rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: For some real number  $c,$  the graphs of the equation  $y=|x-20|+|x+18|$  and the line  $y=x+c$  intersect at exactly one point. What is  $c$ ?
Answer: 18
Problem: What is the smallest perfect square larger than  $1$  with a perfect square number of positive integer factors?
Answer: 36
Problem: A semicircle has diameter  $XY$ . A square  $PQRS$  with side length 12 is inscribed in the semicircle with  $P$  and  $S$  on the diameter. Square  $STUV$  has  $T$  on  $RS$ ,  $U$  on the semicircle, and  $V$  on  $XY$ . What is the area of  $STUV$ ?
Answer: 36
Problem: Determine the number of real roots of the equation ${x^8 -x^7 + 2x^6- 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}= 0}$ 
Answer: 0
Problem: <span style="color:darkred">Find all differentiable functions  $f\colon [0,\infty)\to [0,\infty)$  for which  $f(0)=0$  and  $f^{\prime}(x^2)=f(x)$  for any  $x\in [0,\infty)$  .</span>
Answer:  f(x) = 0
Problem: Find all  $a,b,c \in \mathbb{N}$  such that 
\[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad c^2a|a^3+b^3+c^3.\]

[PS: The original problem was this:
Find all  $a,b,c \in \mathbb{N}$  such that 
\[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad \color{red}{c^2b}|a^3+b^3+c^3.\]
But I think the author meant  $c^2a|a^3+b^3+c^3$ , just because of symmetry]
Answer:  (a, b, c) = (1, 1, 1)
Problem: Dave has a pile of fair standard six-sided dice. In round one, Dave selects eight of the dice and rolls them. He calculates the sum of the numbers face up on those dice to get  $r_1$ . In round two, Dave selects  $r_1$  dice and rolls them. He calculates the sum of the numbers face up on those dice to get  $r_2$ . In round three, Dave selects  $r_2$  dice and rolls them. He calculates the sum of the numbers face up on those dice to get  $r_3$ . Find the expected value of  $r_3$ .
Answer: 343
Problem: Find all real numbers  $a$  for which there exist different real numbers  $b, c, d$  
different from  $a$  such that the four tangents drawn to the curve  $y = \sin (x)$  at the points  $(a, \sin (a)), 
(b, \sin (b)), (c, \sin (c))$  and  $(d, \sin (d))$  form a rectangle.
Answer:  a = n\pi
Problem: Find the smallest positive integer  $n$  for which the polynomial \[x^n-x^{n-1}-x^{n-2}-\cdots -x-1\] has a real root greater than  $1.999$ .

[i]Proposed by James Lin
Answer: 10
Problem: Find the derivative of the solution of the equation  $\ddot{x} = \dot{x}^2 + x^3$  with initial condition  $x(0) = 0$ ,  $\dot{x}(0) = A$  with respect to  $A$  for  $A = 0$ .
Answer: t
Problem: Find the number of ordered pairs of integers  $(a, b)$  that satisfy the inequality

\[
1 < a < b+2 < 10.
\]
Answer: 28
Problem: Find all triples of natural  $ (x,y,n)$  satisfying the condition:
\[ \frac {x! \plus{} y!}{n!} \equal{} 3^n
\]

Define  $ 0! \equal{} 1$ 
Answer:  (1, 2, 1) and (2, 1, 1)
Problem: Determine all sequences  $a_1,a_2,a_3,\dots$  of positive integers that satisfy the equation  $$ (n^2+1)a_{n+1}  - a_n = n^3+n^2+1 $$  for all positive integers  $n$ .
Answer:  a_n = n
Problem: Let  $a, b$ , and  $c$  be integers that satisfy  $2a + 3b = 52$ ,  $3b + c = 41$ , and  $bc = 60$ . Find  $a + b + c$ 
Answer: 25
Problem: What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction.
Answer: 11
Problem: Let  $\{a_n\}$  be a sequence of integers satisfying the following conditions. 


- $a_1=2021^{2021}$
- $0 \le a_k < k$  for all integers  $k \ge 2$
- $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$  is multiple of  $k$  for all positive integers  $k$ .

