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Balkan Mathematical Olympiad 2017 – Problems

May 10, 2017 Leave a comment

Problem 1. Find all ordered pairs of positive integers { (x, y)} such that:

\displaystyle x^3+y^3=x^2+42xy+y^2.

Problem 2. Consider an acute-angled triangle {ABC} with {AB<AC} and let {\omega} be its circumscribed circle. Let {t_B} and {t_C} be the tangents to the circle {\omega} at points {B} and {C}, respectively, and let {L} be their intersection. The straight line passing through the point {B} and parallel to {AC} intersects {t_C} in point {D}. The straight line passing through the point {C} and parallel to {AB} intersects {t_B} in point {E}. The circumcircle of the triangle {BDC} intersects {AC} in {T}, where {T} is located between {A} and {C}. The circumcircle of the triangle {BEC} intersects the line {AB} (or its extension) in {S}, where {B} is located between {S} and {A}.

Prove that {ST}, {AL}, and {BC} are concurrent.

Problem 3. Let {\mathbb{N}} denote the set of positive integers. Find all functions {f:\mathbb{N}\longrightarrow\mathbb{N}} such that

\displaystyle n+f(m)\mid f(n)+nf(m)

for all {m,n\in \mathbb{N}}

Problem 4. On a circular table sit {\displaystyle {n> 2}} students. First, each student has just one candy. At each step, each student chooses one of the following actions:

  • (A) Gives a candy to the student sitting on his left or to the student sitting on his right.
  • (B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right.

At each step, students perform the actions they have chosen at the same time. A distribution of candy is called legitimate if it can occur after a finite number of steps. Find the number of legitimate distributions.

(Two distributions are different if there is a student who has a different number of candy in each of these distributions.)

Source: AoPS

Some of the easy Putnam 2016 Problems

December 11, 2016 Leave a comment

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…

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IMC 2016 – Day 2 – Problem 8

July 28, 2016 2 comments

Problem 8. Let {n} be a positive integer and denote by {\Bbb{Z}_n} the ring of integers modulo {n}. Suppose that there exists a function {f:\Bbb{Z}_n \rightarrow \Bbb{Z}_n} satisfying the following three properties:

  • (i) {f(x) \neq x},
  • (ii) {x = f(f(x))},
  • (iii) {f(f(f(x+1)+1)+1) = x} for all {x \in \Bbb{Z}_n}.

Prove that {n \equiv 2} modulo {4}.

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IMC 2016 – Day 2 – Problem 6

July 28, 2016 1 comment

Problem 6. Let {(x_1,x_2,...)} be a sequence of positive real numbers satisfying {\displaystyle \sum_{n=1}^\infty \frac{x_n}{2n-1}=1}. Prove that

\displaystyle \sum_{k=1}^\infty \sum_{n=1}^k \frac{x_n}{k^2} \leq 2.

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IMC 2016 Problems – Day 2

July 28, 2016 Leave a comment

Problem 6. Let {(x_1,x_2,...)} be a sequence of positive real numbers satisfying {\displaystyle \sum_{n=1}^\infty \frac{x_n}{2n-1}=1}. Prove that

\displaystyle \sum_{k=1}^\infty \sum_{n=1}^k \frac{x_n}{k^2} \leq 2.

Problem 7. Today, Ivan the Confessor prefers continuous functions {f:[0,1]\rightarrow \Bbb{R}} satisfying {f(x)+f(y) \geq |x-y|} for all {x,y \in [0,1]}. Fin the minimum of {\int_0^1 f} over all preferred functions.

Problem 8. Let {n} be a positive integer and denote by {\Bbb{Z}_n} the ring of integers modulo {n}. Suppose that there exists a function {f:\Bbb{Z}_n \rightarrow \Bbb{Z}_n} satisfying the following three properties:

  • (i) {f(x) \neq x},
  • (ii) {f(f(x))=x},
  • (iii) {f(f(f(x+1)+1)+1) = x} for all {x \in \Bbb{Z}_n}.

Prove that {n \equiv 2} modulo {4}.

Problem 9. Let {k} be a positive integer. For each nonnegative integer {n} let {f(n)} be the number of solutions {(x_1,...,x_k) \in \Bbb{Z}^k} of the inequality {|x_1|+...+|x_k| \leq n}. Prove that for every {n \geq 1} we have {f(n-1)f(n+1) \leq f(n)^2}.

Problem 10. Let {A} be a {n \times n} complex matrix whose eigenvalues have absolute value at most {1}. Prove that

\displaystyle \|A^n\| \leq \frac{n}{\ln 2} \|A\|^{n-1}.

(Here {\|B\| = \sup_{\|x\|\leq 1} \|Bx\|} for every {n \times n} matrix {B} and {\|x\| = \sqrt{\sum_{i=1}^n |x_i|^2 }} for every complex vector {x \in \Bbb{C}^n}.)

Official source and more infos here.

Categories: Olympiad, Uncategorized Tags: , ,

IMC 2016 – Day 1 – Problem 3

July 27, 2016 Leave a comment

Problem 3. Let {n} be a positive integer. Also let {a_1,a_2,...,a_n} and {b_1,b_2,...,b_n} be reap numbers such that {a_i+b_i >0} for {i = 1,2,...,n}. Prove that

\displaystyle \sum_{i=1}^n \frac{a_ib_i -b_i^2}{a_i+b_i} \leq \frac{\sum_{i=1}^n a_i \cdot \sum_{i=1}^n b_i - \left(\sum_{i=1}^n b_i \right)^2 }{\sum_{i=1}^n (a_i+b_i)}.

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IMC 2016 – Day 1 – Problem 1

July 27, 2016 Leave a comment

Problem 1. Let {f:[a,b] \rightarrow \Bbb{R}} be continuous on {[a,b]} and differentiable on {(a,b)}. Suppose that {f} has infinitely many zeros, but there is no {x \in (a,b)} with {f(x)=f'(x) = 0}.

  • (a) Prove that {f(a)f(b)=0}.
  • (b) Give an example of such a function.

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