inclusionAI/Ring-lite-rl-data huggingface dataset:
Let  $ a $  ,  $ b $  ,  $ c $  ,  $ d $  ,  $ (a + b + c + 18 + d) $  ,  $ (a + b + c + 18 - d) $  ,  $ (b + c) $  , and  $ (c + d) $  be distinct prime numbers such that  $ a + b + c = 2010 $ ,  $ a $ ,  $ b $ ,  $  c $ ,  $ d \neq 3 $  , and  $ d \le 50 $ . Find the maximum value of the diﬀerence between two of these prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2067}
Complete the conversion between the following number systems: $101101_{(2)} = \_\_\_\_\_\_\_\_\_\_\_\_{(10)}\_\_\_\_\_\_\_\_\_\_\_\_{(7)}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63_{(7)}}
In triangle  $ABC$ ,  $\angle ABC$  is obtuse.  Point  $D$  lies on side  $AC$  such that  $\angle ABD$  is right, and point  $E$  lies on side  $AC$  between  $A$  and  $D$  such that  $BD$  bisects  $\angle EBC$ .  Find  $CE$  given that  $AC=35$ ,  $BC=7$ , and  $BE=5$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In chemical laboratories, the floors are covered with tiles made of acid-resistant ceramics, which have the shape of regular polygons. Which polygons can be used to completely cover the floor without gaps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 4, 6}
Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots, 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. We construct isosceles right triangle $A C D$ with $\angle A D C=90^{\circ}$, where $D, B$ are on the same side of line $A C$, and let lines $A D$ and $C B$ meet at $F$. Similarly, we construct isosceles right triangle $B C E$ with $\angle B E C=90^{\circ}$, where $E, A$ are on the same side of line $B C$, and let lines $B E$ and $C A$ meet at $G$. Find $\cos \angle A G F$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{5}{13}}
There are eleven positive integers $n$ such that there exists a convex polygon with $n$ sides whose angles, in degrees, are unequal integers that are in arithmetic progression. Find the sum of these values of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
Let $\theta=\frac{2\pi}{2015}$, and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form $\frac{b}{2^a}$, where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1441}
Consider a $6 \times 6$ grid of squares. Edmond chooses four of these squares uniformly at random. What is the probability that the centers of these four squares form a square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{561}}