Determine the  $2021^{2022}$ th term of the sequence  $\{a_n\}$ . 
Answer: 0
Problem: A meeting is held at a round table. It is known that 7 women have a woman on their right side, and 12 women have a man on their right side. It is also known that 75% of the men have a woman on their right side. How many people are sitting at the round table?
Answer: 35
Problem: Find all functions  $f:\mathbb{N} \rightarrow \mathbb{N}$ , such that  $f(a)f(a+b)-ab$  is a perfect square for all  $a, b \in \mathbb{N}$ . 
Answer:  f(x) = x
Problem: Let  $I$  be the incenter of a tetrahedron  $ABCD$ , and  $J$  be the center of the exsphere touching the face  $BCD$  containing three remaining faces (outside these faces). The segment  $IJ$  meets the circumsphere of the tetrahedron at point  $K$ . Which of two segments  $IJ$  and  $JK$  is longer?
Answer:  IJ
Problem: The vertices of a right-angled triangle are on a circle of radius  $R$  and the sides of the triangle are tangent to another circle of radius  $r$  (this is the circle that is inside triangle). If the lengths of the sides about the right angles are 16 and 30, determine the value of  $R+r$ .
Answer: 23
Problem: A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?
Answer: 10
Problem: Denote by  $d(n)$  be the biggest prime divisor of  $|n|>1$ . Find all polynomials with integer coefficients satisfy;  $$ P(n+d(n))=n+d(P(n))  $$  for the all  $|n|>1$  integers such that  $P(n)>1$  and  $d(P(n))$  can be defined.
Answer: P(n) = n
Problem: Find all non-negative integer numbers  $n$  for which there exists integers  $a$  and  $b$  such that  $n^2=a+b$  and  $n^3=a^2+b^2.$ 
Answer: 0, 1, 2
Problem: Let n be a positive integer. Find all complex numbers  $x_{1}$ ,  $x_{2}$ , ...,  $x_{n}$  satisfying the following system of equations:
 $x_{1}+2x_{2}+...+nx_{n}=0$ , $x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$ ,
... $x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$ .
Answer:  x_1 = x_2 = \cdots = x_n = 0
Problem: Let  $ABC$  be a triangle with  $\angle BAC=117^\circ$ .  The angle bisector of  $\angle ABC$  intersects side  $AC$  at  $D$ .  Suppose  $\triangle ABD\sim\triangle ACB$ .  Compute the measure of  $\angle ABC$ , in degrees.
Answer: 42^{\circ}
Problem: For  $n > 1$ , let  $a_n$  be the number of zeroes that  $n!$  ends with when written in base  $n$ . Find the maximum value of  $\frac{a_n}{n}$ .
Answer: \frac{1}{2}
Problem: $ABCDEF$  is a hexagon inscribed in a circle such that the measure of  $\angle{ACE}$  is  $90^{\circ}$ . What is the average of the measures, in degrees, of  $\angle{ABC}$  and  $\angle{CDE}$ ?
Answer: 45^	ext{\circ}
Problem: Determine the polynomial  $P\in \mathbb{R}[x]$  for which there exists  $n\in \mathbb{Z}_{>0}$  such that for all  $x\in \mathbb{Q}$  we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]
Answer:  P(x) = ax + b
Problem: A solid rectangular block is formed by gluing together  $N$  congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of  $N$ .
Answer: 384
Problem: The straight line  $l_{1}$  with equation  $x-2y+10 = 0$  meets the circle with equation  $x^2 + y^2 = 100$  at B in the  first quadrant. A line through B, perpendicular to  $l_{1}$  cuts the y-axis at P (0, t). Determine the value of  $t$ .
Answer: t = 20
Problem: Given triangle  $ ABC$  of area 1. Let  $ BM$  be the perpendicular from  $ B$  to the bisector of angle  $ C$ . Determine the area of triangle  $ AMC$ .
Answer: \frac{1}{2}
Problem: Determine the smallest positive integer,  $n$ , which satisfies the equation  $n^3+2n^2 = b$ , where  $b$  is the square of an odd integer.
Answer: 7
Problem: Jeremy has three cups. Cup  $A$  has a cylindrical shape, cup  $B$  has a conical shape, and cup  $C$  has a hemispherical shape. The rim of the cup at the top is a unit circle for every cup, and each cup has the same volume. If the cups are ordered from least height to greatest height, what is the ordering of the cups?
Answer: A, C, B
Problem: Given  $ n$  points arbitrarily in the plane  $ P_{1},P_{2},\ldots,P_{n},$  among them no three points are collinear. Each of  $ P_{i}$  ( $1\le i\le n$ ) is colored red or blue arbitrarily. Let  $ S$  be the set of triangles having  $ \{P_{1},P_{2},\ldots,P_{n}\}$  as vertices, and having the following property: for any two segments  $ P_{i}P_{j}$  and  $ P_{u}P_{v},$  the number of triangles having  $ P_{i}P_{j}$  as side and the number of triangles having  $ P_{u}P_{v}$  as side are the same in  $ S.$  Find the least  $ n$  such that  in  $ S$  there exist two triangles, the vertices of each triangle having the same color.
Answer: 8
Problem: Find a function  $f$  such that  $f(t^2 +t +1) = t$  for all real  $t \ge 0$ 
Answer: f(m) = \frac{-1 + \sqrt{4m - 3}}{2}
Problem: How many ways can the vertices of a cube be colored red or blue so that the color of each vertex is the color of the majority of the three vertices adjacent to it?
Answer: 8
Problem: Let  $G$  be a connected graph and let  $X, Y$  be two disjoint subsets of its vertices, such that there are no edges between them. Given that  $G/X$  has  $m$  connected components and  $G/Y$  has  $n$  connected components, what is the minimal number of connected components of the graph  $G/(X \cup Y)$ ?
Answer: \max(m, n)
Problem:  $P$  is a point inside the triangle  $ABC$ . The line through  $P$  parallel to  $AB$  meets  $AC$   $A_0$  and  $BC$  at  $B_0$ . Similarly, the line through  $P$  parallel to  $CA$  meets  $AB$  at  $A_1$  and  $BC$  at  $C_1$ , and the line through P parallel to BC meets  $AB$  at  $B_2$  and  $AC$  at  $C_2$ . Find the point  $P$  such that  $A_0B_0 = A_1B_1 = A_2C_2$ .
Answer:  \left( \frac{1}{b} + \frac{1}{c} - \frac{1}{a} : \frac{1}{a} + \frac{1}{c} - \frac{1}{b} : \frac{1}{a} + \frac{1}{b} - \frac{1}{c} \right)
Problem: Let  $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$ . Compute the prime  $p$  satisfying  $f(p) = 418{,}195{,}493$ .
Answer: 2
Problem: If  $x$  is a real number satisfying the equation   $$ 9\log_3 x - 10\log_9 x =18 \log_{27} 45, $$   then the value of  $x$  is equal to  $m\sqrt{n}$ , where  $m$  and  $n$  are positive integers, and  $n$  is not divisible by the square of any prime. Find  $m+n$ .
Answer: 140
Problem: Find all triangles  $ABC$  such that  $\frac{a cos A + b cos B + c cos C}{a sin A + b sin B + c sin C} =\frac{a + b + c}{9R}$ , where, as usual,  $a, b, c$  are the lengths of sides  $BC, CA, AB$  and  $R$  is the circumradius.

