Home > Analysis, Geometry, Olympiad, Problem Solving > Some of the easy Putnam 2016 Problems

Some of the easy Putnam 2016 Problems


Here are a few of the problems of the Putnam 2016 contest. I choose to only list problems which I managed to solve. Most of them are pretty straightforward, so maybe the solutions posted here may be very similar to the official ones. You can find a complete list of the problems on other sites, for example here.

A1. Find the smallest integer {j} such that for every polynomial {p} with integer coefficients and every integer {k}, the number

\displaystyle p^{(j)}(k),

that is the {j}-th derivative of {p} evaluated at {k}, is divisible by {2016}.

Hints. Successive derivatives give rise to terms containing products of consecutive numbers. The product of {j} consecutive numbers is divisible by {j!}. Find the smallest number such that {2016 | j!}. Prove that {j-1} does not work by choosing {p = x^{j-1}}. Prove that {j} works by working only on monomials…

A2. Given a positive integer {n} let {M(n)} be the largest integer {m} such that

\displaystyle {m \choose n-1} > { m-1 \choose n}.

Evaluate

\displaystyle \lim_{n \rightarrow \infty} \frac{M(n)}{n}.

Hints: Write formulas for binomial coefficients and simplify what is possible. What remains should be an inequation of degree {2} in terms of {m}. Find the largest positive root of the equation and the largest {m} should be the integer part of that root. Using the fact that the integer part of {x} is equivalent to {x} as {x \rightarrow \infty} we can conclude.

A3. Suppose that {f} is a function from {\Bbb{R} \rightarrow \Bbb{R}} such that

\displaystyle f(x) + f\left( 1-\frac{1}{x} \right) =\arctan x

for all real {x \neq 0}. Find {\int_0^1 f(x) dx}.

Hints: Use a classical trick of replacing {x} by {1-1/x}. This makes the second argument equal to {-1/(x-1)}. Note that making now {x \mapsto -1/(x-1)} the second argument becomes {x}. This will give three equations which allow us to find a formula for {f(x)}.

B1. Let {(x_n)} be a sequence such that {x_0=1} and for {n \geq 0}

\displaystyle x_{n+1} = \ln(e^{x_n}-x_n).

Show that the infinite series {x_0+x_1+x_2+...} converges and find its sum.

Hints: First note that {e^x\geq x+1} and see that {x_n \geq 0}. Exponentiate to find {e^{x_{n+1}} = e^{x_n}-x_n} which will give the monotonicity of the sequence (and the convergence) and also a telescopic expression for {x_n}. Summing {x_n} becomes a simple expresion in terms of the limit of {x_n}.

B3. Suppose that {S} is a finite set of points in the plane such that the area of the triangle {ABC} is at most {1} whenever {A,B,C} are in {S}. Show that there exists a triangle of area {4} that together with its interior covers the set {S}.

Hints: I was surprised to see this, since it is a classical problem… Just pick the triangle {ABC} with vertices in {S} which has the largest area… Then consider the triangle whose midpoints are {A,B,C}. This triangle will contain {S}.

One way to make this more challenging would be to consider a bounded set {S} in the plane, not necessarily finite… I’ll come back in the next days with more solutions. Problem A4 was particularily interesting and I’ll dedicate a post in order to present it.

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