MathDB

3

Part of 2010 Princeton University Math Competition

Problems(7)

2010 PUMaC Algebra A3/B4: 4^x = x^4

Source:

8/20/2011
Let SS be the sum of all real xx such that 4x=x44^x = x^4. Find the nearest integer to SS.
2010 PUMaC Combinatorics A3/B4: maximum regions of 6 circles

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8/21/2011
Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions?
2010 PUMaC Geometry A3: ratio with angle bisector

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8/21/2011
Triangle ABCABC has AB=4AB = 4, AC=5AC = 5, and BC=6BC = 6. An angle bisector is drawn from angle AA, and meets BCBC at MM. What is the nearest integer to 100AMCM100 \frac{AM}{CM}?
geometryratioangle bisector
2010 PUMaC NT A3/B4: n^2-1 product of 3 primes

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8/22/2011
Find the sum of the first 5 positive integers nn such that n2−1n^2 - 1 is the product of 3 distinct primes.
1 is sum of n reciprocals of triangular numbers

Source: 2010 PUMaC Individual Finals A3

8/31/2011
Show that, if n≠2n \neq 2 is a positive integer, that there are nn triangular numbers a1a_1, a2a_2, …\ldots, ana_n such that ∑i=1n1ai=1\displaystyle{\sum_{i=1}^n \frac1{a_i} = 1} (Recall that the kthk^{th} triangular number is k(k+1)2\frac{k(k+1)}2).
number theory unsolvednumber theory
2010 PUMaC Algebra B3: 5th roots of powers of 2

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8/21/2011
Write 125−1=a+b25+c45+d85+e165\displaystyle{\frac{1}{\sqrt[5]{2} - 1} = a + b\sqrt[5]{2} + c\sqrt[5]{4} + d\sqrt[5]{8} + e\sqrt[5]{16}}, with aa, bb, cc, dd, and ee integers. Find a2+b2+c2+d2+e2a^2 + b^2 + c^2 + d^2 + e^2.
2010 PUMaC Ind. Finals B3: digits in reverse order

Source:

8/31/2011
Find (with proof) all natural numbers nn such that, for some natural numbers aa and bb, a≠ba\ne b, the digits in the decimal representations of the two numbers na+1n^a+1 and nb+1n^b+1 are in reverse order.
Digitsdecimal representationnumber theory