rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: One rainy afternoon you write the number  $1$  once, the number  $2$  twice, the number  $3$  three times, and so forth until you have written the number  $99$  ninety-nine times. What is the  $2005$  th digit that you write?
Answer: 1
Problem: For real numbers  $x, y$  and  $z$  it is known that  $x + y = 2$  and  $xy = z^2 + 1$ .
Find the value of the expression  $x^2 + y^2+ z^2$ .
Answer: 2
Problem: Let  $a, b, c$  be positive integers such that  $$ \gcd(a, b) + \text{lcm}(a, b) = \gcd(a, c) + \text{lcm}(a, c). $$  Does it follow from this that  $b = c$ ?
Answer: b = c
Problem: You are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant speed of  $10$  m/s throughout the race. Meanwhile, Al starts running at a constant speed of  $ 1$  m/s. Thereafter, when exactly  $ 1$  second has passed from when Al last changed its speed, Al’s speed instantaneously becomes  $1$  m/s faster, so that Al runs at a constant speed of  $k$  m/s in the kth second of the race. (Start counting seconds at  $ 1$ ). Suppose Al beats you by exactly  $1$  second. How many meters was the race?
Answer: 210
Problem: Solve the system   $\begin {cases} 16x^3 + 4x = 16y + 5 
16y^3 + 4y = 16x + 5 \end{cases}$ 
Answer:  x = y
Problem: A farmer buys a batch of trees, which he wishes to plant in a square grid. For example, if he had  $25$  trees, then he could plant them as shown below.

[asy]
size(3cm,0);
dot((0,0));
dot((0,1));
dot((0,2));
dot((0,3));
dot((0,4));
dot((1,0));
dot((1,1));
dot((1,2));
dot((1,3));
dot((1,4));
dot((2,0));
dot((2,1));
dot((2,2));
dot((2,3));
dot((2,4));
dot((3,0));
dot((3,1));
dot((3,2));
dot((3,3));
dot((3,4));
dot((4,0));
dot((4,1));
dot((4,2));
dot((4,3));
dot((4,4));
[/asy]

However, the farmer finds that he cannot plant his trees in a square grid. If he had  $20$  more trees, or if he had  $39$  fewer trees, then he could plant his trees in a square grid. How many trees did the farmer buy?
Answer: 880
Problem: You are bargaining with a salesperson for the price of an item. Your first offer is  $a$  dollars and theirs is  $b$  dollars. After you raise your offer by a certain percentage and they lower their offer by the same percentage, you arrive at an agreed price. What is that price, in terms of  $a$  and  $b$ ?
Answer: \frac{2ab}{a + b}
Problem: Three real numbers  $x, y, z$  are such that  $x^2 + 6y = -17, y^2 + 4z = 1$  and  $z^2 + 2x = 2$ . What is the value of  $x^2 + y^2 + z^2$ ?
Answer: 14
Problem: What is the least positive integer  $n$  for which  $9n$  is a perfect square and  $12n$  is a perfect cube?
Answer: 144
Problem: Find all triples  $(a, b, c)$  of positive integers such that $a^2 + b^2 = n\cdot lcm(a, b) + n^2$  $b^2 + c^2 = n \cdot lcm(b, c) + n^2$  $c^2 + a^2 = n \cdot lcm(c, a) + n^2$ for some positive integer  $n$ .
Answer: (a, b, c) = (n, n, n)
Problem: Let  $n$  be the number of ordered quadruples  $(x_1,x_2,x_3,x_4)$  of positive odd integers that satisfy  $\sum_{i=1}^4 x_i=98.$   Find  $\frac n{100}.$ 
Answer: 248.04
Problem: A finite set  $\{a_1, a_2, ... a_k\}$  of positive integers with  $a_1 < a_2 < a_3 < ... < a_k$  is named  *alternating* if  $i+a$  for  $i = 1, 2, 3, ..., k$  is even. The empty set is also considered to be alternating. The number of alternating subsets of  $\{1, 2, 3,..., n\}$  is denoted by  $A(n)$ .
Develop a method to determine  $A(n)$  for every  $n \in N$  and calculate hence  $A(33)$ .
Answer: 5702887
Problem: Find the smallest positive integer  $n,$  such that    $3^k+n^k+ (3n)^k+ 2014^k$  is a perfect square for all natural numbers  $k,$  but not a perfect cube, for all natural numbers  $k.$ 
Answer: 2
Problem: Let  $\triangle ABC$  be a triangle with circumcenter  $O$  satisfying  $AB=13$ ,  $BC = 15$ , and  $AC = 14$ .  Suppose there is a point  $P$  such that  $PB \perp BC$  and  $PA \perp AB$ .  Let  $X$  be a point on  $AC$  such that  $BX \perp OP$ . What is the ratio  $AX/XC$ ?
Answer: \frac{169}{225}
Problem: $ABC$  is an isosceles triangle with  $AB=AC$ . Point  $D$  is constructed on AB such that  $\angle{BCD}=15^\circ$ . Given that  $BC=\sqrt{6}$  and  $AD=1$ , find the length of  $CD$ .
Answer: \sqrt{13 - 6\sqrt{3}}
Problem: Suppose that a sequence  $(a_n)_{n=1}^{\infty}$  of integers has the following property: For all  $n$  large enough (i.e.  $n \ge N$  for some  $N$  ),  $a_n$  equals the number of indices  $i$ ,  $1 \le i < n$ , such that  $a_i + i \ge n$ . Find the maximum possible number of integers which occur infinitely many times in the sequence.
Answer:  2
Problem: Let  $\mathbb{Z}$  denote the set of all integers. Find all polynomials  $P(x)$  with integer coefficients that satisfy the following property:

