rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Let  $l$  be the tangent line at the point  $(t,\ t^2)\ (0<t<1)$  on the parabola  $C: y=x^2$ . Denote by  $S_1$  the area of the part enclosed by  $C,\ l$  and the  $x$ -axis, denote by  $S_2$  of the area of the part enclosed by  $C,\ l$  and the line  $x=1$ . Find the minimum value of  $S_1+S_2$ .
Answer: \frac{1}{3}
Problem: Find the largest possible value of  $k$  for which  $3^{11}$  is expressible as the sum of  $k$  consecutive positive integers.
Answer: 486
Problem: Find all integers  $k$  such that both  $k + 1$  and  $16k + 1$  are perfect squares.
Answer: k = 0 \text{ and } k = 3
Problem: Suppose  $a$  is a real number such that  $3a + 6$  is the greatest integer less than or equal to  $a$  and  $4a + 9$  is the least integer greater than or equal to  $a$ . Compute  $a$ .
Answer: -3
Problem: Find the functions  $ f:\mathbb{N}\longrightarrow\mathbb{N} $  that satisfy the following relation:  $$  \gcd\left( x,f(y)\right)\cdot\text{lcm}\left(f(x), y\right) = \gcd (x,y)\cdot\text{lcm}\left( f(x), f(y)\right) ,\quad\forall x,y\in\mathbb{N} .  $$  
Answer:  f(x) = x
Problem: Determine the number of real solutions  $a$  to the equation:
\[ \left[\,\frac{1}{2}\;a\,\right]+\left[\,\frac{1}{3}\;a\,\right]+\left[\,\frac{1}{5}\;a\,\right] = a. \]
Here, if  $x$  is a real number, then  $[\,x\,]$  denotes the greatest integer that is less than or equal to  $x$ .
Answer: 0
Problem: Let  $ABCD$  be a convex quadrilateral such that  $AB + BC = 2021$  and  $AD = CD$ . We are also given that  $\angle ABC = \angle  CDA = 90^o$ . Determine the length of the diagonal  $BD$ .
Answer: \frac{2021 \sqrt{2}}{2}
Problem: Let  $ABCD$  be a parallelogram with  $\angle BAC = 45^\circ,$  and  $AC > BD .$  Let  $w_1$  and  $w_2$  be two circles with diameters  $AC$  and  $DC,$  respectively. The circle  $w_1$  intersects  $AB$  at  $E$  and the circle  $w_2$  intersects  $AC$  at  $O$  and  $C$ , and  $AD$  at  $F.$  Find the ratio of areas of triangles  $AOE$  and  $COF$  if  $AO = a,$  and  $FO = b .$ 
Answer: \frac{a^2}{b^2}
Problem: For real numbers  $x$  and  $y$  we define  $M(x, y)$  to be the maximum of the three numbers  $xy$ ,  $(x- 1)(y - 1)$ , and  $x + y - 2xy$ . Determine the smallest possible value of  $M(x, y)$  where  $x$  and  $y$  range over all real numbers satisfying  $0 \le x, y \le 1$ .
Answer: 1
Problem: At Ignus School there are  $425$  students. Of these students  $351$  study mathematics,  $71$  study Latin, and  $203$  study chemistry. There are  $199$  students who study more than one of these subjects, and  $8$  students who do not study any of these subjects. Find the number of students who study all three of these subjects.
Answer: 9
Problem: Suppose that the real numbers  $a_{1},a_{2},...,a_{2002}$  satisfying  $\frac{a_{1}}{2}+\frac{a_{2}}{3}+...+\frac{a_{2002}}{2003}=\frac{4}{3}$  $\frac{a_{1}}{3}+\frac{a_{2}}{4}+...+\frac{a_{2002}}{2004}=\frac{4}{5}$  $...$  $\frac{a_{1}}{2003}+\frac{a_{2}}{2004}+...+\frac{a_{2002}}{4004}=\frac{4}{4005}$ 
Evaluate the sum  $\frac{a_{1}}{3}+\frac{a_{2}}{5}+...+\frac{a_{2002}}{4005}$ .
