+23 Roger used 15 small Christmas trees * 14 fairy lights/small Christmas tree = 210 fairy lights on small Christmas trees.
The 6 big Christmas trees each had 14 fairy lights/small Christmas tree + 14 fairy lights = 28 fairy lights.
In total, Roger used 210 fairy lights + 6 * 28 fairy lights = 278 fairy lights.
If Roger used 1/3 of the fairy lights to decorate the small Christmas trees, he had 3 * 278 fairy lights = 834 fairy lights at first.
So the answer is 834
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+23 Use the angle bisector theorem. The angle bisector theorem states that the ratio of the lengths of two segments that are bisected by an angle bisector is equal to the ratio of the lengths of the other two segments that are bisected by the angle bisector. In this case, the two segments that are bisected by the angle bisector are AC and BC. The other two segments are AM and CM. So, the ratio of AM to CM is equal to the ratio of AC to BC.
Find the length of AM. The length of AM can be found using the Pythagorean theorem. AB is a diameter of the circle, so AB = 10. The length of AC is 8, so the length of AM is equal to 6. The ratio of AM to CM is equal to the ratio of AC to BC, which is 1:1. So, the length of CM is equal to 6.
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+23 There are 21 solutions to the equation u + v + w + x + y + z = 10 where u, v, w, x, y, and z are nonnegative integers, and x is at most 2 or y is at least 3.
To solve this problem, we can use the following cases:
Case 1: x=0 and y>=3. In this case, the only solution is u+v+w+z=10. There is 1 solution in this case.
Case 2: x=1 and y>=3. In this case, the possible solutions are u+v+w+z=9, u+v+w+z=8, and u+v+w+z=7. There are 3 solutions in this case.
Case 3: x=2 and y>=3. In this case, the possible solutions are u+v+w+z=8, u+v+w+z=7, and u+v+w+z=6. There are 3 solutions in this case.
Case 4: x=0 and y<3. In this case, the possible solutions are u+v+w+x+y+z=10, u+v+w+x+y+z=9, u+v+w+x+y+z=8, u+v+w+x+y+z=7, u+v+w+x+y+z=6, u+v+w+x+y+z=5, u+v+w+x+y+z=4, and u+v+w+x+y+z=3. There are 8 solutions in this case.
The total number of solutions is 1+3+3+8=21
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+23 To find BC and BZ in triangle AB, we can use the angle bisector theorem.
The angle bisector theorem states that in a triangle, if a line divides one side into two segments, the ratio of the lengths of those segments is equal to the ratio of the lengths of the other two sides.
In triangle AB, let's label BC as "x" and BZ as "y".
According to the angle bisector theorem, we can set up the following ratios:
AY / CY = AB / BC 16 / 16 = 16 / x
Simplifying the equation: 1 = 16 / x
Cross-multiplying: x = 16
So, BC = 16.
Now, let's find BZ using the same theorem:
AY / CZ = AB / BZ 16 / 16 = 16 / y
Simplifying the equation: 1 = 16 / y
Cross-multiplying: y = 16
So, BZ = 16.
Therefore, BC = 16 and BZ = 16 in triangle AB.
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