Answer:  a = b = c
Problem: Find all functions  $f : \mathbb R \to \mathbb R$  for which
\[f(a - b) f(c - d) + f(a - d) f(b - c) \leq  (a - c) f(b - d),\]
for all real numbers  $a, b, c$  and  $d$ . Note that there is only one occurrence of  $f$  on the right hand side!
Answer:  f(x) = 0 for all x or f(x) = x for all x.
Problem: For a positive integer  $n$  let  $t_n$  be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with  $n$  elements. For example,  $t_3 = 1$ . Find a closed form formula for  $t_n$  and determine the last digit of  $t_{2022}$ .

(I also give here that  $t_4 = 10$ , for a reader to check his/her understanding of the problem statement.)
Answer: 1
Problem: We consider a fixed point  $P$  in the interior of a fixed sphere $.$  We construct three segments  $PA, PB,PC$ , perpendicular two by two $,$  with the vertexes  $A, B, C$  on the sphere $.$  We consider the vertex  $Q$  which is opposite to  $P$  in the parallelepiped (with right angles) with  $PA, PB, PC$  as edges $.$  Find the locus of the point  $Q$  when  $A, B, C$  take all the positions compatible with our problem.
Answer: a sphere centered at P with radius R
Problem: Let  $N{}$  be a fixed positive integer,  $S{}$  be the set  $\{1, 2,\ldots , N\}$  and  $\mathcal{F}$  be the set of functions  $f:S\to S$  such that  $f(i)\geqslant i$  for all  $i\in S.$  For each  $f\in\mathcal{F}$  let  $P_f$  be the unique polynomial of degree less than  $N{}$  satisfying  $P_f(i) = f(i)$  for all  $i\in S.$  If  $f{}$  is chosen uniformly at random from  $\mathcal{F}$  determine the expected value of  $P_f'(0)$  where\[P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.\]
Answer: \frac{N+1}{2}
Problem: Let  $p>q$  be primes, such that  $240 \nmid p^4-q^4$ . Find the maximal value of  $\frac{q} {p}$ .
Answer: \frac{2}{3}
Problem: Initially, on the blackboard are written all natural numbers from  $1$  to  $20$ . A move consists of selecting  $2$  numbers  $a<b$  written on the blackboard such that their difference is at least  $2$ , erasing these numbers and writting  $a+1$  and  $b-1$  instead. What is the maximum numbers of moves one can perform?
Answer: 9
Problem: What is the largest positive integer  $n$  for which there is a unique integer  $k$  such that  $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$ ?
Answer: 112
Problem: Let  $ABC$  be an isosceles triangle with  $AC=BC$ , let  $M$  be the midpoint of its side  $AC$ , and let  $Z$  be the line through  $C$  perpendicular to  $AB$ . The circle through the points  $B$ ,  $C$ , and  $M$  intersects the line  $Z$  at the points  $C$  and  $Q$ . Find the radius of the circumcircle of the triangle  $ABC$  in terms of  $m = CQ$ .
Answer: m
Problem: Let  $ABC$  be a triangle with circumcenter  $O$  such that  $AC = 7$ . Suppose that the circumcircle of  $AOC$  is tangent to  $BC$  at  $C$  and intersects the line  $AB$  at  $A$  and  $F$ . Let  $FO$  intersect  $BC$  at  $E$ . Compute  $BE$ .
Answer: \frac{7}{2}
Problem: There are two round tables with  $n{}$  dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any  $2n$  pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break  $n{}$  of those new friendships. For which  $n{}$  is the good wizard able to achieve his goal no matter what the evil sorcerer does?
Answer:  n
Problem: A survey of participants was conducted at the Olympiad.  $ 90\%$  of the participants liked the first round,  $60\%$  of the participants liked the second round,  $90\%$  of the participants liked the opening of the Olympiad. Each participant was known to enjoy at least two of these three events. Determine the percentage of participants who rated all three events positively.
Answer: 40\%
Problem: For arbiterary integers  $n,$  find the continuous function  $f(x)$  which satisfies the following equation. \[\lim_{h\rightarrow 0}\frac{1}{h}\int_{x-nh}^{x+nh}f(t) dt=2f(nx).\] Note that  $x$  can range all real numbers and  $f(1)=1.$ 
Answer:  f(x) = x
Problem: Two circles intersect at the points  $C$  and  $D$ . The straight lines  $CD$  and  $BYXA$  intersect at the point  $Z$ . Moreever, the straight line  $WB$  is tangent to both of the circles. Suppose  $ZX=ZY$  and  $AB\cdot AX=100$ . Find the value of  $BW$ .
Answer: 10
Problem: Let  $p$  be a prime number. Find all polynomials  $P$  with integer coefficients with the following properties: $(a)$   $P(x)>x$  for all positive integers  $x$ . $(b)$  The sequence defined by  $p_0:=p$ ,  $p_{n+1}:=P(p_n)$  for all positive integers  $n$ , satisfies the property that for all positive integers  $m$  there exists some  $l\geq 0$  such that  $m\mid p_l$ . 
Answer:  P(x) = x + 1 \text{ and } P(x) = x + p
Problem: Find all integers  $n$  for which both  $4n + 1$  and  $9n + 1$  are perfect squares.
Answer: 0
Problem: Emily starts with an empty bucket. Every second, she either adds a stone to the bucket or removes a stone from the bucket, each with probability  $\frac{1}{2}$ . If she wants to remove a stone from the bucket and the bucket is currently empty, she merely does nothing for that second (still with probability  $\hfill \frac{1}{2}$ ). What is the probability that after  $2017$  seconds her bucket contains exactly  $1337$  stones?
Answer: \frac{\binom{2017}{340}}{2^{2017}}
Problem: Hi guys,