Answer: 1 - \frac{1}{4005^{2}}
Problem: Two of the numbers  $a+b, a-b, ab, a/b$  are positive, the other two are negative. Find the sign of  $b$ 
Answer: b < 0
Problem: In base- $2$  notation, digits are  $0$  and  $1$  only and the places go up in powers of  $-2$ . For example,  $11011$  stands for  $(-2)^4+(-2)^3+(-2)^1+(-2)^0$  and equals number  $7$  in base  $10$ . If the decimal number  $2019$  is expressed in base  $-2$  how many non-zero digits does it contain ?
Answer: 6
Problem: Given positive integers  $a, b,$  and  $c$  with  $a + b + c = 20$ .
Determine the number of possible integer values for  $\frac{a + b}{c}.$ 
Answer: 6
Problem: Describe in geometric terms the set of points  $(x,y)$  in the plane such that  $x$  and  $y$  satisfy the condition  $t^2+yt+x\ge 0$  for all  $t$  with  $-1\le t\le 1$ .
Answer:  x = \frac{y^2}{4}
Problem: There exists complex numbers  $z=x+yi$  such that the point  $(x, y)$  lies on the ellipse with equation  $\frac{x^2}9+\frac{y^2}{16}=1$ . If  $\frac{z-1-i}{z-i}$  is real, compute  $z$ .

*(Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 3)*
Answer: \pm \frac{3\sqrt{15}}{4} + i
Problem: Suppose that  $ a + b = 20$ ,  $b + c = 22$ , and  $c + a = 2022$ . Compute  $\frac{a-b}{c-a}$  .
Answer: 1000
Problem: If  $f,g: \mathbb{R} \to \mathbb{R}$  are functions that satisfy  $f(x+g(y)) = 2x+y $   $\forall x,y \in \mathbb{R}$ , then determine  $g(x+f(y))$ .
Answer: \frac{x}{2} + y
Problem: Let  $N$  be the number of complex numbers  $z$  with the properties that  $|z|=1$  and  $z^{6!}-z^{5!}$  is a real number. Find the remainder when  $N$  is divided by  $1000$ .

Answer: 440
Problem: Find the least number of elements of a finite set  $A$  such that there exists a function  $f : \left\{1,2,3,\ldots \right\}\rightarrow A$  with the property: if  $i$  and  $j$  are positive integers and  $i-j$  is a prime number, then  $f(i)$  and  $f(j)$  are distinct elements of  $A$ .
Answer: 4
Problem: Find the least  $n$  such that any subset of  ${1,2,\dots,100}$  with  $n$  elements has 2 elements with a diﬀerence of 9.
Answer: 51
Problem: Let  $S$  be the set of all real numbers  $x$  such that  $0 \le x \le 2016 \pi$  and  $\sin x < 3 \sin(x/3)$ .  The set  $S$  is the union of a finite number of disjoint intervals.  Compute the total length of all these intervals.
Answer: 1008\pi
Problem: Let  $M$  be the midpoint of side  $AC$  of the triangle  $ABC$ . Let  $P$  be a point on the side  $BC$ . If  $O$  is the point of intersection of  $AP$  and  $BM$  and  $BO = BP$ , determine the ratio  $\frac{OM}{PC}$  .
Answer: \frac{1}{2}
Problem: Determine all pairs  $ (a,b)$  of real numbers such that  $ 10, a, b, ab$  is an arithmetic progression.
Answer: (4, -2) \text{ or } (2.5, -5)
Problem: Two lines intersect inside a unit square, splitting it into four regions. Find the maximum product of the areas of the four regions.

[i]Proposed by Nathan Ramesh
Answer: \frac{1}{256}
Problem: Let  $n$  be an integer. We consider  $s (n)$ , the sum of the  $2001$  powers of  $n$  with the exponents  $0$  to  $2000$ . So  $s (n) = \sum_{k=0}^{2000}n ^k$  . What is the unit digit of  $s (n)$  in the decimal system?
Answer: 1
Problem: Consider the set  $\mathcal{T}$  of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let  $\triangle \in \mathcal{T}$  be the triangle with least perimeter. If  $a^{\circ}$  is the largest angle of  $\triangle$  and  $L$  is its perimeter, determine the value of  $\frac{a}{L}$ .
