Olympiad problems

2022 SEEMOUS, 2

Tags: analysis , functional equation , continuity , palic , differentiation , integration

Let $a, b, c \in \mathbb{R}$ be such that $$a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1.$$ We say that a function $f$ is a [i]Palić function[/i] if $f: \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous and satisfies $$f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz)$$ for all $x, y, z \in \mathbb{R}.$ Prove that any Palić function is infinitely many times differentiable and find all Palić functions.

2007 F = Ma, 3

Tags: analytic geometry , function , differentiation

The coordinate of an object is given as a function of time by $x = 8t - 3t^2$, where $x$ is in meters and $t$ is in seconds. Its average velocity over the interval from $ t = 1$ to $t = 2 \text{ s}$ is $ \textbf{(A)}\ -2\text{ m/s}\qquad\textbf{(B)}\ -1\text{ m/s}\qquad\textbf{(C)}\ -0.5\text{ m/s}\qquad\textbf{(D)}\ 0.5\text{ m/s}\qquad\textbf{(E)}\ 1\text{ m/s} $

2006 Petru Moroșan-Trident, 3

Tags: function , calculus , primitive , differentiation

Let be a differentiable function $ f:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} , $ and a primitive $ F:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} $ of it such that $ F=f+f\cdot f. $ Show that: [b]a)[/b] $ f $ is nondecreasing. [b]b)[/b] $ \lim_{x\to\infty } f(x)/x =1/2 $ [i]Vasile Solovăstru[/i]

2022 ISI Entrance Examination, 2

Tags: calculus , differentiation , function

Consider the function $$f(x)=\sum_{k=1}^{m}(x-k)^{4}~, \qquad~ x \in \mathbb{R}$$ where $m>1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.

1997 VJIMC, Problem 3

Tags: calculus , differentiation

Let $c_1,c_2,\ldots,c_n$ be real numbers such that $$c_1^k+c_2^k+\ldots+c_n^k>0\qquad\text{for all }k=1,2,\ldots$$Let us put $$f(x)=\frac1{(1-c_1x)(1-c_2x)\cdots(1-c_nx)}.$$$z\in\mathbb C$ Show that $f^{(k)}(0)>0$ for all $k=1,2,\ldots$.