here's the topic of the "**Oliforum contest**", the competition for training that i'm arranging from some weeks    :)  .
i'll try to sumarize all details about it, be carefully.
**1-** The contest is made of 2 rounds. 
**2-** Each round is made of some problems (from 3 to 5..), and i will give points 0 to 7: 6 for all correct solutions, 7 for *only* elegants and correct solutions (similar to ML contest). 
**3-** It is tought for high-school student, and also for undergraduate or higher student, but you cannot use non-elementary-theory, otherwise the solution is invalid.
**4-** All begins from italian-oliforum and the classification is unique.
**5-** The first round starts on 17:00 of 28 september and it ends on 17:00 of 30 seprember, so you have 48 hours :) .
        **NB.** All times (date and hours) are according to *ROME MERIDIAN (+2GTM),*please attention!
**6-** Enrollment is not necessary, it sufficies to send me solution.
**7-** Time of arriving solution is not important.
**8-** *Way of sending solutions*.
-          You need to send solutions to both email address: pao_leo88@hotmail.it and leonettipaolo@gmail.com in a *unique pdf file* (so, with attachment)  of a right size;
-          The pdf file must be written with *Latex* (or a way that I can understand);
-          You *rename the pdf file*as your nickname (the same of yours in ML);
-          You need to write nothing in the email;
-          Try to be *clear* in solutions!
 
{If *(and only if)* you are not able to create a pdf in latex model, you can send me a email in .doc format with solutions, as latex or in a way that i (and all correctors) can understand, or also with private message to me}.

Later, together with texts, I'll make a list below this topic where i'll write all names of who sent me solutions in right way.

For all doubts and questions, ask me in private messages :wink: 

 

Thanks for the attentions and good luck
Answer:  p = 3
Problem: There are a family of  $5$  siblings. They have a pile of at least  $2$  candies and are trying to split them up
amongst themselves. If the  $2$  oldest siblings share the candy equally, they will have  $1$  piece of candy left over.
If the  $3$  oldest siblings share the candy equally, they will also have  $1$  piece of candy left over. If all  $5$  siblings
share the candy equally, they will also have  $1$  piece left over. What is the minimum amount of candy required
for this to be true?
Answer: 31
Problem: Find all functions  $ f: \mathbb{N} \to \mathbb{Z} $  satisfying   $$  n \mid f\left(m\right) \Longleftrightarrow m \mid \sum\limits_{d \mid n}{f\left(d\right)}  $$   holds for all positive integers  $ m,n $ 
Answer:  f(n) = 0
Problem: Let  $a$  and  $b$  be integer solutions to  $17a+6b=13$ . What is the smallest possible positive value for  $a-b$ ?
Answer: 17
Problem: A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace?
Answer: 65
Problem: The positive reals  $x_1, x_2, ... , x_n$  have harmonic mean  $1$ . Find the smallest possible value of  $x_1 + \frac{x_2 ^2}{2} + \frac{x_3 ^3}{3} + ... + \frac{x_n ^n}{n}$ .
Answer: H_n
Problem: A positive integer  $N$  has  $20$  digits when written in base  $9$  and  $13$  digits when written in base  $27$ . How many digits does  $N$  have when written in base  $3$ ?
Answer: 39
Problem: For a positive integer  $n$ , two payers  $A$  and  $B$  play the following game: Given a pile of  $s$  stones, the players take turn alternatively with  $A$  going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of  $n$  stones. The winner is the one who takes the last stone. Assuming both  $A$  and  $B$  play perfectly, for how many values of  $s$  the player  $A$  cannot win?
Answer: n-1
Problem: Find all polynomials  $P(x)$  with integer coefficients such that for all positive number  $n$  and prime  $p$  satisfying  $p\nmid nP(n)$ , we have  $ord_p(n)\ge ord_p(P(n))$ .
Answer:  P(x) = c
Problem: Let  $ABC$  be a triangle in the plane with  $AB = 13$ ,  $BC = 14$ ,  $AC = 15$ . Let  $M_n$  denote the smallest possible value of  $(AP^n + BP^n + CP^n)^{\frac{1}{n}}$  over all points  $P$  in the plane. Find  $\lim_{n \to \infty} M_n$ .
Answer: 8.125
Problem: Let  $S$  be the increasing sequence of positive integers whose binary representation has exactly  $8$  ones. Let  $N$  be the  $1000^{th}$  number in  $S$ . Find the remainder when  $N$  is divided by  $1000$ .
Answer: 32
Problem: For any positive integer n, evaluate   $$ \sum_{i=0}^{\lfloor \frac{n+1}{2} \rfloor}
 {n-i+1 \choose i} $$  , where   $\lfloor n \rfloor$   is the greatest integer less than or equal to  $n$  .