Answer: 8
Problem: Pete has some trouble slicing a 20-inch (diameter) pizza. His first two cuts (from center to circumference of the pizza) make a 30º slice. He continues making cuts until he has gone around the whole pizza, each time trying to copy the angle of the previous slice but in fact adding 2º each time. That is, he makes adjacent slices of 30º, 32º, 34º, and so on. What is the area of the smallest slice?
Answer: 5\pi
Problem: Heather and Kyle need to mow a lawn and paint a room. If Heather does both jobs by herself, it will take her a total of nine hours. If Heather mows the lawn and, after she finishes, Kyle paints the room, it will take them a total of eight hours. If Kyle mows the lawn and, after he finishes, Heather paints the room, it will take them a total of seven hours. If Kyle does both jobs by himself, it will take him a total of six hours. It takes Kyle twice as long to paint the room as it does for him to mow the lawn. The number of hours it would take the two of them to complete the two tasks if they worked together to mow the lawn and then worked together to paint the room is a fraction  $\tfrac{m}{n}$ where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .
Answer: 41
Problem: find the smallest integer  $n\geq1$  such that the equation :  $$ a^2+b^2+c^2-nd^2=0  $$   
has  $(0,0,0,0)$  as unique solution .
Answer:  n = 4
Problem: Find all triplets  $(a,b,p)$  where  $a,b$  are positive integers and  $p$  is a prime number satisfying:  $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ 
Answer:  (2, 2, 2)
Problem: Let  $\triangle ABC$  satisfies  $\cos A:\cos B:\cos C=1:1:2$ , then  $\sin A=\sqrt[s]{t}$ ( $s\in\mathbb{N},t\in\mathbb{Q^+}$  and  $t$  is an irreducible fraction). Find  $s+t$ .
Answer: \frac{19}{4}
Problem: Tina writes four letters to her friends Silas, Jessica, Katie, and Lekan. She prepares an envelope for Silas, an envelope for Jessica, an envelope for Katie, and an envelope for Lekan. However, she puts each letter into a random envelope. What is the probability that no one receives the letter they are supposed to receive?
Answer: \frac{3}{8}
Problem: Find the largest  $2$ -digit number  $N$  which is divisible by  $4$ , such that all integral powers of  $N$  end with  $N$ .
Answer: 76
Problem: Nine people sit in three rows of three chairs each. The probability that two of them, Celery and Drum, sit next to each other in the same row is  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Find  $100m+n$ .
Answer: 209
Problem: Jenny places 100 pennies on a table, 30 showing heads and 70 showing tails.  She chooses 40 of the pennies at random (all different) and turns them over.  That is, if a chosen penny was showing heads, she turns it to show tails; if a chosen penny was showing tails, she turns it to show heads.  At the end, what is the expected number of pennies showing heads?
Answer: 46
Problem:  $ g(x)$  is a differentiable function for  $ 0\leq x\leq \pi$  and  $ g'(x)$  is a continuous function for  $ 0\leq x\leq \pi$ .
Let  $ f(x) \equal{} g(x)\sin x.$  Find  $ g(x)$  such that  $ \int_0^{\pi} \{f(x)\}^2dx \equal{} \int_0^{\pi}\{f'(x)\}^2dx$ .
Answer: g(x) = c
Problem: Today, Ivan the Confessor prefers continuous functions  $f:[0,1]\to\mathbb{R}$  satisfying  $f(x)+f(y)\geq |x-y|$  for all pairs  $x,y\in [0,1]$ . Find the minimum of  $\int_0^1 f$  over all preferred functions.

(Proposed by Fedor Petrov, St. Petersburg State University)
Answer: \frac{1}{4}
Problem: If  $x$ ,  $y$ ,  $k$  are positive reals such that \[3=k^2\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right)+k\left(\dfrac{x}{y}+\dfrac{y}{x}\right),\] find the maximum possible value of  $k$ .
