solve the following: Consider a tetrahedron with two isoscel...

Prompt

solve the following: Consider a tetrahedron with two isosceles triangle faces with side lengths $5\sqrt{10}, \,5\sqrt{10},$ and $10$ and two isosceles triangle faces with side lengths $5\sqrt{10},\, 5\sqrt{10},$ and $18.$ The four vertices of the tetrahedron lie on a sphere with center $S,$ and the four faces of the tetrahedron are tangent to a sphere with center $R.$ The distance $RS$ can be written as $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find ${}m + n.$ Find the number of seven-tuple $(a_1, a_2, a_3, a_4, a_5, a_6, a_7)$ that follows the conditions. $\{a_1, a_2, a_3, a_4, a_5, a_6, a_7\} \subset \{1,2,3\}$ $a_1+a_2+a_3+a_4+a_5+a_6+a_7$ is a multiple of $3$. $a_1 a_2 a_4 + a_2 a_3 a_5 + a_3 a_4 a_6 + a_4 a_5 a_7 + a_5 a_6 a_1 + a_6 a_7 a_2$ is a multiple of $3$. Positive real numbers x, y, and z satisfy the following equations: xyz = 3, (x − y)(y − z)(z − x) = 4, (x + y)(y + z)(z + x) = 40. Compute the minimum possible value for x Let ABC be a triangle with centroid G and circumcenter O. Suppose that the orthocenter of triangle AGO lies on line BC. Given that AB = 11 and AC = 13, compute BC. In an equiangular pentagon, the sum of the squares of the side lengths equals $308$, and the sum of the squares of the diagonal lengths equals $800$. The square of the perimeter of the pentagon can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$. Let (S) be the set of all ordered pairs ((x,y)) of nonnegative integers (0 \le x \le 19) and (0 \le y \le 2). Compute the number of permutations ((x_1,y_1),(x_2,y_2),\dots,(x_{60},y_{60})) of the elements of (S) such that: (y_1=2) and (y_{60}=0); for all integers (1 \le i \le 59), exactly one of the following holds: (x_i=x_{i+1}) and (|y_i-y_{i+1}|=1); (y_i=y_{i+1}) and (x_i-x_{i+1}) is (-1) or (19). A positive integer $n$ is called solitary if, for any nonnegative integers $a$ and $b$ such that $a+b=n$, either $a$ or $b$ contains the digit $1$ when written in base $10$. Determine the number of solitary integers less than $10^{2026}$ Three people are each initially given the number 0. Every hour, they each simultaneously flip a fair coin: for each person, if their coin lands as heads, they increase their number by one; otherwise, they don’t change their number. What is the expected number of hours until the first occurrence of the three numbers all being distinct? Let ABC be a triangle with orthocenter H. The internal angle bisector of ∠BAC meets the circumcircles of triangles ABH, ACH, and ABC again at points P, Q, and M, respectively. Suppose that points A, P, Q, and M are distinct and lie on the internal angle bisector of ∠BAC in that order. Given that AP = 4, AQ = 5, and BC = 7, compute AM