Answer:  F_{n+1}
Problem: Given that  $\{a_n\}$  is a sequence of integers satisfying the following condition for all positive integral values of  $n$ :  $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2016$ . Find all possible values of  $a_1$  and  $a_2$ 
Answer:  (a_1, a_2) = (0, 2016), (-14, 70), (-69, 15), (-2015, 1), (2016, 0), (70, -14), (15, -69), (1, -2015)
Problem: Let  $ u_1$ ,  $ u_2$ ,  $ \ldots$ ,  $ u_{1987}$  be an arithmetic progression with  $ u_1 \equal{} \frac {\pi}{1987}$  and the common difference  $ \frac {\pi}{3974}$ . Evaluate
\[ S \equal{} \sum_{\epsilon_i\in\left\{ \minus{} 1, 1\right\}}\cos\left(\epsilon_1 u_1 \plus{} \epsilon_2 u_2 \plus{} \cdots \plus{} \epsilon_{1987} u_{1987}\right)
\]
Answer: 0
Problem: Basil needs to solve an exercise on summing two fractions  $\dfrac{a}{b}$  and  $\dfrac{c}{d}$ , where   $a$ ,  $b$ ,  $c$ ,  $d$  are some non-zero real numbers. But instead of summing he performed multiplication  (correctly). It appears that Basil's answer coincides with the correct answer to given exercise. Find the value of   $\dfrac{b}{a} + \dfrac{d}{c}$ .
Answer: 1
Problem: Find out how many positive integers  $n$  not larger than  $2009$  exist such that the last digit of  $n^{20}$  is  $1$ .
Answer: 804
Problem: The sequence of functions  $f_n:[0,1]\to\mathbb R$   $(n\ge2)$  is given by  $f_n=1+x^{n^2-1}+x^{n^2+2n}$ . Let  $S_n$  denote the area of the figure bounded by the graph of the function  $f_n$  and the lines  $x=0$ ,  $x=1$ , and  $y=0$ . Compute  $$ \lim_{n\to\infty}\left(\frac{\sqrt{S_1}+\sqrt{S_2}+\ldots+\sqrt{S_n}}n\right)^n. $$  
Answer: e^{\frac{1}{2}}
Problem: Determine all pairs of positive integers  $(x, y)$  satisfying the equation  $p^x - y^3 = 1$ , where  $p$  is a given prime number.
Answer:  (1, 1)  and  (2, 2)
Problem: Let  $S$  be the set of all 3-tuples  $(a, b, c)$  of positive integers such that  $a + b + c = 2013$ . Find   $$ \sum_{(a,b,c)\in S} abc. $$  
Answer: \binom{2015}{5}
Problem: What is the smallest possible natural number  $n$  for which the equation  $x^2 -nx + 2014 = 0$  has integer roots?
Answer: 91
Problem: Calvin makes a number. He starts with  $1$ , and on each move, he multiplies his current number by  $3$ , then adds  $5$ . After  $10$  moves, find the sum of the digits (in base  $10$ ) when Calvin's resulting number is expressed in base  $9$ .
Answer: 21
Problem: The perpendicular bisectors of the sides  $AB$  and  $CD$  of the rhombus  $ABCD$  are drawn. It turned out that they divided the diagonal  $AC$  into three equal parts. Find the altitude of the rhombus if  $AB = 1$ .
Answer: \frac{\sqrt{3}}{2}
Problem: Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region).  What is the maximum number of resulting regions?
Answer: 22
Problem: Geetha wants to cut a cube of size  $4 \times 4\times 4$  into  $64$  unit cubes (of size $1\times 1\times 1$ ). Every cut must be straight, and parallel to a face of the big cube. What
is the minimum number of cuts that Geetha needs?
Note: After every cut, she can rearrange the pieces before cutting again. At every
cut, she can cut more than one pieces as long as the pieces are on a straight line.
Answer: 9
Problem: Find all real numbers  $a$  such that the inequality  $3x^2 + y^2 \ge  -ax(x + y)$  holds for all real numbers  $x$  and  $y$ .
Answer: -2 \leq a \leq 6