rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: Find all real  $x$  that satisfy the equation   $$ \frac{1}{x+1}+\frac{1}{x+2}=\frac{1}{x} $$  *2015 CCA Math Bonanza Lightning Round #2.2*
Answer: \pm \sqrt{2}
Problem: Let  $k$  be a positive real number. In the   $X-Y$  coordinate plane, let  $S$  be the set of all points of the form  $(x,x^2+k)$  where  $x\in\mathbb{R}$ . Let  $C$  be the set of all circles whose center lies in  $S$ , and which are tangent to  $X$ -axis. Find the minimum value of  $k$  such that any two circles in  $C$  have at least one point of intersection.
Answer: \frac{1}{4}
Problem: Determine all functions  $f: \mathbb{R} \rightarrow \mathbb{R}$  such that for all  $x$ ,  $y \in \mathbb{R}$  hold:
      $$ f(xf(y)-yf(x))=f(xy)-xy $$
Answer: f(x) = x \text{ or } f(x) = |x|
Problem: Find the smallest prime  $p$  such that the digits of  $p$  (in base 10) add up to a prime number greater than  $10$ .
Answer: 29
Problem: Let  $A,B$  be adjacent vertices of a regular  $n$ -gon ( $n\ge5$ ) with center  $O$ . A triangle  $XYZ$ , which is congruent to and initially coincides with  $OAB$ , moves in the plane in such a way that  $Y$  and  $Z$  each trace out the whole boundary of the polygon, with  $X$  remaining inside the polygon. Find the locus of  $X$ .
Answer: n
Problem: A permutation  $(a_1, a_2, a_3, \dots, a_{100})$  of  $(1, 2, 3, \dots, 100)$  is chosen at random. Denote by  $p$  the probability that  $a_{2i} > a_{2i - 1}$  for all  $i \in \{1, 2, 3, \dots, 50\}$ . Compute the number of ordered pairs of positive integers  $(a, b)$  satisfying  $\textstyle\frac{1}{a^b} = p$ .
Answer: 6
Problem: For positive integers  $a>b>1$ , define
\[x_n = \frac {a^n-1}{b^n-1}\] 
Find the least  $d$  such that for any  $a,b$ , the sequence  $x_n$  does not contain  $d$  consecutive prime numbers.
Answer: 3
Problem: Solve the equation  $[x] \cdot \{x\} = 1991x$ . (Here  $[x]$  denotes the greatest integer less than or equal to  $x$ , and  $\{x\}=x-[x]$ .)
Answer: 0 \text{ or } -\frac{1}{1992}
Problem: A function  $f(x)$  satisfies  $f(x)=f\left(\frac{c}{x}\right)$   for some real number  $c(>1)$  and all positive number  $x$ . 

If  $\int_1^{\sqrt{c}} \frac{f(x)}{x} dx=3$ , evaluate  $\int_1^c \frac{f(x)}{x} dx$ 
Answer: 0
Problem: Find all values of the positive integer  $k$  that has the property:
There are no positive integers  $a,b$  such that the expression  $A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}$  is a composite positive number.
Answer:  k = 1
Problem: Find the largest positive integer  $k$  such that  $\phi ( \sigma ( 2^k)) = 2^k$ . ( $\phi(n)$  denotes the number of positive integers that are smaller than  $n$  and relatively prime to  $n$ , and  $\sigma(n)$  denotes the sum of divisors of  $n$ ). As a hint, you are given that  $641|2^{32}+1$ .
Answer: 31
Problem: In the coordinate plane, denote by  $ S(a)$  the area of the region bounded by the line passing through the point  $ (1,\ 2)$  with the slope  $ a$  and the parabola  $ y\equal{}x^2$ . When  $ a$  varies in the range of  $ 0\leq a\leq 6$ , find the value of  $ a$  such that  $ S(a)$  is minimized.
Answer:  a = 2
Problem: Let  $p$  prime and  $m$  a positive integer. Determine all pairs  $( p,m)$  satisfying the equation:  $ p(p+m)+p=(m+1)^3$ 
Answer: (2, 1)
Problem: Let  $a_1$ ,  $a_2$ ,  $a_3$ ,  $a_4$ , and  $a_5$  be random integers chosen independently and uniformly from the set  $\{ 0, 1, 2, \dots, 23 \}$ .  (Note that the integers are not necessarily distinct.)  Find the probability that
\[
  \sum_{k=1}^{5} \operatorname{cis} \Bigl( \frac{a_k \pi}{12} \Bigr) = 0.
