Compute tan-1(1) -tan-1(1/3) + tan-1(1/5) – tan-1(1/7) + … .
Category Archives: problem
2011-24 (n-k) choose k
Evaluate the sum \[ \sum_{k=0}^{[n/2]} (-4)^{n-k} \binom{n-k}{k} ,\] where [x] denotes the greatest integer less than or equal to x.
2011-23 Constant Function
Let \(f:\mathbb{R}^n\to \mathbb{R}^{n-1}\) be a function such that for each point a in \(\mathbb{R}^n\), the limit $$\lim_{x\to a} \frac{|f(x)-f(a)|}{|x-a|}$$ exists. Prove that f is a constant function.
2011-22 Seoul Subway Line 2
In Seoul Subway Line 2, subway stations are placed around a circular subway line. Assume that each segment of Seoul Subway Line 2 has a fixed price. Suppose that you hid money at each subway station so that the sum of the money is only enough for one roundtrip around Seoul Subway Line 2.
Prove that there is a station that you can start and take a roundtrip tour of Seoul Subway Line 2 while paying each segment by the money collected at visited stations.
2011-21 Zeros
For a nonnegative integer n, let \(F_n(x)=\sum_{m=0}^n \frac{(-2)^m (2n-m)! \Gamma(x+1)}{m! (n-m)! \Gamma(x-m+1)}\). Find all x such that Fn(x)=0.
2011-20 Double infinite series
For a real number x, let d(x)=minn:integer (x-n)2. Evaluate the following double infinite series:
. . . + 8 d(x/8)+4 d(x/4) + 2 d(x/2) + d(x) + d(2x) / 2 + d(4x)/4 + d(8x)/8 + . . .
2011-19 Irreducible polynomial
Find all n≥2 such that the polynomial xn-xn-1-xn-2-…-x-1 is irreducible over the rationals.
2011-18 Continuous Function and Differentiable Function
Let f(x) be a continuous function on I=[a,b], and let g(x) be a differentiable function on I. Let g(a)=0 and c≠0 a constant. Prove that if
|g(x) f(x)+c g′(x)|≤|g(x)| for all x∈I,
then g(x)=0 for all x∈I.
2011-17 Infinitely many solutions
Let f(n) be the maximum positive integer m such that the sum of all positive divisors of m is less than or equal to n. Find all positive integers k such that there are infinitely many positive integers n satisfying the equation n-f(n)=k.
2011-16 Odd Sets with Even Intersection
Let A1, A2, A3, …, An be finite sets such that |Ai| is odd for all 1≤i≤n and |Ai∩Aj| is even for all 1≤i<j≤n. Prove that it is possible to pick one element ai in each set Ai so that a1, a2, …,an are distinct.