside it such that  $\angle ATB  = \angle BTC = 120^o$ . A circle centered at point  $E$  passes through the midpoints of the sides of  $ABC$ . For  $B, T, E$  collinear, find angle  $ABC$ .
Answer: 30^{\circ}
Problem: Peter and Vasil together thought of ten 5-degree polynomials. Then, Vasil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Vasil could have called out?
Answer: 50
Problem: Find all real numbers  $a$  for which the equation  $x^2a- 2x + 1 = 3 |x|$  has exactly three distinct real solutions in  $x$ .
Answer: \frac{1}{4}
Problem: Let  $\triangle ABC$  be inscribed in circle  $O$  with  $\angle ABC = 36^\circ$ .  $D$  and  $E$  are on the circle such that  $\overline{AD}$  and  $\overline{CE}$  are diameters of circle  $O$ . List all possible positive values of  $\angle DBE$  in degrees in order from least to greatest.
Answer: 36^
\circ, 144^
\circ
Problem: Let trapezoid  $ABCD$  inscribed in a circle  $O$ ,  $AB||CD$ . Tangent at  $D$  wrt  $O$  intersects line  $AC$  at  $F$ ,  $DF||BC$ . If  $CA=5, BC=4$ , then find  $AF$ .
Answer: 4
Problem: While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards.  Let  $a$  be the probability that the cards are of different ranks.  Compute  $\lfloor 1000a\rfloor$ .
Answer: 941
Problem: Let  $ABCD$  be a rectangle. Circles with diameters  $AB$  and  $CD$  meet at points  $P$  and  $Q$  inside the rectangle such that  $P$  is closer to segment  $BC$  than  $Q$ . Let  $M$  and  $N$  be the midpoints of segments  $AB$  and  $CD$ . If  $\angle MPN = 40^\circ$ , find the degree measure of  $\angle BPC$ .
Answer: 80^	ext{\circ}
Problem: Let  $t$  and  $n$  be fixed integers each at least  $2$ . Find the largest positive integer  $m$  for which there exists a polynomial  $P$ , of degree  $n$  and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each  $k = 0,1, ..., m$ .
Answer: n
Problem: Which natural numbers cannot be presented in that way:  $[n+\sqrt{n}+\frac{1}{2}]$ ,  $n\in\mathbb{N}$  $[y]$  is the greatest integer function.
Answer: perfect squares
Problem: Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible?
Answer: 144
Problem: 2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.
Answer:  11
Problem: There are  $2022$  signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length  $1011$  whose each term is either red or blue, Auto can always remove  $1011$  signs from the line so that the remaining  $1011$  signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.
Answer: 2^{1011}
Problem: A convex pentagon  $P=ABCDE$  is inscribed in a circle of radius  $1$ . Find the maximum area of  $P$  subject to the condition that the chords  $AC$  and  $BD$  are perpendicular.
Answer: 1 + \frac{3\sqrt{3}}{4}
Problem: Find all functions  $f:\mathbb{R}^+\to\mathbb{R}^+$  such that for any real numbers  $a, b, c, d >0$  satisfying  $abcd=1$ ,\[(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d)\] holds true.

*(Swiss Mathematical Olympiad 2011, Final round, problem 4)*
Answer:  f(x) = x
Problem: A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("flipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it). Thus, Student 3 will close the third locker, open the sixth, close the ninth. . . . Student 5 then goes through and "flips"every 5th locker. This process continues with all students with odd numbers  $n<100$  going through and "flipping" every  $n$ th locker. 