\]
(Here  $\operatorname{cis} \theta$  means  $\cos \theta + i \sin \theta$ .)


Answer: \frac{35}{2 \cdot 24^3}
Problem: Suppose that  $n^2-2m^2=m(n+3)-3$ . Find all integers  $m$  such that all corresponding solutions for  $n$  will *not* be real.
Answer: \{-2, -1, 0\}
Problem: Find the maximum value of the positive real number  $k$  such that the inequality  $$ \frac{1}{kab+c^2} +\frac{1} {kbc+a^2} +\frac{1} {kca+b^2}  \geq \frac{k+3}{a^2+b^2+c^2}   $$  holds for all positive real numbers  $a,b,c$  such that  $a^2+b^2+c^2=2(ab+bc+ca).$ 
Answer: k = 2
Problem: Consider  $n$  children in a playground, where  $n\ge 2$ . Every child has a coloured hat, and every pair of children is joined by a coloured ribbon. For every child, the colour of each ribbon held is different, and also different from the colour of that child’s hat. What is the minimum number of colours that needs to be used?
Answer:  n
Problem: In the following figure, the bigger wheel has circumference  $12$ m and the inscribed wheel has circumference  $8 $ m.  $P_{1}$  denotes a point on the bigger wheel and  $P_{2}$  denotes a point on the smaller wheel. Initially  $P_{1}$  and  $P_{2}$  coincide as in the figure. Now we roll the wheels on a smooth surface and the smaller wheel also rolls in the bigger wheel smoothly. What distance does the bigger wheel have to roll so that the points will be together again?
Answer: 24
Problem: A rogue spaceship escapes.  $54$  minutes later the police leave in a spaceship in hot pursuit. If the police spaceship travels  $12\%$  faster than the rogue spaceship along the same route, how many minutes will it take for the police to catch up with the rogues?
Answer: 450
Problem: Let  $k$  and  $n$  be positive integers. Consider an array of  $2\left(2^n-1\right)$  rows by  $k$  columns. A  $2$ -coloring of the elements of the array is said to be *acceptable* if any two columns agree on less than  $2^n-1$  entries on the same row.

Given  $n$ , determine the maximum value of  $k$  for an acceptable  $2$ -coloring to exist.
Answer:  k = 2^n
Problem: How many sequences  $ a_1,a_2,...,a{}_2{}_0{}_0{}_8$  are there such that each of the numbers  $ 1,2,...,2008$  occurs once in the sequence, and  $ i \in (a_1,a_2,...,a_i)$  for each  $ i$  such that  $ 2\le i \le2008$ ?
Answer: 2^{2007}
Problem: The number  $2^{29}$  has a  $9$ -digit decimal representation that contains all but one of the  $10$  (decimal) digits. Determine which digit is missing
Answer: 4
Problem: What is the smallest positive integer  $t$  such that there exist integers  $x_{1},x_{2}, \cdots, x_{t}$  with \[{x_{1}}^{3}+{x_{2}}^{3}+\cdots+{x_{t}}^{3}=2002^{2002}\;\;?\]
Answer: 4
Problem: Amanda has the list of even numbers  $2, 4, 6, \dots 100$  and Billy has the list of odd numbers  $1, 3, 5, \dots 99$ . Carlos creates a list by adding the square of each number in Amanda's list to the square of the corresponding number in Billy's list. Daisy creates a list by taking twice the product of corresponding numbers in Amanda's list and Billy's list. What is the positive difference between the sum of the numbers in Carlos's list and the sum of the numbers in Daisy's list?
Answer: 50
Problem: How many non-intersecting pairs  of  paths we have from (0,0) to (n,n) so that path can move two ways:top or right?
Answer: \binom{2n-2}{n-1}^2 - \binom{2n-2}{n-2}^2
Problem: What is the area of a triangle with side lengths  $17$ ,  $25$ , and  $26$ ?
Answer: 204
Problem: In a room, each person is an painter and/or a musician.  $2$  percent of the painters are musicians, and  $5$  percent of the musicians are painters. Only one person is both an painter and a musician. How many people are in the room?
Answer: 69
Problem: Let  $a_n$  be a sequence defined by some  $a_0$  and the recursion  $a_{n+1} = a_n + 2 \cdot  3^n$  for  $n \ge 0$ . 
Determine all rational values of  $a_0$  such that  $a^j_k / a^k_j$  is an integer for all integers  $j$  and  $k$  with  $0 < j < k$ .
