Return your final response within \boxed{}. If Jeff picks one letter randomly from the alphabet, what is the probability that the letter is in the word `PROBABILITY'?The answer to this question is \boxed{\frac{9}{26}}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{\frac{9}{26}}


Return your final response within \boxed{}. Let  $ABCD$  be a square with the point of intersection  $O$  of the diagonals and let  $P,\ Q,\ R,\ S$  be the points which are on the segments  $OA,\ OB,\ OC,\ OD$ , respectively such that  $OP=3,\ OQ=5,\ OR=4$ . If the points of intersection of the lines  $AB$  and  $PQ$ , the lines  $BC$  and  $QR$ , the lines  $CD$  and  $RS$  are collinear, then find the length of the segment  $OS$ .The answer to this question is \boxed{\frac{60}{23}}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{\frac{60}{23}}


Return your final response within \boxed{}.  $a,b,c$  are positive numbers such that  $ a^2 + b^2 + c^2 = 2abc + 1 $ . Find the maximum value of
\[ (a-2bc)(b-2ca)(c-2ab) \]The answer to this question is \boxed{\dfrac{1}{8}}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{\dfrac{1}{8}}


Return your final response within \boxed{}. Let  $ABC$  be a triangle with  $AB = 9$ ,  $BC = 10$ , and  $CA = 17$ . Let  $B'$  be the reflection of the point  $B$  over the line  $CA$ . Let  $G$  be the centroid of triangle  $ABC$ , and let  $G'$  be the centroid of triangle  $AB'C$ . Determine the length of segment  $GG'$ .The answer to this question is \boxed{\frac{48}{17}}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{\frac{48}{17}}


Return your final response within \boxed{}. There are  $17$  students in Marek's class, and all of them took a test. Marek's score was  $17$  points higher than the arithmetic mean of the scores of the other students. By how many points is Marek's score higher than the arithmetic mean of the scores of the entire class? Justify your answer.The answer to this question is \boxed{16}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{16}


Return your final response within \boxed{}. James is six years older than Louise. Eight years from now, James will be four times as old as Louise was four years before now. What is the sum of their current ages?The answer to this question is \boxed{26}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{26}


Return your final response within \boxed{}. When the expression $(2x^4+3x^3+x-14)(3x^{10}-9x^7+9x^4+30)-(x^2+5)^7$ is expanded, what is the degree of the resulting polynomial?The answer to this question is \boxed{14}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{14}


Return your final response within \boxed{}. Let $d$ be a positive number such that when $109$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$.The answer to this question is \boxed{71}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{71}


Return your final response within \boxed{}. Let $a_n=6^{n}+8^{n}$ . Determine the remainder on dividing $a_{83}$ by $49$ .The answer to this question is \boxed{35}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{35}


Return your final response within \boxed{}. An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.
The answer to this question is The value of \( N \) is \boxed{144}.. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: The value of \( N \) is \boxed{144}.


Return your final response within \boxed{}. Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $m$ and $n$ be relatively prime positive integers such that $\dfrac mn$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $m+n$.The answer to this question is \boxed{41}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{41}


Return your final response within \boxed{}. 1.  In the rotating frame $S^{\prime}$, show that the effective field can be approximated by

$$
\boldsymbol{B}_{\text {eff }} \approx b \boldsymbol{i}^{\prime},
$$

which is commonly known as rotating wave approximation. What is the precession frequency $\Omega$ in frame $S^{\prime}$ ?The answer to this question is \boxed{\omega}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{\omega}


Return your final response within \boxed{}. The arithmetic mean, geometric mean, and harmonic mean of $a$, $b$, $c$ are $8$, $5$, $3$ respectively. What is the value of $a^2+b^2+c^2$?The answer to this question is \boxed{326}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{326}


Return your final response within \boxed{}. Compute $0.18\div0.003.$The answer to this question is \boxed{60}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{60}


Return your final response within \boxed{}. 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   $x,y,z,t\in Q.$ The answer to this question is \boxed{f(x) = c} for some constant \( c \in \mathbb{Q}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{f(x) = c} for some constant \( c \in \mathbb{Q}