For any infinite sequence  $a_1$ ,  $a_2$ ,  $\dotsc$  of integers in which each integer in  $\mathbb{Z}$  appears exactly once, there exist indices  $i < j$  and an integer  $k$  such that  $a_i +a_{i+1} +\dotsb +a_j = P(k)$ .
Answer:  P(x) = ax + b
Problem: The sides of triangle are  $x$ ,  $2x+1$  and  $x+2$  for some positive rational  $x$ . Angle of triangle is  $60$  degree. Find perimeter
Answer: 9
Problem: Square  $CASH$  and regular pentagon  $MONEY$  are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have?
Answer: 8
Problem: Let  $ABCD$  be a square. Let  $E, Z$  be points on the sides  $AB, CD$  of the square respectively, such that  $DE\parallel BZ$ . Assume that the triangles  $\triangle EAD, \triangle ZCB$  and the parallelogram  $BEDZ$  have the same area. 

If the distance between the parallel lines  $DE$  and  $BZ$  is equal to  $1$ , determine the area of the square.
Answer: 4
Problem: Find the number of different quadruples  $(a, b, c, d)$  of positive integers such that  $ab =cd = a + b + c + d - 3$ .
Answer: 5
Problem: Let  $\mathcal{P}$  be a parabola that passes through the points  $(0, 0)$  and  $(12, 5)$ . Suppose that the directrix of  $\mathcal{P}$  takes the form  $y = b$ . (Recall that a parabola is the set of points equidistant from a point called the focus and line called the directrix) Find the minimum possible value of  $|b|$ .
Answer: 7.2
Problem: Find all the triplets of real numbers  $(x , y , z)$  such that : $y=\frac{x^3+12x}{3x^2+4}$  ,  $z=\frac{y^3+12y}{3y^2+4}$  ,  $x=\frac{z^3+12z}{3z^2+4}$ 
Answer: (0, 0, 0), (2, 2, 2), (-2, -2, -2)
Problem: For a given integer  $k \geq 1$ , find all  $k$ -tuples of positive integers  $(n_1,n_2,...,n_k)$  with  $\text{GCD}(n_1,n_2,...,n_k) = 1$  and  $n_2|(n_1+1)^{n_1}-1$ ,  $n_3|(n_2+1)^{n_2}-1$ , ... ,  $n_1|(n_k+1)^{n_k}-1$ .
Answer: (1, 1, \ldots, 1)
Problem: Let  $ \alpha_1$ ,  $ \alpha_2$ ,  $ \ldots$ ,  $ \alpha_{2008}$  be real numbers. Find the maximum value of \[ \sin\alpha_1\cos\alpha_2 \plus{} \sin\alpha_2\cos\alpha_3 \plus{} \cdots \plus{} \sin\alpha_{2007}\cos\alpha_{2008} \plus{} \sin\alpha_{2008}\cos\alpha_1\]
Answer: 1004
Problem: For each nonnegative integer  $n$  we define  $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$ . Find the greatest common divisor of the numbers  $A_0,A_1,\ldots, A_{1999}$ .
Answer: 7
Problem: The parabola  $\mathcal P$  given by equation  $y=x^2$  is rotated some acute angle  $\theta$  clockwise about the origin such that it hits both the  $x$  and  $y$  axes at two distinct points.  Suppose the length of the segment  $\mathcal P$  cuts the  $x$ -axis is  $1$ .  What is the length of the segment  $\mathcal P$  cuts the  $y$ -axis?
Answer: 2
Problem: Suppose that  $x$  and  $y$  are real numbers that satisfy the system of equations $2^x-2^y=1$  $4^x-4^y=\frac{5}{3}$ Determine  $x-y$ 
Answer: 2
Problem: Let  $R=(8,6)$ . The lines whose equations are  $8y=15x$  and  $10y=3x$  contain points  $P$  and  $Q$ , respectively, such that  $R$  is the midpoint of  $\overline{PQ}$ . The length of  $PQ$  equals  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .
Answer: 67
Problem: Does there exists a function  $f : \mathbb{N} \longrightarrow \mathbb{N}$ \begin{align*} f \left( m+ f(n) \right) = f(m) +f(n) + f(n+1) \end{align*}
for all  $m,n \in \mathbb{N}$  ?
Answer:  f \equiv 0
Problem: The number  $16^4+16^2+1$  is divisible by four distinct prime numbers. Compute the sum of these four primes.
Answer: 264
Problem: Suppose  $a,b,c,x,y,z$  are pairwisely different real numbers. How many terms in the following can be  $1$  at most:   $$ \begin{aligned}
&ax+by+cz,&&&&ax+bz+cy,&&&&ay+bx+cz,
&ay+bz+cx,&&&&az+bx+cy,&&&&az+by+cx?
\end{aligned} $$  
Answer: 2
Problem:  $a_0, a_1, \ldots, a_{100}$  and  $b_1, b_2,\ldots, b_{100}$  are sequences of real numbers, for which the property holds: for all  $n=0, 1, \ldots, 99$ , either  $$ a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n, $$  or  $$ a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n. $$  Given  $a_{100}\leq a_0$ , what is the maximal value of  $b_1+b_2+\cdots+b_{100}$ ?
Answer: 50
Problem: Determine all integers  $a$  and  $b$  such that
\[(19a + b)^{18} + (a + b)^{18} + (a + 19b)^{18}\]
is a perfect square.
Answer: (a, b) = (0, 0)
Problem: Find the smallest positive integer  $j$  such that for every polynomial  $p(x)$  with integer coefficients and for every integer  $k,$  the integer 
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]
(the  $j$ -th derivative of  $p(x)$  at  $k$ ) is divisible by  $2016.$ 
Answer: 8
Problem: Let  $a,b,c,d$  be positive integers such that the number of pairs  $(x,y) \in (0,1)^2$  such that both  $ax+by$  and  $cx+dy$  are integers is equal with 2004. If  $\gcd (a,c)=6$  find  $\gcd (b,d)$ .
Answer: 2
Problem: 
	 Determine the number of integers  $2 \le n \le 2016$  such that  $n^n-1$  is divisible by  $2$ ,  $3$ ,  $5$ ,  $7$ .
Answer: 9
Problem: Find all polynomials  $f$  with real coefficients such that for all reals  $x, y, z$  such that  $x+y+z =0$ , the following relation holds:   $$ f(xy) + f(yz) + f(zx) = f(xy + yz + zx). $$  
Answer: f(x) = cx
Problem: Alex has an  $20 \times 16$  grid of lightbulbs, initially all off. He has  $36$  switches, one for each row and column. Flipping the switch for the  $i$ th row will toggle the state of each lightbulb in the  $i$ th row (so that if it were on before, it would be off, and  vice versa). Similarly, the switch for the  $j$ th column will toggle the state of each bulb in the  $j$ th column. Alex makes some (possibly empty) sequence of switch flips, resulting in some configuration of the lightbulbs and their states. How many distinct possible configurations of lightbulbs can Alex achieve with such a sequence? Two configurations are distinct if there exists a lightbulb that is on in one configuration and off in another.
Answer: 2^{35}
Problem: Given a  $3 \times 3$  symmetric real matrix  $A$ , we define  $f(A)$  as a  $3 \times 3$  matrix with the same eigenvectors of  $A$  such that if  $A$  has eigenvalues  $a$ ,  $b$ ,  $c$ , then  $f(A)$  has eigenvalues  $b+c$ ,  $c+a$ ,  $a+b$  (in that order). We define a sequence of symmetric real  $3\times3$  matrices  $A_0, A_1, A_2, \ldots$  such that  $A_{n+1} = f(A_n)$  for  $n \geq 0$ . If the matrix  $A_0$  has no zero entries, determine the maximum number of indices  $j \geq 0$  for which the matrix  $A_j$  has any null entries.
