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Puzzles

11 December

There are five 3-digit numbers whose digits are all either 1 or 2 and who do not contain two 2s in a row: 111, 112, 121, 211, and 212.
How many 14-digit numbers are there whose digits are all either 1 or 2 and who do not contain two 2s in a row?

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10 December

A line is tangent to a curve if the line touches the curve at exactly one point.
The line \(y=-160\,000\) is tangent to the parabola \(y=x^2-ax\). What is \(a\)?

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9 December

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10. Today's number is the largest number you can make with the digits in the red boxes.
++= 20
+ + ÷
+= 0
+ ×
÷×= 12
=
22
=
6
=
2

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Tags: numbers, grids

8 December

The equation \(x^5 - 7x^4 - 27x^3 + 175x^2 + 218x = 840\) has five real solutions. What is the product of all these solutions?

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7 December

What is the area of the largest triangle that fits inside a regular hexagon with area 952?

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6 December

There are 21 three-digit integers whose digits are all non-zero and whose digits add up to 8.
How many positive integers are there whose digits are all non-zero and whose digits add up to 8?

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5 December

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10. Today's number is the product of the numbers in the red boxes.
×÷= 15
+ + +
×÷= 14
×÷= 27
=
9
=
5
=
5

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Tags: numbers, grids

4 December

The last three digits of \(5^5\) are 125.
What are the last three digits of \(5^{2,022,000,000}\)?

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