Answer: a_0 = 1
Problem: Find all sets of real numbers  $\{a_1,a_2,\ldots, _{1375}\}$  such that
\[2 \left( \sqrt{a_n - (n-1)}\right) \geq a_{n+1} - (n-1), \quad \forall n \in \{1,2,\ldots,1374\},\]
and
\[2 \left( \sqrt{a_{1375}-1374} \right) \geq a_1 +1.\]
Answer: a_n = n \text{ for all } n \in \{1, 2, \ldots, 1375\}
Problem: Find the sum of all the real values of x satisfying  $(x+\frac{1}{x}-17)^2$   $= x + \frac{1}{x} + 17.$ 
Answer: 35
Problem: Find all naturals  $k$  such that  $3^k+5^k$  is the power of a natural number with the exponent  $\ge 2$ .
Answer: k = 1
Problem: Let  $I$  be a nonempty open subinterval of the set of positive real numbers. For which even  $n \in \mathbb{N}$  are there injective function  $f: I \to \mathbb{R}$  and positive function  $p: I \to \mathbb{R}$ , such that for all  $x_1 , \ldots , x_n \in I$ ,
\[ f \left( \frac{1}{2} \left( \frac{x_1+\cdots+x_n}{n}+\sqrt[n]{x_1 \cdots x_n} \right) \right)=\frac{p(x_1)f(x_1)+\cdots+p(x_n)f(x_n)}{p(x_1)+\cdots+p(x_n)} \]
holds?
Answer: all even n \in \mathbb{N}
Problem: $m$  boys and  $n$  girls ( $m>n$ ) sat across a round table, supervised by a teacher, and they did a game, which went like this. At first, the teacher pointed a boy to start the game. The chosen boy put a coin on the table. Then, consecutively in a clockwise order, everyone did his turn. If the next person is a boy, he will put a coin to the existing pile of coins. If the next person is a girl, she will take a coin from the existing pile of coins. If there is no coin on the table, the game ends. Notice that depending on the chosen boy, the game could end early, or it could go for a full turn. If the teacher wants the game to go for at least a full turn, how many possible boys could be chosen?
Answer: m - n
Problem: There exist two distinct positive integers, both of which are divisors of  $10^{10}$ , with sum equal to  $157$ .  What are they?
Answer: 32, 125
Problem: Determine the sum of absolute values for the complex roots of  $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$ 
Answer: 8
Problem: Let  $\{X_n\}$  and  $\{Y_n\}$  be sequences defined as follows: \[X_0=Y_0=X_1=Y_1=1,\] \begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*} Let  $k$  be the largest integer that satisfies all of the following conditions: [list=i][*]  $|X_i-k|\leq 2007$ , for some positive integer  $i$ ;
[*]  $|Y_j-k|\leq 2007$ , for some positive integer  $j$ ; and
[*]  $k<10^{2007}.$ [/list] Find the remainder when  $k$  is divided by  $2007$ .
Answer:  k \mod 2007
Problem: Given an acute triangle  $ABC$  with altituties AD and BE. O circumcinter of  $ABC$ .If o lies on the segment DE then find the value of  $sinAsinBcosC$ 
Answer: \frac{1}{2}
Problem: Find all triples  $(p,q,n)$  that satisfy
\[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\]
where  $p,q$  are odd primes and  $n$  is an positive integer.
Answer: (3, 3, n)
Problem: Compute the number of positive integer divisors of  $2121$  with a units digit of  $1$ .
Answer: 4
Problem: Find all positive integers  $x$  such that  $2x+1$  is a perfect square but none of the integers  $2x+2, 2x+3, \ldots, 3x+2$  are perfect squares.
Answer: 4
Problem: Let  $ABCD$  be a parallelogram. Let  $E$  be the midpoint of  $AB$  and  $F$  be the midpoint of  $CD$ . Points  $P$  and  $Q$  are on segments  $EF$  and  $CF$ , respectively, such that  $A, P$ , and  $Q$  are collinear. Given that  $EP = 5$ ,  $P F = 3$ , and  $QF = 12$ , find  $CQ$ .
Answer: 8
Problem: each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of ways of colouring the grid. find the largest k so that  $3^k$  divides P.
Answer: 1009
Problem: If positive integers,  $a,b,c$  are pair-wise co-prime, and, \[\ a^2|(b^3+c^3), b^2|(a^3+c^3), c^2|(a^3+b^3) \] find  $a,b,$  and  $c$ 
Answer: (1, 1, 1)
Problem: Solve the equation  $m^{gcd(m,n)} = n^{lcm(m,n)}$  in positive integers, where gcd( $m, n$ ) – greatest common
divisor of  $m,n$ , and lcm( $m, n$ ) – least common multiple of  $m,n$ .