Answer: 2
Problem: Let  $a_1, a_2,...,a_{2018}$  be a sequence of numbers such that all its elements are elements of a set  $\{-1,1\}$ . Sum   $$ S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j $$   can be negative and can also be positive. Find the minimal value of this sum 
Answer: -1009
Problem: What is the largest two-digit integer for which the product of its digits is  $17$  more than their sum?
Answer: 74
Problem: Let  $\alpha > 0$  be a real number. Compute the limit of the sequence  $\{x_n\}_{n\geq 1}$  defined by   $$ x_n=\begin{cases} \sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right),& \text{when}\ n>\frac{1}{\alpha} 0,& \text{when}\ n\leq \frac{1}{\alpha}\end{cases} $$  
Answer: \frac{1}{2}
Problem: Find all positive integers  $k$  such that for any positive integer  $n$ ,  $2^{(k-1)n+1}$  does not divide  $\frac{(kn)!}{n!}$ .
Answer: 2
Problem: The roots of the polynomial  $x^3 - \frac32 x^2 - \frac14 x + \frac38 = 0$  are in arithmetic progression. What are they?
Answer: x = -\frac{1}{2}, \frac{1}{2}, \frac{3}{2}
Problem: Determine all positive integer  $a$  such that the equation  $2x^2 - 30x + a = 0$  has two prime roots, i.e. both roots are prime numbers.
Answer: 52
Problem: Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number.  This number is exactly nine times the product Sarah should have obtained.  What is the sum of the two-digit number and the three-digit number?
Answer: 126
Problem: Let  $a,b,c$  be the roots of  $x^3-9x^2+11x-1=0$ , and let  $s=\sqrt{a}+\sqrt{b}+\sqrt{c}$ .  Find  $s^4-18s^2-8s$ .
Answer: -37
Problem: Consider a triangle  $ABC$  with  $BC = 3$ . Choose a point  $D$  on  $BC$  such that  $BD = 2$ . Find the value of
\[AB^2 + 2AC^2 - 3AD^2.\]
Answer: 6
Problem: Find the greatest positive integer  $m$  with the following property:
For every permutation  $a_1, a_2, \cdots, a_n,\cdots$  of the set of positive integers, there exists positive integers  $i_1<i_2<\cdots <i_m$  such that  $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$  is an arithmetic progression with an odd common difference.
Answer: 3
Problem:  Kevin writes the multiples of three from  $1$  to  $100$  on the whiteboard. How many digits does he write?
Answer: 63
Problem: Define  $ P(\tau ) = (\tau + 1)^3$  . If  $x + y = 0$ , what is the minimum possible value of  $P(x) + P(y)$ ?
Answer: 2
Problem: Solve in the set of real numbers the equation \[ 3x^3 \minus{} [x] \equal{} 3,\] where  $ [x]$  denotes the integer part of  $ x.$ 
Answer: x = \sqrt[3]{\frac{4}{3}}
Problem: Define an ordered quadruple of integers  $(a, b, c, d)$  as interesting if  $1 \le a<b<c<d \le 10$ , and  $a+d>b+c$ . How many ordered quadruples are there?
Answer: 80
Problem: Let  $a$  and  $b$  be positive integers such that  $a>b$  and the difference between  $a^2+b$  and  $a+b^2$  is prime. Compute all possible pairs  $(a,b)$ .

