Puzzles

What is the area of the largest area triangle that has one side of length 32 and one side of length 19?

The equation \(352x^3-528x^2+90=0\) has three distinct real-valued solutions.

Today's number is the number of integers \(a\) such that the equation \(352x^3-528x^2+a=0\) has three distinct real-valued solutions.

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.

The digital product of a number is computed by multiplying together all of its digits. For example, the digital product of 6273 is 252.

Today's number is the smallest number whose digital product is 252.

Each clue in this crossnumber is formed of two parts connected by a logical connective: and means that both parts are true; nand means that at most one part is true; or means that at least one part is true; nor means that neither part is true; xor means that exactly one part is true; xnor means that either both parts are false or both parts are true. No number starts with 0.

The odd numbers are written in a pyramid.

What is the mean of the numbers in the 19th row?

You start at the point marked A in the picture below. You want to get to the point marked B. You may travel to the right, upwards, or to the left along the black lines, but you cannot pass along the same line segment more than once.

Today's number is the total number of possible routes to get from A to B.

The diagram to the left shows three circles and two triangles. The three circles all meet at one point. The vertices of the smaller red triangle are at the centres of the circles. The lines connecting the vertices of the larger blue triangle to the point where all three circles meet are diameters of the three circles.

The area of the smaller red triangle is 226. What is the area of the larger blue triangle?