# 2021-13 Not convex

Prove or disprove the following:

There exist an infinite sequence of functions $$f_n: [0, 1] \to \mathbb{R} , n=1, 2, \dots$$ ) such that

(1) ( f_n(0) = f_n(1) = 0 ) for any ( n ),

(2) ( f_n(\frac{a+b}{2}) \leq f_n(a) + f_n(b) ) for any ( a, b \in [0, 1] ),

(3) ( f_n – c f_m ) is not identically zero for any ( c \in \mathbb{R} ) and ( n \neq m ).

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# 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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# 2018-16 A convex function

Find the minimum $$m$$ (if it exists) such that every convex function $$f:[-1,1]\to[-1,1]$$ has a constant $$c$$ such that $\int_{-1}^1 \lvert f(x)-c\rvert \,dx \le m.$

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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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# 2012-15 Functional Equation

Let $$n$$ be a fixed positive integer. Find all functions $$f:\mathbb{R}\to\mathbb{R}$$ satisfying $f(x^{n+1}-y^{n+1})=(x-y)[f(x)^n+f(x)^{n-1}f(y)+\cdots+f(x)f(y)^{n-1}+f(y)^n].$

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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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# 2011-18 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(xf(x)+c g′(x)|≤|g(x)| for all x∈I,

then g(x)=0 for all x∈I.

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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 $a_n=\sum_{k=0}^{\infty} \frac{k^n}{2^{k+1}}.$