Answer: (2, 1)
Problem:  $A_1,A_2,\cdots,A_8$  are fixed points on a circle. Determine the smallest positive integer  $n$  such that among any  $n$  triangles with these eight points as vertices, two of them will have a common side.
Answer:  n = 9
Problem: Let  $x_1$ ,  $x_2$ , …,  $x_{10}$  be 10 numbers.  Suppose that  $x_i + 2 x_{i + 1} = 1$  for each  $i$  from 1 through 9.  What is the value of  $x_1 + 512 x_{10}$ ?
Answer: 171
Problem: A single elimination tournament is held with  $2016$  participants. In each round, players pair up to play games with each other. There are no ties, and if there are an odd number of players remaining before a round then one person will get a bye for the round. Find the minimum number of rounds needed to determine a winner.

[i]Proposed by Nathan Ramesh
Answer: 11
Problem: What is the largest integer that must divide  $n^5-5n^3+4n$  for all integers  $n$ ?
Answer: 120
Problem: The student population at one high school consists of freshmen, sophomores, juniors, and seniors.  There are 25 percent more freshmen than juniors, 10 percent fewer sophomores than freshmen, and 20 percent of the students are seniors. If there are 144 sophomores, how many students attend the school?
Answer: 540
Problem: Find all functions  $f:\mathbb{N}^* \rightarrow \mathbb{N}$  such that



- $f(x \cdot y) = f(x) + f(y)$
- $f(30) = 0$
- $f(x)=0$  always when the units digit of  $x$  is  $7$

Answer:  f(n) = 0 \quad \forall n \in \mathbb{N}^*
Problem: Let  $p$  be a prime number, and define a sequence by:  $x_i=i$  for  $i=,0,1,2...,p-1$  and  $x_n=x_{n-1}+x_{n-p}$  for  $n \geq p$ 
Find the remainder when  $x_{p^3}$  is divided by  $p$ .
Answer: p-1
Problem: Let  $n$  be a positive integer. Determine the smallest positive integer  $k$  with the following property: it is possible to mark  $k$  cells on a  $2n \times 2n$  board so that there exists a unique partition of the board into  $1 \times 2$  and  $2 \times 1$  dominoes, none of which contain two marked cells.

