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Problems 2

The document presents five mathematical problems involving real numbers, composite integers, ordered triplets, triangle geometry, and circle intersections. Each problem requires finding specific values or counts based on given conditions. The problems involve concepts such as factors, area calculations, and properties of geometric shapes.

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42 views1 page

Problems 2

The document presents five mathematical problems involving real numbers, composite integers, ordered triplets, triangle geometry, and circle intersections. Each problem requires finding specific values or counts based on given conditions. The problems involve concepts such as factors, area calculations, and properties of geometric shapes.

Uploaded by

ayushagrules
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Problems

1. Real numbers p, q ∈ [ 56 , 1] satisfy pq = p + q − 1. The smallest possible value


of p3 + q 3 can be expressed as m n , where m and n are relatively prime positive
integers. Find m + n.
√
2. A composite integer k is denoted as fair√if and only√ if 30k cannot be
simplified. For example, 4 is not fair, because 120 = 2 30. Let N be the sum
of all fair 1 ≤ k ≤ 200. Find N .

3. Find the number of ordered triplets of positive integers (a, b, c) that sat-
isfy the property that a + 2b + 3c is a factor of 30.

4. Let I be the incenter of △ABC. If IA = IB = IC, there exists a point


P inside △ABC so that the shortest distances from P to AB, BC, and AC are
3, 5, and 7, respectively. The square of the area of [ABC] can be expressed as
m
n , where m and n are relatively prime positive integers. Find m + n.

5. Circles ω1 and ω2 intersect at points P and Q. The tangent to both ω1


and ω2 closer to P intersects QP at A. The radius of ω1 is 6 and the radius of
ω2 is 8. If AP = 3, find the value of ⌊10P Q⌋.

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