Answer: \frac{\sqrt{7} - 1}{2}
Problem: What is the sum of all distinct values of  $x$  that satisfy  $x^4-x^3-7x^2+13x-6=0$ ?
Answer: 0
Problem: What is the smallest number that could be the date of the first Saturday after the second Monday following the second Thursday of a month?
Answer: 17
Problem: Find all the real numbers  $ \alpha$  satisfy the following property: for any positive integer  $ n$  there exists an integer  $ m$  such that  $ \left |\alpha\minus{}\frac{m}{n}\right|<\frac{1}{3n}$ .
Answer: all real numbers
Problem: The first term of a sequence is  $2014$ . Each succeeding term is the sum of the cubes of the digits of the previous term. What is the  $2014$  th term of the sequence?
Answer: 370
Problem: Find all triples  $ (x,y,z)$  of real numbers which satisfy the simultaneous equations

\[ x \equal{} y^3 \plus{} y \minus{} 8\]

\[y \equal{} z^3 \plus{} z \minus{} 8\]

\[ z \equal{} x^3 \plus{} x \minus{} 8.\]
Answer:  (x, y, z) = (2, 2, 2)
Problem: Determine the lowest positive integer n such that following statement is true:
If polynomial with integer coefficients gets value 2 for n different integers, 
then it can't take value 4 for any integer.
Answer:  n = 4
Problem: The expressions  $A=1\times2+3\times4+5\times6+\cdots+37\times38+39$  and  $B=1+2\times3+4\times5+\cdots+36\times37+38\times39$  are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers  $A$  and  $B$ .
Answer: 722
Problem: Shirley went to the store planning to buy  $120$  balloons for  $10$  dollars. When she arrived, she was surprised to nd that the balloons were on sale for  $20$  percent less than expected. How many balloons could Shirley buy for her  $10$  dollars?
Answer: 150
Problem: Mike has two similar pentagons. The first pentagon has a perimeter of  $18$  and an area of  $8 \frac{7}{16}$  . The second pentagon has a perimeter of  $24$ . Find the area of the second pentagon.
Answer: 15
Problem: Let  $S$  be the set of positive integer divisors of  $20^9.$  Three numbers are chosen independently and at random from the set  $S$  and labeled  $a_1,a_2,$  and  $a_3$  in the order they are chosen. The probability that both  $a_1$  divides  $a_2$  and  $a_2$  divides  $a_3$  is  $\frac mn,$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m.$ 
Answer: 77
Problem:  $100$  distinct natural numbers  $a_1, a_2, a_3, \ldots, a_{100}$  are written on the board. Then, under each number  $a_i$ , someone wrote a number  $b_i$ , such that  $b_i$  is the sum of  $a_i$  and the greatest common factor of the other  $99$  numbers. What is the least possible number of distinct natural numbers that can be among  $b_1, b_2, b_3, \ldots, b_{100}$ ?
Answer: 100
Problem: Find all positive integers  $n$  such that the product of all positive divisors of  $n$  is  $24^{240}$ .
Answer:  n = 24^5
Problem: The sum of all of the interior angles of seven polygons is  $180\times17$ . Find the total number of sides of the polygons.
Answer: 31
Problem: Given triangle  $ABC$ , let  $D$ ,  $E$ ,  $F$  be the midpoints of  $BC$ ,  $AC$ ,  $AB$  respectively and let  $G$  be the centroid of the triangle. For each value of  $\angle BAC$ , how many non-similar triangles are there in which  $AEGF$  is a cyclic quadrilateral?
Answer: 2
Problem: Let  $n{}$  be a positive integer. What is the smallest sum of digits that  $5^n + 6^n + 2022^n$  can take?
Answer: 8
Problem: Let  $X$  be a set with  $n$  elements. Find the largest number of subsets of  $X$ , each with  $3$  elements, so that no two of them are disjoint.
Answer: \binom{n-1}{2}
Problem: Find all function  $f,g: \mathbb{Q} \to \mathbb{Q}$  such that
\[\begin{array}{l}
 f\left( {g\left( x \right) - g\left( y \right)} \right) = f\left( {g\left( x \right)} \right) - y  
 g\left( {f\left( x \right) - f\left( y \right)} \right) = g\left( {f\left( x \right)} \right) - y  
 \end{array}\]
for all  $x,y \in \mathbb{Q}$ .
