Tag Archives: function

2018-20 Almost Linear Function

Let \(f:\mathbb R\to\mathbb R\) be a function such that \[ -1\le f(x+y)-f(x)-f(y)\le 1\] for all reals \(x\), \(y\). Does there exist a constant \(c\) such that \( \lvert f(x)-cx\rvert \le 1\) for all reals \(x\)?

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2015-20 Dense function

Prove or disprove the following statement:

There exists a function \( f : \mathbb{R} \to \mathbb{R} \) such that

(1) \( f \equiv 0 \) almost everywhere, and

(2) for any nonempty open interval \(I\), \( f(I) = \mathbb{R} \).

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2012-13 functions for an inequality

Determine all nonnegative functions f(x,y) and g(x,y) such that \[ \left(\sum_{i=1}^n a_i b_i \right)^2 \le \left( \sum_{i=1}^n f(a_i,b_i)\right) \left(\sum_{i=1}^n g(a_i,b_i)\right) \le \left(\sum_{i=1}^n a_i^2\right) \left(\sum_{i=1}^n b_i^2\right)\] for all reals \(a_i\), \(b_i\) and all positive integers n.

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2012-8 Non-fixed points

Let X be a finite non-empty set. Suppose that there is a function \(f:X\to X\) such that \( f^{20120407}(x)=x\) for all \(x\in X\). Prove that the number of elements x in X such that \(f(x)\neq x\) is divisible by 20120407.

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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.

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2011-3 Counting functions

Let us write \([n]=\{1,2,\ldots,n\}\). Let \(a_n\) be the number of all functions \(f:[n]\to [n]\) such that \(f([n])=[k]\) for some positive integer \(k\). Prove that \[a_n=\sum_{k=0}^{\infty} \frac{k^n}{2^{k+1}}.\]

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