Geometry Warmups

Geometry Warmups: Level 5 Challenges


In a triangle ABCABC A=84,\angle A =84^\circ, C=78.\angle C=78^\circ. Points DD and EE are taken on the sides ABAB and BC,BC, so that ACD=48,\angle ACD =48^\circ, CAE=63.\angle CAE =63^\circ. What is the measure (in degrees) of CDE\angle CDE ?

A parabola passes through the following points

(1,1),(0,0),(12,14),(1,1),(75,35). \left( -1,1 \right) ,\left( 0,0 \right) ,\left( \dfrac { 1 }{ 2 } ,\dfrac { 1 }{ 4 } \right) ,\left( 1,1 \right) ,\left( -\dfrac { 7 }{ 5 } ,-\dfrac { 3 }{ 5 } \right).

It also passes through the point: (289240,ab),(\dfrac { 289 }{ 240 } ,\dfrac { a }{ b } ), where a,ba,b are positive coprime integers. Find a+ba+b

Consider a glass in the shape of an inverted truncated right cone (i.e. frustrum). The radius of the base is 4, the radius of the top is 9, and the height is 7. There is enough water in the glass such that when it is tilted the water reaches from the tip of the base to the edge of the top. The proportion of the water in the cup as a ratio of the cup's volume can be expressed as the fraction mn \frac{m}{n} , for relatively prime integers mm and nn. Compute m+nm+n.

A square sheet of paper has its center marked with point OO. A random point is uniformly chosen on the square paper, and this point is labeled PP. The paper is then folded so that point PP coincides with point OO. Let the probability that a pentagon is formed after the fold be NN. Find the value of 1000N\lfloor 1000N \rfloor.

The two large gray circles are congruent, and each is half the diameter of the largest circle. All circles that appear to be tangent to each other are indeed tangent to each other.

radius of blue circleradius of red circle=ab, \frac{\text{radius of blue circle}}{\text{radius of red circle}} = \frac{a}{b},
where aa and bb are coprime positive integers. Find a+ba+b.


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