Answer:  f(x) = cx } \text{ and } g(x) = \frac{x}{c}
Problem: Let  $F$  be a finite field having an odd number  $m$  of elements. Let  $p(x)$  be an irreducible (i.e. nonfactorable) polynomial over  $F$  of the form   $$ x^2+bx+c, ~~~~~~ b,c \in F. $$   For how many elements  $k$  in  $F$  is  $p(x)+k$  irreducible over  $F$ ?
Answer: n = \frac{m-1}{2}
Problem: Let  $x,\ y$  be real numbers. For a function  $f(t)=x\sin t+y\cos t$ , draw the domain of the points  $(x,\ y)$  for which the following inequality holds.
\[\left|\int_{-\pi}^{\pi} f(t)\cos t\ dt\right|\leq \int_{-\pi}^{\pi} \{f(t)\}^2dt.\]
Answer:  x^2 + \left(y - \frac{1}{2}\right)^2 \geq \frac{1}{4}
Problem: Some blue and red circular disks of identical size are packed together to form a triangle. The top level has one disk and each level has 1 more disk than the level above it. Each disk not at the bottom level touches two disks below it and its colour is blue if these two disks are of the same colour. Otherwise its colour is red.

Suppose the bottom level has 2048 disks of which 2014 are red. What is the colour of the disk at the top?
Answer: blue
Problem: Find all positive integer  $n$  such that  $$ \frac{\sin{n\theta}}{\sin{\theta}} - \frac{\cos{n\theta}}{\cos{\theta}} = n-1 $$  holds for all  $\theta$  which are not integral multiples of  $\frac{\pi}{2}$  
Answer: n = 1
Problem: Let  $S_{n}=\{1,n,n^{2},n^{3}, \cdots \}$ , where  $n$  is an integer greater than  $1$ . Find the smallest number  $k=k(n)$  such that there is a number which may be expressed as a sum of  $k$  (possibly repeated) elements in  $S_{n}$  in more than one way. (Rearrangements are considered the same.)
Answer: k(n) = n + 1
Problem: Find all monic polynomials  $P(x)=x^{2023}+a_{2022}x^{2022}+\ldots+a_1x+a_0$  with real coefficients such that  $a_{2022}=0$ ,  $P(1)=1$  and all roots of  $P$  are real and less than  $1$ .
Answer:  P(x) = x^{2023}
Problem: Find all triplets of positive rational numbers  $(m,n,p)$  such that the numbers  $m+\frac 1{np}$ ,  $n+\frac 1{pm}$ ,  $p+\frac 1{mn}$  are integers.
Answer: \left(\frac{1}{2}, \frac{1}{2}, 4\right), \left(\frac{1}{2}, 1, 2\right), (1, 1, 1)
Problem: Find all integers  $n>1$  such that any prime divisor of  $n^6-1$  is a divisor of  $(n^3-1)(n^2-1)$ .
Answer: n = 2
Problem: A positive integer is called *cool* if it can be expressed in the form  $a!\cdot b!+315$  where  $a,b$  are positive integers. For example,  $1!\cdot 1!+315=316$  is a cool number. Find the sum of all cool numbers that are also prime numbers.

[i]Proposed by Evan Fang
Answer: 317
Problem: A rectangular wooden block has a square top and bottom, its volume is  $576$ , and the surface area of its vertical sides is  $384$ . Find the sum of the lengths of all twelve of the edges of the block.
Answer: 112
Problem: Find all ordered pairs  $(a,b)$ , of natural numbers, where  $1<a,b\leq 100$ , such that $\frac{1}{\log_{a}{10}}+\frac{1}{\log_{b}{10}}$  is a natural number.
Answer: (2, 5), (5, 2), (4, 25), (25, 4), (10, 10), (20, 5), (5, 20), (10, 100), (100, 10), (20, 50), (50, 20), (25, 40), (40, 25), (100, 100)
Problem: The cost of producing each item is inversely proportional to the square root of the number of items produced. The cost of producing ten items is  $ \$ 2100 $. If items sell for $  \ $30$  each, how many items need to be sold so that the producers break even?