Answer: (m, n) = (1, 1)
Problem: Several irrational numbers are written on a blackboard. It is known that for every two numbers  $ a$  and  $ b$  on the blackboard, at least one of the numbers  $ a\over b\plus{}1$  and  $ b\over a\plus{}1$  is rational. What maximum number of irrational numbers can be on the blackboard?
Answer: 3
Problem: Let  $p_1,p_2,p_3$  be primes with  $p_2\neq p_3$  such that  $4+p_1p_2$  and  $4+p_1p_3$  are perfect squares. Find all possible values of  $p_1,p_2,p_3$ .
Answer:  (7, 11, 3) and (7, 3, 11)
Problem: Queen Jack likes a 5-card hand if and only if the hand contains only queens and jacks.  Considering all possible 5-card hands that can come from a standard 52-card deck, how many hands does Queen Jack like?
Answer: 56
Problem: On  $5\times 5$  squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
Answer: 24
Problem: You can determine all 4-ples  $(a,b, c,d)$  of real numbers, which solve the following equation system  $\begin{cases} ab + ac = 3b + 3c 
bc + bd = 5c + 5d 
ac + cd = 7a + 7d 
ad + bd = 9a + 9b \end{cases} $ 
Answer:  (3, 5, 7, 9)
Problem: A  $9\times  1$  rectangle is divided into unit squares. A broken line, from the lower left to the upper right corner, goes through
all  $20$  vertices of the unit squares and consists of   $19$  line segments. How many such lines are there?
Answer: 92378
Problem: Aisha went shopping. At the first store she spent  $40$  percent of her money plus four dollars. At the second store she spent  $50$  percent of her remaining money plus  $5$  dollars. At the third store she spent  $60$  percent of her remaining money plus six dollars. When Aisha was done shopping at the three stores, she had two dollars left. How many dollars did she have with her when she started shopping?
Answer: 90
Problem: An Indian raga has two kinds of notes: a short note, which lasts for  $1$  beat and a long note, which lasts for  $2$  beats. For example, there are  $3$  ragas which are  $3$  beats long;  $3$  short notes, a short note followed by a long note, and a long note followed by a short note. How many Indian ragas are 11 beats long?
Answer: 144
Problem: Let  $c$  be a complex number.  Suppose there exist distinct complex numbers  $r$ ,  $s$ , and  $t$  such that for every complex number  $z$ , we have
\[
  (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]	
Compute the number of distinct possible values of  $c$ .
Answer: 4
Problem: Two vector fields  $\mathbf{F},\mathbf{G}$  are defined on a three dimensional region  $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$ .  $$ \mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0) $$  Evaluate the following integral.
\[\iiint_{W} (\mathbf{G}\cdot \text{curl}(\mathbf{F}) - \mathbf{F}\cdot \text{curl}(\mathbf{G})) dV\]
Answer: 0
Problem: Find  all functions   $f:Q\rightarrow Q$  such that  \[ f(x+y)+f(y+z)+f(z+t)+f(t+x)+f(x+z)+f(y+t)\ge 6f(x-3y+5z+7t) \]  for all   $x,y,z,t\in Q.$ 
Answer: f(x) = c for some constant c \in \mathbb{Q}
Problem: Solve the following system of equations in  $\mathbb{R}$  where all square roots are non-negative:
 $
\begin{matrix}

a  - \sqrt{1-b^2} + \sqrt{1-c^2} = d 
b -  \sqrt{1-c^2} + \sqrt{1-d^2} = a 
c -  \sqrt{1-d^2} + \sqrt{1-a^2} = b 
d -  \sqrt{1-a^2} + \sqrt{1-b^2} = c 

\end{matrix}
$ 
Answer:  a = b = c = d
Problem: Determine all triples  $(x,y,z)$  of real numbers  that satisfy all of the following three equations:  $$ \begin{cases}  \lfloor x \rfloor + \{y\} =z  \lfloor y \rfloor + \{z\} =x  \lfloor z \rfloor + \{x\} =y  \end{cases} $$  
Answer: (x, y, z) \text{ such that } x = y = z
Problem: Consider the base 27 number 
\[
  n = ABCDEFGHIJKLMNOPQRSTUVWXYZ ,
\]	
where each letter has the value of its position in the alphabet.  What remainder do you get when you divide  $n$  by 100?  (The remainder is an integer between 0 and 99, inclusive.)