Answer: k = 2n
Problem: On a closed track, clockwise, there are five boxes  $A, B, C, D$  and  $E$ , and the length of the track section between boxes  $A$  and  $B$  is  $1$  km, between  $B$  and  $C$  -  $5$  km, between  $C$  and  $D$  -  $2$  km, between  $D$  and  $E$  -  $10$  km, and between  $E$  and  $A$  -  $3$  km. On the track, they drive in a clockwise direction, the race always begins and ends in the box. What box did you start from if the length of the race was exactly  $1998$  km?
Answer: E
Problem: Solve the group of equations: \[\begin{cases} (x + y)^3 = z  (y + z)^3 = x  (z + x)^3 = y \end{cases}\]
Answer: (x, y, z) \in \{(0, 0, 0), (\frac{\sqrt{2}}{4}, \frac{\sqrt{2}}{4}, \frac{\sqrt{2}}{4}), (-\frac{\sqrt{2}}{4}, -\frac{\sqrt{2}}{4}, -\frac{\sqrt{2}}{4})\}
Problem: Determine all real numbers  $ a$  such that the inequality  $ |x^2 \plus{} 2ax \plus{} 3a|\le2$  has exactly one solution in  $ x$ .
Answer: 1, 2
Problem: A trapezoid has area  $2\, m^2$  and the sum of its diagonals is  $4\,m$ . Determine the height of this trapezoid.
Answer: \sqrt{2} \, \text{m}
Problem: Suppose that  $x,$   $y,$  and  $z$  are three positive numbers that satisfy the equations  $xyz=1,$   $x+\frac{1}{z}=5,$  and  $y+\frac{1}{x}=29.$  Then  $z+\frac{1}{y}=\frac{m}{n},$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n.$ 
Answer: 5
Problem: The  $30$  edges of a regular icosahedron are distinguished by labeling them  $1,2,\dots,30.$  How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?
Answer: 2^{20} \times 3^{10}
Problem: An alarm clock runs 4 minutes slow every hour. It was set right  $ 3 \frac{1}{2}$  hours ago. Now another clock which is correct shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon?
Answer: 14
Problem: All the squares of a board of  $(n+1)\times(n-1)$  squares
are painted with **three  colors** such that, for any two different
columns and any two different rows, the 4 squares in their
intersections they don't have all the same color. Find the
greatest possible value of  $n$ .
Answer: 5
Problem: Four circles of radius  $1$  with centers  $A,B,C,D$  are in the plane in such a way that each circle is tangent to two others. A fifth circle passes through the center of two of the circles and is tangent to the other two.
Find the possible values of the area of the quadrilateral  $ABCD$ .
Answer: 3, 4
Problem: Let  $ P(x)$  be a nonzero polynomial such that, for all real numbers  $ x$ ,  $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$ . Determine the maximum possible number of real roots of  $ P(x)$ .
Answer: 4
Problem: Find all quadruplets  $(a, b, c, d)$  of real numbers satisfying the system
                                            $(a + b)(a^2 + b^2) = (c + d)(c^2 + d^2)$                                             $(a + c)(a^2 + c^2) = (b + d)(b^2 + d^2)$                                             $(a + d)(a^2 + d^2) = (b + c)(b^2 + c^2)$ (Slovakia)
Answer: (a, b, c, d) = (0, 0, 0, 0)
Problem: On a circle, 2022 points are chosen such that distance between two adjacent points is always the same. There are  $k$  arcs, each having endpoints on chosen points, with different lengths. Arcs do not contain each other. What is the maximum possible number of  $k$ ?
Answer: 1011
Problem: Find the area of a square inscribed in an equilateral triangle, with one edge of the square on an edge of the triangle, if the side length of the triangle is  $ 2\plus{}\sqrt{3}$ .
Answer: \frac{12 + 6\sqrt{3}}{16}
Problem: If, in a triangle of sides  $a, b, c$ , the incircle has radius  $\frac{b+c-a}{2}$ , what is the magnitude of  $\angle A$ ?
Answer: \frac{\pi}{2}
Problem:  $ f(x)$  is a given polynomial whose degree at least 2. Define the following polynomial-sequence:  $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$ , for all  $ n \in N$ . Let  $ r_n$  be the average of  $ g_n(x)$ 's roots. If  $ r_{19}\equal{}99$ , find  $ r_{99}$ .
Answer: 99
Problem: Terence Tao is playing rock-paper-scissors. Because his mental energy is focused on solving the twin primes conjecture, he uses the following very simple strategy: 

   ·He plays rock first. 
   ·On each subsequent turn, he plays a different move than the previous one, each with probability ½. 