How many lockers are open after this process?
Answer: 10
Problem: Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is  $3$  jumps east and  $10$  jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at Frist, located  $2$  jumps east and  $4$  jumps north of Fine Hall. How many ways can he get to McCosh without going through Frist?
Answer: 181
Problem: Find all polynomials  $f(x)$  with real coefficients having the property  $f(g(x))=g(f(x))$  for every polynomial  $g(x)$  with real coefficients.
Answer:  f(x) = x
Problem: An alphabet consists of  $n$  letters. What is the maximal length of a word if we know that any two consecutive letters  $a,b$  of the word are different and that the word cannot be reduced to a word of the kind  $abab$  with  $a\neq b$  by removing letters.
Answer: 2n - 1
Problem: In a tennis tournament, each competitor plays against every other competitor, and there are no draws. Call a group of four tennis players ``ordered'' if there is a clear winner and a clear loser (i.e., one person who beat the other three, and one person who lost to the other three.) Find the smallest integer  $n$  for which any tennis tournament with  $n$  people has a group of four tennis players that is ordered.
Answer: 8
Problem: The sides  $x$  and  $y$  of a scalene triangle satisfy  $x + \frac{2\Delta }{x}=y+ \frac{2\Delta }{y}$  , where  $\Delta$  is the area of the triangle. If  $x = 60, y = 63$ , what is the length of the largest side of the triangle?
Answer: 87
Problem: Given is a simple graph with  $239$  vertices, such that it is not bipartite and each vertex has degree at least  $3$ . Find the smallest  $k$ , such that each odd cycle has length at most  $k$ .
Answer: k = 3
Problem: A number is called *capicua* if when it is written in decimal notation, it can be read equal from left to right as from right to left; for example:  $8, 23432, 6446$ . Let  $x_1<x_2<\cdots<x_i<x_{i+1},\cdots$  be the sequence of all capicua numbers. For each  $i$  define  $y_i=x_{i+1}-x_i$ . How many distinct primes contains the set  $\{y_1,y_2, \ldots\}$ ?
Answer: 2
Problem: Set  $T\subset\{1,2,\dots,n\}^3$  has the property that for any two triplets  $(a,b,c)$  and  $(x,y,z)$  in  $T$ , we have  $a<b<c$ , and also, we know that at most one of the equalities  $a=x$ ,  $b=y$ ,  $c=z$  holds. Maximize  $|T|$ .
Answer: \binom{n}{3}
Problem: Find all polynomials  $W$  with real coefficients possessing the following property: if  $x+y$  is a rational number, then  $W(x)+W(y)$  is rational.
Answer:  W(x) = ax + b where a and b are rational numbers.
Problem: One out of every seven mathematicians is a philosopher, and one out of every nine philosophers is a mathematician. Are there more philosophers or mathematicians?
Answer: p > m
Problem: Let  $a$  and  $b$  be two real numbers and let $M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$ . Find the values of  $a$  and  $b$  for which  $M(a,b)$  is minimal.
Answer: a = -\frac{1}{3}, b = -\frac{1}{3}
Problem: let  $m,n$  be natural number with  $m>n$  . find all such pairs of  $(m,n) $ such that $gcd(n+1,m+1)=gcd(n+2,m+2) =..........=gcd(m, 2m-n) = 1 $ 
Answer: (m, n) = (n+1, n) \text{ for all } n \in \mathbb{N}
Problem: Given a natural number  $k$  and a real number  $a (a>0)$ ,  find the maximal value of  $a^{k_1}+a^{k_2}+\cdots +a^{k_r}$ , where  $k_1+k_2+\cdots +k_r=k$  ( $k_i\in \mathbb{N} ,1\le r \le k$ ).
Answer: a^k
Problem: Find all pairs  $(a,b)$  of different positive integers that satisfy the equation  $W(a)=W(b)$ , where  $W(x)=x^{4}-3x^{3}+5x^{2}-9x$ .
Answer:  (1, 2) and (2, 1)
Problem: Let  $p=4k+3$  be a prime number. Find the number of different residues mod p of  $(x^{2}+y^{2})^{2}$  where  $(x,p)=(y,p)=1.$ 
Answer: \frac{p-1}{2}
Problem: Let the tangent line passing through a point  $A$   outside the circle with center  $O$  touches the circle at  $B$  and  $C$ . Let  $[BD]$  be the diameter of the circle. Let the lines  $CD$  and  $AB$  meet at  $E$ . If the lines  $AD$  and  $OE$  meet at  $F$ , find  $|AF|/|FD|$ .
Answer: \frac{1}{2}
Problem: The cards in a stack of  $2n$  cards are numbered consecutively from  $1$  through  $2n$  from top to bottom. The top  $n$  cards are removed, kept in order, and form pile  $A$ . The remaining cards form pile  $B$ . The cards are then restacked by taking cards alternately from the tops of pile  $B$  and  $A$ , respectively. In this process, card number  $(n+1)$  becomes the bottom card of the new stack, card number  $1$  is on top of this card, and so on, until piles  $A$  and  $B$  are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named *magical*. Find the number of cards in the magical stack in which card number  $131$  retains its original position.
Answer: 130
Problem: Congruent unit circles intersect in such a way that the center of each circle lies on the circumference of the other. Let  $R$  be the region in which two circles overlap. Determine the perimeter of  $R.$ 
Answer: \frac{2\pi}{3}
Problem: For two quadratic trinomials  $P(x)$  and  $Q(x)$  there is a linear function  $\ell(x)$  such that  $P(x)=Q(\ell(x))$  for all real  $x$ . How many such linear functions  $\ell(x)$  can exist?
Answer: 2
Problem: For  $f(x)=x^4+|x|,$  let  $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$ 

