BCML

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2641   
What value of $a$ satisfies $27x^3 - 16\sqrt{2}=(3x-2\sqrt{2})(9x^2 + 12x\sqrt{2}+a)$?

2642   
If the first $25$ positive integers are multiplied together, in how many zeros does the product terminate?

2643   
What is the smallest positive number $x$ for which $\left(16^\sqrt{2}\right)^x$ represents a positive integer?

2644   
Of the pairs of positive integers $(x, y)$ that satisfies $3x+7y=188$, which ordered pair has the least positive difference $x-y$?

2645   

What is the smallest positive integer greater than $5$ which leaves a remainder of $5$ when divided by each of $6$, $7$, $8$, and $9$?


2646   
What are all the ordered pairs of positive numbers $(x, y)$ for which $x=\sqrt{2y}$ and $y=\sqrt{x}$?

2647   
How many minutes past 4 o'clock are the hands of a standard 12-hour clock first perpendicular to each other?

2648   
I first drove 16 km at 48 km/hr, then I drove 20 km at 40 km/hr, and finally I drove 24 km at 36 km/hr. What was my average speed in km/hr, for the entire trip?

2649   
How many different triangles have vertices selected from the seven points (-4, 0), (-2, 0), (0,0), (2,0), (4,0), (0,2), and (0,4)?

2650   

Three circular cylinders are strapped together as shown. The cross-section of each cylinder is a circle of radius 1. Presuming that the strap used to bind the cylinders together has no thickness and no extra length, how long is the binding strap?


2651   
What are the coordinates of the reflection of (6,0) across the graph of $y=3x$?

2652   
If the area of a circle's inscribed square is 60, what is the area of its circumscribed square?

2653   
What are all values of $x$ for which $log_x\sqrt{x+12}>1$?

2654   
Determine the units digit of the sum $0!+1!+2!+\cdots+n!+\cdots+20!$?

2655   
What is the area of a trapezoid the lengths of whose bases are 10 and 16, and the lengths of whose legs are 8 and 10?

2656   

A diagonal of a square intersects a segment that connects one vertex of the square to the midpoint of an opposite side, as shown. If the length of the shorter section of the diagonal is 2, what is the area of the square?


2657   
What real value of $x$ satisfies $\sqrt{5x} - \sqrt{2x} = 5-2$?

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