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Question: Find the last three digits of the product of the positive roots of $\sqrt{1995}x^{\log_{1995}x}=x^2.$
Answer: Okay, so I need to find the last three digits of the product of the positive roots of the equation √(1995) * x^(log_{1995} x) = x². Hmm, let's see. First, let me try to understand the equation and fig... |
Question: For how many pairs of consecutive integers in $\{1000,1001,1002^{}_{},\ldots,2000\}$ is no carrying required when the two integers are added?
Answer: Okay, so the problem is asking: How many pairs of consecutive integers in the set {1000, 1001, 1002, ..., 2000} require no carrying when the two integers are ad... |
Question: Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i + 1}$ , replace the midpoint triangle of every face of $P_{i}$ by an outward-pointin... |
Question: Set $A$ consists of $m$ consecutive integers whose sum is $2m,$ and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is 99. Find $m.$
Answer: Okay, so I need to solve this problem where ther... |
Question: Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of ... |
Question: Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n_{}$ but do not divide $n_{}$ ?
Answer: Okay, let's try to figure out this problem. The question is: Let \( n = 2^{31} \times 3^{19} \). How many positive integer divisors of \( n^2 \) are less than \( n \) but do not divide \( ... |
Question: Given that $\sum_{k=1}^{35}\sin 5k=\tan \frac mn,$ where angles are measured in degrees, and $m_{}$ and $n_{}$ are relatively prime positive integers that satisfy $\frac mn<90,$ find $m+n.$
Answer: Okay, so I need to find the sum of sin 5k from k=1 to 35, and express that as the tangent of some fraction m/n, ... |
Question: The solid shown has a square base of side length $s$ . The upper edge is parallel to the base and has length $2s$ . All other edges have length $s$ . Given that $s=6\sqrt{2}$ , what is the volume of the solid? [asy] import three; size(170); pathpen = black+linewidth(0.65); pointpen = black; currentprojection ... |
Question: A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is... |
"Question: Let $A_1A_2A_3\\ldots A_{12}$ be a dodecagon ( $12$ -gon). Three frogs initially sit at $(...TRUNCATED) |
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