Find the value of   $\frac{I_1}{I_2}.$ 
Answer: 2
Problem: Find all  prime number   $p$ ,  such that  there exist an infinite number of  positive integer  $n$   satisfying the following condition: $p|n^{ n+1}+(n+1)^n.$ (September 29, 2012, Hohhot)
Answer:  2
Problem: For certain ordered pairs  $(a,b)$  of real numbers, the system of equations \begin{eqnarray*} && ax+by =1 &&x^2+y^2=50\end{eqnarray*} has at least one solution, and each solution is an ordered pair  $(x,y)$  of integers.  How many such ordered pairs  $(a,b)$  are there?
Answer: 18
Problem: A mustache is created by taking the set of points  $(x, y)$  in the  $xy$ -coordinate plane that satisfy  $4 + 4 \cos(\pi x/24) \le y \le 6 + 6\cos(\pi x/24)$  and  $-24 \le x \le 24$ .  What is the area of the mustache?
Answer: 96
Problem: For every positive integer  $n$ , determine the biggest positive integer  $k$  so that  $2^k |\ 3^n+1$ 
Answer:  k = 2
Problem:  Determine twice differentiable functions  $f: \mathbb{R} \rightarrow \mathbb{R}$  which verify relation 

    \[
    \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
    \]
Answer:  f(x) = C
Problem: What is the maximum possible value of  $k$  for which  $2013$  can be written as a sum of  $k$  consecutive positive integers?
Answer: 61
Problem: Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\).
Answer: 6
Problem: Find all positive integers  $n$  for which the number  $1!+2!+3!+\cdots+n!$  is a perfect power of an integer.
Answer:  n = 1 and n = 3
Problem: Each of  $2k+1$  distinct 7-element subsets of the 20 element set intersects with exactly  $k$  of them. Find the maximum possible value of  $k$ .
Answer: 2
Problem: Find all positive integer number  $n$  such that  $[\sqrt{n}]-2$  divides  $n-4$  and  $[\sqrt{n}]+2$  divides  $n+4$ . Note:  $[r]$  denotes the integer part of  $r$ .
Answer:  n = 2, 4, 11, 20, 31, 36, 44 , and n = a^2 + 2a - 4 for a \in \mathbb{N} \setminus \{1, 2\}.
Problem: Let   ${ a\uparrow\uparrow b = {{{{{a^{a}}^a}^{\dots}}}^{a}}^{a}} $ , where there are  $ b $  a's in total. That is  $ a\uparrow\uparrow b $  is given by the recurrence \[  a\uparrow\uparrow b = \begin{cases} a & b=1 a^{a\uparrow\uparrow (b-1)} & b\ge2\end{cases} \] What is the remainder of  $  3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3)) $  when divided by  $ 60 $ ?
Answer: 27
Problem: Seongcheol has  $3$  red shirts and  $2$  green shirts, such that he cannot tell the difference between his three red shirts and he similarly cannot tell the difference between his two green shirts. In how many ways can he hang them in a row in his closet, given that he does not want the two green shirts next to each other?
Answer: 6
Problem: For which digits  $a$  do exist integers  $n \geq 4$  such that each digit of  $\frac{n(n+1)}{2}$  equals  $a \ ?$ 
Answer: 5 \text{ and } 6
Problem: Find all value of  $ n$  for which there are nonzero real numbers  $ a, b, c, d$  such that after expanding and collecting similar terms, the polynomial  $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$  has exactly  $ n$  nonzero coefficients.
Answer: 0, 50, 100, 101
Problem: The polynomial  $x^3-kx^2+20x-15$  has  $3$  roots, one of which is known to be  $3$ . Compute the greatest possible sum of the other two roots.
Answer: 5
Problem: Find all triples  $m$ ;  $n$ ;  $l$  of natural numbers such that  $m + n = gcd(m; n)^2$ ;  $m + l = gcd(m; l)^2$ ;  $n + l = gcd(n; l)^2$ :
Answer:  (2, 2, 2)
Problem: The bases of a trapezoid have lengths 10 and 21, and the legs have lengths  $\sqrt{34}$  and  $3 \sqrt{5}$ .  What is the area of the trapezoid?
Answer: \frac{93}{2}
Problem: The lengths of the three sides  $a, b, c$  with  $a \le b \le c$ , of a right triangle is an integer. Find all the sequences  $(a, b, c)$  so that the values of perimeter and area of the triangle are the same.
Answer:  (5, 12, 13)
Problem: The triangle  $K_2$  has as its vertices the feet of the altitudes of a non-right triangle  $K_1$ . Find all possibilities for the sizes of the angles of  $K_1$  for which the triangles  $K_1$  and  $K_2$  are similar.
Answer: A = 60^
\circ, B = 60^
\circ, C = 60^
\circ
Problem: If  $a$  and  $b$  satisfy the equations  $a +\frac1b=4$  and  $\frac1a+b=\frac{16}{15}$ , determine the product of all possible values of  $ab$ .

Answer: 1
Problem: Four men are each given a unique number from  $1$  to  $4$ , and four women are each given a unique number from  $1$  to  $4$ . How many ways are there to arrange the men and women in a circle such that no two men are next to each other, no two women are next to each other, and no two people with the same number are next to each other? Note that two configurations are considered to be the same if one can be rotated to obtain the other one.
Answer: 12
Problem: Find all polynomials  $P$  with integer coefficients such that  $P (0)\ne  0$  and   $$ P^n(m)\cdot P^m(n) $$   is a square of an integer for all nonnegative integers  $n, m$ .

*Remark:* For a nonnegative integer  $k$  and an integer  $n$ ,  $P^k(n)$  is defined as follows:  $P^k(n) = n$  if  $k = 0$  and  $P^k(n)=P(P(^{k-1}(n))$  if  $k >0$ .

Proposed by Adrian Beker.
Answer: P(x) = x + 1
Problem: Let   $p$  a prime number,  $p\geq 5$ . Find the number of polynomials of the form
\[ x^p + px^k + p x^l + 1, \quad k > l, \quad k, l \in \left\{1,2,\dots,p-1\right\},  \] which are irreducible in  $\mathbb{Z}[X]$ .
Answer: 2 \binom{\frac{p-1}{2}}{2}
Problem: Find all a positive integers  $a$  and  $b$ , such that  $$ \frac{a^b+b^a}{a^a-b^b} $$  is an integer
Answer:  (a, b) = (2, 1) } and (a, b) = (1, 2)
Problem: In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three
5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as
necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile.
Find the number of diﬀerent positive weights of chemicals that Gerome could measure.

Answer: 63
Problem: In a committee there are  $n$  members. Each pair of members are either friends or enemies. Each committee member has exactly three enemies. It is also known that for each committee member, an enemy of his friend is automatically his own enemy. Find all possible value(s) of  $n$ 
Answer: n = 4, 6
Problem: Determine the maximum number of bishops that we can place in a  $8 \times 8$  chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop. 

Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the  $2$  main diagonals and the ones parallel to those ones.
Answer: 16
Problem: In each square of a  $5 \times 5$  board one of the numbers  $2, 3, 4$  or  $5$  is written so that the the sum of all the numbers in each row, in each column and on each diagonal is always even. How many ways can we fill the board?