Answer: 0
Problem: Find the solutions of positive integers for the system  $xy + x + y = 71$  and  $x^2y + xy^2 = 880$ .
Answer: (x, y) = (11, 5) 	ext{ or } (5, 11)
Problem: Amandine and Brennon play a turn-based game, with Amadine starting.
On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers.
For example, if  $3, 5$  have been selected so far, only  $1, 2, 4, 7$  are available to be picked;
if only  $3$  has been selected so far, all numbers not divisible by three are eligible.
A player loses immediately if they select the integer  $1$ .

Call a number  $n$  *feminist* if  $\gcd(n, 6) = 1$  and if Amandine wins if she starts with  $n$ . Compute the sum of the *feminist* numbers less than  $40$ .
Answer: 192
Problem: Ryan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and he chooses whether to weight heads or tails more with equal probability. Brice flips his modified coin twice and it lands up heads both times. The probability that the coin lands up heads on the next flip can be expressed in the form  $\tfrac{p}{q}$  for positive integers  $p, q$  satisfying  $\gcd(p, q) = 1$ , what is  $p + q$ ?

Answer: 8
Problem: What is the largest positive integer  $n < 1000$  for which there is a positive integer  $m$  satisfying \[\text{lcm}(m,n) = 3m \times \gcd(m,n)?\]

Answer: 972
Problem: Determine all real solutions  $x, y, z$  of the following system of equations:  $\begin{cases}
x^3 - 3x = 4 - y 
2y^3 - 6y = 6 - z 
3z^3 - 9z = 8 - x\end{cases}$ 
Answer: x = y = z = 2
Problem: Determine the maximum possible number of points you can place in a rectangle with lengths  $14$  and  $28$  such that any two of those points are more than  $10$  apart from each other.
Answer: 8
Problem: Find all triples of complex numbers  $(x, y, z)$  for which  $$ (x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0 $$  
Answer: (x, y, z) such that x + y + z = 0 and xyz = 0
Problem: Two people play the following game: there are  $40$  cards numbered from  $1$  to  $10$  with  $4$  different signs. At the beginning they are given  $20$  cards each. Each turn one player either puts a card on the table or removes some cards from the table, whose sum is  $15$ . At the end of the game, one player has a  $5$  and a  $3$  in his hand, on the table there's a  $9$ , the other player has a card in his hand. What is it's value?
Answer: 8
Problem: In trapezoid  $ABCD$  holds  $AD \mid \mid BC$ ,  $\angle ABC = 30^{\circ}$ ,  $\angle BCD = 60^{\circ}$  and  $BC=7$ . Let  $E$ ,  $M$ ,  $F$  and  $N$  be midpoints of sides  $AB$ ,  $BC$ ,  $CD$  and  $DA$ , respectively. If  $MN=3$ , find  $EF$ 
Answer: 3.5
Problem: There is a regular  $17$ -gon  $\mathcal{P}$  and its circumcircle  $\mathcal{Y}$  on the plane. 
The vertices of  $\mathcal{P}$  are coloured in such a way that  $A,B \in \mathcal{P}$  are of different colour, if the shorter arc connecting  $A$  and  $B$  on  $\mathcal{Y}$  has  $2^k+1$  vertices, for some  $k \in \mathbb{N},$  including  $A$  and  $B.$  
What is the least number of colours which suffices?
Answer: 4
Problem: Find the smallest positive integer  $k$  which is representable in the form  $k=19^n-5^m$  for some positive integers  $m$  and  $n$ .
Answer: 14
Problem: Alexis has  $2020$  paintings, the  $i$ th one of which is a  $1\times i$  rectangle for  $i = 1, 2, \ldots, 2020$ . Compute the smallest integer  $n$  for which they can place all of the paintings onto an  $n\times n$  mahogany table without overlapping or hanging off the table.
Answer: 1430
Problem: Let  $p$  be an odd prime number, and let  $\mathbb{F}_p$  denote the field of integers modulo  $p$ . Let  $\mathbb{F}_p[x]$  be the ring of polynomials over  $\mathbb{F}_p$ , and let  $q(x) \in \mathbb{F}_p[x]$  be given by  $q(x) = \sum_{k=1}^{p-1} a_k x^k$  where  $a_k = k^{(p-1)/2}$  mod  $p$ . Find the greatest nonnegative integer  $n$  such that  $(x-1)^n$  divides  $q(x)$  in  $\mathbb{F}_p[x]$ .