Return your final response within \boxed{}. The factors of $x^4+64$ are:
$\textbf{(A)}\ (x^2+8)^2\qquad\textbf{(B)}\ (x^2+8)(x^2-8)\qquad\textbf{(C)}\ (x^2+2x+4)(x^2-8x+16)\\ \textbf{(D)}\ (x^2-4x+8)(x^2-4x-8)\qquad\textbf{(E)}\ (x^2-4x+8)(x^2+4x+8)$The answer to this question is \boxed{E}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{E}


Return your final response within \boxed{}. Let  $ S \equal{} \{1,2,3,\cdots ,280\}$ . Find the smallest integer  $ n$  such that each  $ n$ -element subset of  $ S$  contains five numbers which are pairwise relatively prime.The answer to this question is \boxed{217}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{217}


Return your final response within \boxed{}. An environmental agency needs to hire a number of new employees so that 85 of the new employees will be able to monitor water pollution, 73 of the new employees will be able to monitor air pollution, and exactly 27 of the new employees will be able to monitor both. (These 27 are included in the 85 and 73 mentioned above.) What is the minimum number of employees that need to be hired?The answer to this question is \boxed{131}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{131}


Return your final response within \boxed{}. How many positive integer multiples of 1001 can be expressed in the form $10^{j} - 10^{i}$ , where $i$ and $j$ are integers and $0\leq i < j \leq 99$ ?The answer to this question is \boxed{784}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{784}


Return your final response within \boxed{}. Solve in the set of real numbers, the system:  $$ x(3y^2+1)=y(y^2+3) $$    $$ y(3z^2+1)=z(z^2+3) $$    $$ z(3x^2+1)=x(x^2+3) $$  The answer to this question is \boxed{(1, 1, 1), (-1, -1, -1), (0, 0, 0)}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{(1, 1, 1), (-1, -1, -1), (0, 0, 0)}


Return your final response within \boxed{}. How many square units are in the area of the parallelogram with vertices at (0, 0), (6, 0), (2, 8) and (8, 8)?The answer to this question is \boxed{48}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{48}


Return your final response within \boxed{}. For all positive integers  $n$ , let
\[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute  $f(2019) - f(2018)$ .  Here  $\varphi(n)$  denotes the number of positive integers less than or equal to  $n$  which are relatively prime to  $n$ .The answer to this question is \boxed{11431}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{11431}


Return your final response within \boxed{}. Adam and Simon start on bicycle trips from the same point at the same time. Adam travels east at 8mph and Simon travels south at 6mph. After how many hours are they 60 miles apart?The answer to this question is \boxed{6}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{6}


Return your final response within \boxed{}. The quadratic polynomial $P(x),$ with real coefficients, satisfies
\[P(x^3 + x) \ge P(x^2 + 1)\]for all real numbers $x.$  Find the sum of the roots of $P(x).$The answer to this question is \boxed{4}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{4}


Return your final response within \boxed{}. Prove that for natural number  $n$  it's possible to find complex numbers  $\omega_1,\omega_2,\cdots,\omega_n$  on the unit circle that  $$ \left\lvert\sum_{j=1}^{n}\omega_j\right\rvert=\left\lvert\sum_{j=1}^{n}\omega_j^2\right\rvert=n-1 $$  iff  $n=2$  occurs.The answer to this question is \boxed{ n = 2 }. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{ n = 2 }


Return your final response within \boxed{}. Let  $ d_1,d_2 \ldots, d_r$  be the positive divisors of  $ n$
 $ 1\equal{}d_1<d_2< \ldots <d_r\equal{}n$
If  $ (d_7)^2 \plus{} (d_{15})^2\equal{} (d_{16})^2$  find all posible values of  $ d_{17}$ The answer to this question is \boxed{28}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{28}


Return your final response within \boxed{}. An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence
\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \]for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.The answer to this question is \boxed{2}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{2}


Return your final response within \boxed{}. A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas?The answer to this question is \boxed{256}. Based on the answer and the constraints of the thought chain length, you should deduce the most logical reasoning process. Note: During the thought process, you should pretend not to have seen the answer, but you must rationally infer the correct answer mentioned earlier based on the content of the thought chain.

Answer: \boxed{256}