Clarification. A  $5\times 5$  board has exactly  $18$  diagonals of different sizes. In particular, the corners are size  $ 1$  diagonals.
Answer: 2^{39}
Problem: Let  $F_n$  be the nth Fibonacci number, defined by  $F_1 = F_2 = 1$  and  $F_n = F_{n-1} + F_{n-2}$  for  $n > 2$ . Let  $A_0, A_1, A_2,\cdots$  be a sequence of points on a circle of radius  $1$  such that the minor arc from  $A_{k-1}$  to  $A_k$  runs clockwise and such that
\[\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+1}^2+1}\]
for  $k \geq 1$ , where  $\mu(XY )$  denotes the radian measure of the arc  $XY$  in the clockwise direction. What is the limit of the radian measure of arc  $A_0A_n$  as  $n$  approaches infinity?
Answer: 2\pi
Problem: A fair coin is to be tossed  $10$  times. Let  $i/j$ , in lowest terms, be the probability that heads never occur on consecutive tosses. Find  $i+j$ .
Answer: 73
Problem: Given that nonzero reals  $a,b,c,d$  satisfy  $a^b=c^d$  and  $\frac{a}{2c}=\frac{b}{d}=2$ , compute  $\frac{1}{c}$ .
Answer: 16
Problem: BdMO National 2016 Higher Secondary 

<u>**Problem 4:**</u>
Consider the set of integers  $  \left \{ 1, 2, ......... , 100 \right \} $ . Let  $   \left \{ x_1, x_2, ......... , x_{100} \right \}$  be some arbitrary arrangement of the integers  $ \left \{ 1, 2, ......... , 100 \right \}$ , where all of the  $x_i$  are different. Find the smallest possible value of the sum, $S = \left | x_2 - x_1 \right | + \left | x_3 - x_2  \right | + ................+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | $ .
Answer: 198
Problem: Suppose  $x>1$  is a real number such that  $x+\tfrac 1x = \sqrt{22}$ .  What is  $x^2-\tfrac1{x^2}$ ?
Answer: 18
Problem: Let  $ P_1$  be a regular  $ r$ -gon and  $ P_2$  be a regular  $ s$ -gon  $ (r\geq s\geq 3)$  such that each interior angle of  $ P_1$  is  $ \frac {59}{58}$  as large as each interior angle of  $ P_2$ . What's the largest possible value of  $ s$ ?
Answer: 117
Problem: We have an empty equilateral triangle with length of a side  $l$ . We put the triangle, horizontally, over a sphere of radius  $r$ . Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of  $l$  and  $r$ )?
Answer: \sqrt{r^2 + \frac{l^2}{4}}
Problem: Find all primes  $p,q,r$  such that  $qr-1$  is divisible by  $p$ ,  $pr-1$  is divisible by  $q$ ,  $pq-1$  is divisible by  $r$ .
Answer: (2, 3, 5) and its permutations.
Problem: On a circumference of a unit radius, take points  $A$  and  $B$  such that section  $AB$  has length one.  $C$  can be any point on the longer arc of the circle between  $A$  and  $B$ . How do we take  $C$  to make the perimeter of the triangle  $ABC$  as large as possible?
Answer: 3
Problem: At certain store, a package of 3 apples and 12 oranges costs 5 dollars, and a package of 20 apples and 5 oranges costs 13 dollars. Given that apples and oranges can only be bought in these two packages, what is the minimum nonzero amount of dollars that must be spent to have an equal number of apples and oranges?
Answer: 64
Problem: Consider a regular cube with side length  $2$ .  Let  $A$  and  $B$  be  $2$  vertices that are furthest apart.  Construct a sequence of points on the surface of the cube  $A_1$ ,  $A_2$ ,  $\ldots$ ,  $A_k$  so that  $A_1=A$ ,  $A_k=B$  and for any  $i = 1,\ldots, k-1$ , the distance from  $A_i$  to  $A_{i+1}$  is  $3$ .  Find the minimum value of  $k$ .
Answer: 7
Problem: There is infinite sequence of composite numbers  $a_1,a_2,...,$  where  $a_{n+1}=a_n-p_n+\frac{a_n}{p_n}$  ;  $p_n$  is smallest prime divisor of  $a_n$ . It is known, that  $37|a_n$  for every  $n$ .
Find possible values of  $a_1$ 
Answer: 37^2
Problem: Let  $n$  be a positive integer. A frog starts on the number line at  $0$ . Suppose it makes a finite sequence of hops, subject to two conditions:  	
- The frog visits only points in  $\{1, 2, \dots, 2^n-1\}$ , 	each at most once.
- The length of each hop is in  $\{2^0, 2^1, 2^2, \dots\}$ . 	(The hops may be either direction, left or right.)
 Let  $S$  be the sum of the (positive) lengths of all hops in the sequence. What is the maximum possible value of  $S$ ?
Answer: \frac{4^n - 1}{3}