Answer: 10 \sqrt[3]{49}
Problem: Define a "digitized number" as a ten-digit number  $a_0a_1\ldots a_9$  such that for  $k=0,1,\ldots, 9$ ,  $a_k$  is equal to the number of times the digit  $k$  occurs in the number.  Find the sum of all digitized numbers.
Answer: 6210001000
Problem: Find the positive constant  $c_0$  such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for  $c>c_0$  and diverges for  $0<c<c_0$ .
Answer: \frac{1}{e}
Problem: Solve following system equations:
\[\left\{ \begin{array}{c}
3x+4y=26\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \  
\sqrt{x^2+y^2-4x+2y+5}+\sqrt{x^2+y^2-20x-10y+125}=10\  \end{array}
\right.\ \ \] 



Answer: (x, y) = (6, 2)
Problem: Let  $a_0$ ,  $a_1$ ,  $a_2$ ,  $\dots$  be an infinite sequence of real numbers such that  $a_0 = \frac{4}{5}$  and
\[
  a_{n} = 2 a_{n-1}^2 - 1
\]
for every positive integer  $n$ .  Let  $c$  be the smallest number such that for every positive integer  $n$ , the product of the first  $n$  terms satisfies the inequality
\[
  a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.
\]
What is the value of  $100c$ , rounded to the nearest integer?
Answer: 167
Problem: A set  $\mathcal{S}$  of distinct positive integers has the following property: for every integer  $x$  in  $\mathcal{S},$  the arithmetic mean of the set of values obtained by deleting  $x$  from  $\mathcal{S}$  is an integer.  Given that 1 belongs to  $\mathcal{S}$  and that 2002 is the largest element of  $\mathcal{S},$  what is the greatet number of elements that  $\mathcal{S}$  can have?
Answer: 30
Problem: Determine the number of integers  $ n$  with  $ 1 \le n \le N\equal{}1990^{1990}$  such that  $ n^2\minus{}1$  and  $ N$  are coprime.
Answer: 591 \times 1990^{1989}
Problem: Let  $\mathcal{F}$  be a family of subsets of  $\{1,2,\ldots, 2017\}$  with the following property: if  $S_1$  and  $S_2$  are two elements of  $\mathcal{F}$  with  $S_1\subsetneq S_2$ , then  $|S_2\setminus S_1|$  is odd. Compute the largest number of subsets  $\mathcal{F}$  may contain.
Answer: 2 \binom{2017}{1008}
Problem: Let  $ l_1,\ l_2$  be the tangent and nomal line respectively at the point  $ (p,\ \ln (p \plus{} 1))$  on the curve  $ C: y \equal{} \ln (x \plus{} 1)$ .
Denote by  $ T_i\ (i \equal{} 1,\ 2)$  the areas bounded by  $ l_i\ (i \equal{} 1,\ 2), C$  and the  $ y$  axis respectively. Find the limit  $ \lim_{p\rightarrow 0} \frac {T_2}{T_1}$ .
Answer: -1
Problem: Take the number   $2^{2004}$  and calculate the sum   $S$  of all its digits. Then the sum of all the digits of   $S$  is calculated to obtain   $R$ . Next, the sum of all the digits of  $R$ is calculated   and so on until a single digit number is reached. Find it.  (For example if we take    $2^7=128$ , we find that    $S=11,R=2$ . So in this case of    $2^7$  the searched digit will be  $2$ ).
Answer: 1
Problem: It is known that  $\frac{7}{13} + \sin \phi = \cos \phi$  for some real  $\phi$ . What is sin  $2\phi$ ?
Answer: \frac{120}{169}
Problem: Compute the number of real numbers  $t$  such that \[t = 50 \sin(t - \lfloor t \rfloor).\] Here  $\lfloor \cdot\rfloor$  denotes the greatest integer function.
Answer: 50
Problem: Perimeter of triangle  $ABC$  is  $1$ . Circle  $\omega$  touches side  $BC$ , continuation of side  $AB$  at  $P$  and continuation of side  $AC$  in  $Q$ . Line through midpoints  $AB$  and  $AC$  intersects circumcircle of  $APQ$  at  $X$  and  $Y$ .
Find length of  $XY$ . 

Answer: \frac{1}{2}
Problem: Find the smallest real  $\alpha$ , such that for any convex polygon  $P$  with area  $1$ , there exist a point  $M$  in the plane, such that the area of convex hull of  $P\cup Q$  is at most  $\alpha$ , where  $Q$  denotes the image of  $P$  under central symmetry with respect to  $M$ .
Answer: \frac{4}{3}
Problem: **Problem 2**
Determine all pairs  $(n, m)$  of positive integers satisfying the equation  $$ 5^n = 6m^2 + 1\ .  $$  
Answer:  (2, 2)
Problem: It's pouring down rain, and the amount of rain hitting point  $(x,y)$  is given by   $$ f(x,y)=|x^3+2x^2y-5xy^2-6y^3|. $$  If you start at the origin  $(0,0)$ , find all the possibilities for  $m$  such that  $y=mx$  is a straight line along which you could walk without any rain falling on you.
Answer: -1, \frac{1}{2}, -\frac{1}{3}
Problem: Given a triangle  $ABC$  with integer side lengths, where  $BD$  is an angle bisector of  $\angle ABC$ ,  $AD=4$ ,  $DC=6$ , and  $D$  is on  $AC$ , compute the minimum possible perimeter of  $\triangle ABC$ .
Answer: 25
Problem: Which configurations of five (not necessarily distinct) points  $p_1 ,\ldots, p_5$  on the circle  $x^2 +y^2 =1$  maximize the sum of the ten distances  $$ \sum_{i<j} d(p_i, p_j)? $$  
Answer: 10 \left( \sin \frac{\pi}{5} + \sin \frac{2\pi}{5} \right)
Problem: Find all real triples  $(a,b,c)$ , for which   $$ a(b^2+c)=c(c+ab) $$     $$ b(c^2+a)=a(a+bc) $$     $$ c(a^2+b)=b(b+ca). $$  
Answer: (a, b, c) = (x, x, x) \text{ for any } x \in \mathbb{R}
Problem: Let  $ABC$  be a triangle with area  $K$ . Points  $A^*$ ,  $B^*$ , and  $C^*$  are chosen on  $AB$ ,  $BC$ , and  $CA$  respectively such that  $\triangle{A^*B^*C^*}$  has area  $J$ . Suppose that \[\frac{AA^*}{AB}=\frac{BB^*}{BC}=\frac{CC^*}{CA}=\frac{J}{K}=x\] for some  $0<x<1$ . What is  $x$ ?
Answer: \frac{1}{3}
Problem: Find the maximum possible value of  $n$ , such that the integers  $1,2,\ldots n$  can be filled once each in distinct cells of a  $2015\times 2015$  grid and satisfies the following conditions:
[list]
[*] For all  $1\le i \le n-1$ , the cells with  $i$  and  $i+1$  share an edge. Cells with  $1$  and  $n$  also share an edge. In addition, no other pair of numbers share an edge.
[*] If two cells with  $i<j$  in them share a vertex, then  $\min\{j-i,n+i-j\}=2$ .
Answer: \frac{2016^2}{2}
Problem: How many ten-digit whole numbers satisfy the following property: they have only  $2$  and  $5$  as digits, and there are no consecutive  $2$ 's in the number (i.e. any two  $2$ 's are separated by at least one  $5$ )?
Answer: 144
Problem: What is the sum of all the integers  $n$  such that  $\left|n-1\right|<\pi$ ?
Answer: 7
Problem: Given a square  $ABCD,$  with  $AB=1$  mark the midpoints  $M$  and  $N$  of  $AB$  and  $BC,$  respectively. A lasar beam shot from  $M$  to  $N,$  and the beam reflects of  $BC,CD,DA,$  and comes back to  $M.$  This path encloses a smaller area inside square  $ABCD.$  Find this area. 
Answer: \frac{1}{2}