MathDB

12

Part of 2009 AIME Problems

Problems(2)

Right triangle with a circle

Source: 2009 AIME I #12

3/18/2009
In right △ABC \triangle ABC with hypotenuse AB‾ \overline{AB}, AC \equal{} 12, BC \equal{} 35, and CD‾ \overline{CD} is the altitude to AB‾ \overline{AB}. Let ω \omega be the circle having CD‾ \overline{CD} as a diameter. Let I I be a point outside △ABC \triangle ABC such that AI‾ \overline{AI} and BI‾ \overline{BI} are both tangent to circle ω \omega. The ratio of the perimeter of △ABI \triangle ABI to the length AB AB can be expressed in the form mn \displaystyle\frac{m}{n}, where m m and n n are relatively prime positive integers. Find m\plus{}n.
ratiogeometryperimeterinradiustrapezoidAMCAIME
Distinct Pairs with Distinct Sums

Source: AIME 2009II Problem 12

4/2/2009
From the set of integers {1,2,3,…,2009} \{1,2,3,\ldots,2009\}, choose k k pairs {ai,bi} \{a_i,b_i\} with ai<bi a_i<b_i so that no two pairs have a common element. Suppose that all the sums a_i\plus{}b_i are distinct and less than or equal to 2009 2009. Find the maximum possible value of k k.
AMC2009 AIME IIAIME