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Geometry Warmups
Mathematics is filled with shapes that are kaleidoscopic in variety. Wielded since ancient times, the power of geometry helps us examine and measure these shapes.
In a triangle \(ABC\) \(\angle A =84^\circ,\) \(\angle C=78^\circ.\) Points \(D\) and \(E\) are taken on the sides \(AB\) and \(BC,\) so that \(\angle ACD =48^\circ,\) \(\angle CAE =63^\circ.\) What is the measure (in degrees) of \(\angle CDE\) ?
by
Calvin Lin
A parabola passes through the following points
\[ \left( -1,1 \right) ,\left( 0,0 \right) ,\left( \dfrac { 1 }{ 2 } ,\dfrac { 1 }{ 4 } \right) ,\left( 1,1 \right) ,\left( -\dfrac { 7 }{ 5 } ,-\dfrac { 3 }{ 5 } \right). \]
It also passes through the point: \((\dfrac { 289 }{ 240 } ,\dfrac { a }{ b } ),\) where \(a,b\) are positive coprime integers. Find \(a+b\)
by
Michael Mendrin
by
Garrett Webb
A square sheet of paper has its center marked with point \(O\). A random point is uniformly chosen on the square paper, and this point is labeled \(P\). The paper is then folded so that point \(P\) coincides with point \(O\). Let the probability that a pentagon is formed after the fold be \(N\). Find the value of \(\lfloor 1000N \rfloor\).
by
Daniel Liu
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.
\[ \frac{\text{radius of blue circle}}{\text{radius of red circle}} = \frac{a}{b}, \]
where \(a\) and \(b\) are coprime positive integers. Find \(a+b\).
by
Matt Enlow