Answer: \frac{p-1}{2}
Problem: Find the maximum possible value of  $k$  for which there exist distinct reals  $x_1,x_2,\ldots ,x_k $  greater than  $1$  such that for all  $1 \leq i, j \leq k$ ,  $$ x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}. $$
Answer: 4
Problem: For finite sets  $A,M$  such that  $A \subseteq M \subset \mathbb{Z}^+$ , we define   $$ f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}. $$   Given a positive integer  $k$ , we call  $M$  *k-colorable* if it is possible to color the subsets of  $M$  with  $k$  colors so that for any  $A \subseteq M$ , if  $f_M(A)\neq A$  then  $f_M(A)$  and  $A$  have different colors.
Determine the least positive integer  $k$  such that every finite set  $M \subset\mathbb{Z}^+$  is k-colorable.
Answer: k = 2
Problem: Let  $f_{0}(x)=x$ , and for each  $n\geq 0$ , let  $f_{n+1}(x)=f_{n}(x^{2}(3-2x))$ . Find the smallest real number that is at least as large as
\[ \sum_{n=0}^{2017} f_{n}(a) + \sum_{n=0}^{2017} f_{n}(1-a)\]
for all  $a \in [0,1]$ .
Answer: 2018
Problem: Determine the number of solutions of the simultaneous equations  $ x^2 \plus{} y^3 \equal{} 29$  and  $ \log_3 x \cdot \log_2 y \equal{} 1.$ 
Answer: 2
Problem: Let  $d$  be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than  $d$  degrees. What is the minimum possible value for  $d$ ?
Answer: 120
Problem: For any integer  $r \geq 1$ , determine the smallest integer  $h(r) \geq 1$  such that for any partition of the set  $\{1, 2, \cdots, h(r)\}$  into  $r$  classes, there are integers  $a \geq  0 \ ; 1 \leq x \leq y$ , such that  $a + x, a + y, a + x + y$  belong to the same class.
Answer: 2r
Problem: Given that a duck found that  $5-2\sqrt{3}i$  is one of the roots of  $-259 + 107x - 17x^2 + x^3$ , what is the sum of the real parts of the other two roots?
Answer: 7
Problem: Let  $ \theta_1, \theta_2,\ldots , \theta_{2008}$  be real numbers. Find the maximum value of
 $ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$ 
Answer: 1004
Problem: The summation  $\sum_{k=1}^{360} \frac{1}{k \sqrt{k+1} + (k+1)\sqrt{k}}$  is the ratio of two relatively prime positive integers  $m$  and  $n$ . Find  $m + n$ .
Answer: 37
Problem: The polynomial  $-400x^5+2660x^4-3602x^3+1510x^2+18x-90$  has five rational roots. Suppose you guess a rational number which could possibly be a root (according to the rational root theorem). What is the probability that it actually is a root?
Answer: \frac{1}{36}
Problem: Determine all integers  $a$  for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions  $x, \;y$ .
Answer:  a = 2 and a = -2
Problem: Terry drove along a scenic road using  $9$  gallons of gasoline.  Then Terry went onto the freeway and used  $17$  gallons of gasoline.  Assuming that Terry gets  $6.5$  miles per gallon better gas mileage on the freeway than on the scenic road, and Terry’s average gas mileage for the entire trip was  $30$  miles per gallon, find the number of miles Terry drove.
Answer: 780
Problem: Let  $a$  and  $b$  be natural numbers such that  $2a-b$ ,  $a-2b$   and  $a+b$   are  all distinct squares. What  is the smallest possible value of  $b$  ?
Answer: 3
Problem: Arthur stands on a circle drawn with chalk in a parking lot.  It is sunrise and there are birds in the trees nearby.  He stands on one of five triangular nodes that are spaced equally around the circle, wondering if and when the aliens will pick him up and carry him from the node he is standing on.  He flips a fair coin  $12$  times, each time chanting the name of a nearby star system.  Each time he flips a head, he walks around the circle, in the direction he is facing, until he reaches the next node in that direction.  Each time he flips a tail, he reverses direction, then walks around the circle until he reaches the next node in that new direction.  After  $12$  flips, Arthur finds himself on the node at which he started.  He thinks this is fate, but Arthur is quite mistaken.  If  $a$  and  $b$  are relatively prime positive integers such that  $a/b$  is the probability that Arthur flipped exactly  $6$  heads, find  $a+b$ .
Answer: 1255