What is the probability that his 5th move will be rock?
Answer: \frac{1}{2}
Problem: There are  $27$  cards, each has some amount of ( $1$  or  $2$  or  $3$ ) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a *match* such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a *match* be cause they have distinct amount of shapes, distinct shapes but the same color of shapes.
What is the maximum number of cards that we can choose such that non of the triples make a *match*?
Answer: 9
Problem: **Evaluate**

  $\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\ sinx\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} dx$ 
Answer: \frac{\pi}{4}
Problem: Suppose  $a,b$  are positive real numbers such that  $a+a^2 = 1$  and  $b^2+b^4=1$ .  Compute  $a^2+b^2$ .
Answer: 1
Problem:  Find all real parameters  $p$  for which the equation  $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$ has three distinct real roots which are sides of a right triangle.

Answer:  p = \sqrt{2}
Problem: Find the area of quadrilateral  $ABCD$  if: two opposite angles are right;two sides which form right angle are of equal length and sum of lengths of other two sides is  $10$ 
Answer: 25
Problem: Two sequences  $\{a_i\}$  and  $\{b_i\}$  are defined as follows:  $\{ a_i \} = 0, 3, 8, \dots, n^2 - 1, \dots$  and  $\{ b_i \} = 2, 5, 10, \dots, n^2 + 1, \dots $ . If both sequences are defined with  $i$  ranging across the natural numbers, how many numbers belong to both sequences?
Answer: 0
Problem: Given that in a triangle  $ABC$ ,  $AB=3$ ,  $BC=4$  and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of  $AC$ .
Answer: 5
Problem: Find all positive integers  $b$  with the following property: there exists positive integers  $a,k,l$  such that  $a^k + b^l$  and  $a^l + b^k$  are divisible by  $b^{k+l}$  where  $k \neq l$ .
Answer: 1
Problem: Find all positive integers  $m$  and  $n$  such that  $1 + 5 \cdot 2^m = n^2$ .
Answer: (4, 9)
Problem: For each integer  $1\le j\le 2017$ , let  $S_j$  denote the set of integers  $0\le i\le 2^{2017} - 1$  such that  $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$  is an odd integer. Let  $P$  be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1\le j\le 2017} \left(1 - \prod_{i\in S_j} x_i\right).\]
Compute the remainder when
\[ \sum_{\left(x_0, \ldots, x_{2^{2017} - 1}\right)\in\{0, 1\}^{2^{2017}}} P\left(x_0, \ldots, x_{2^{2017} - 1}\right)\]
is divided by  $2017$ .
Answer: 2
Problem: Lines  $l_1$  and  $l_2$  both pass through the origin and make first-quadrant angles of  $\frac{\pi}{70}$  and  $\frac{\pi}{54}$  radians, respectively, with the positive x-axis. For any line  $l$ , the transformation  $R(l)$  produces another line as follows:  $l$  is reflected in  $l_1$ , and the resulting line is reflected in  $l_2$ . Let  $R^{(1)}(l)=R(l)$  and  $R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)$ . Given that  $l$  is the line  $y=\frac{19}{92}x$ , find the smallest positive integer  $m$  for which  $R^{(m)}(l)=l$ .
Answer: 945
Problem: Players A and B play a game. They are given a box with  $n=>1$  candies. A starts first. On a move, if in the box there are  $k$  candies, the player chooses positive integer  $l$  so that  $l<=k$  and  $(l, k) =1$ , and eats  $l$  candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of  $n$ .
Answer:  n \equiv 1 \pmod{2}