Please act as a match teacher and solve the math problem step by step.
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# Question:
Let T be a rational number. Compute $\\sin ^{2} \\frac{T \\pi}{2}+\\sin ^{2} \\frac{(5-T) \\pi}{2}$.
# Reasoning:
Note that $\\sin \\frac{(5-T) \\pi}{2}=\\cos \\left(\\frac{\\pi}{2}-\\frac{(5-T) \\pi}{2}\\right)=\\cos \\left(\\frac{T \\pi}{2}-2 \\pi\\right)=\\cos \\frac{T \\pi}{2}$. Thus the desired quantity is $\\sin ^{2} \\frac{T \\pi}{2}+\\cos ^{2} \\frac{T \\pi}{2}=\\mathbf{1}$ (independent of $T$ ).
# Answer:
The answer is: \\boxed{1}
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# Question:
Let $T=11$. Compute the value of $x$ that satisfies $\\sqrt{20+\\sqrt{T+x}}=5$.
# Reasoning:
Squaring each side gives $20+\\sqrt{T+x}=25$, thus $\\sqrt{T+x}=5$, and $x=25-T$. With $T=11$, $x=14$.
# Answer:
The answer is: \\boxed{14}
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# Question:
The sum of the interior angles of an $n$-gon equals the sum of the interior angles of a pentagon plus the sum of the interior angles of an octagon. Compute $n$.
# Reasoning:
Using the angle sum formula, $180^{\\circ} \\cdot(n-2)=180^{\\circ} \\cdot 3+180^{\\circ} \\cdot 6=180^{\\circ} \\cdot 9$. Thus $n-2=9$, and $n=11$.
# Answer:
The answer is: \\boxed{11}
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# Question:
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# Reasoning: