Olympiad problems

Today's calculation of integrals, 882

Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

2010 Contests, 3

Tags: function , trigonometry , complex number , calculus , inequalities , algebra

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.

2010 Princeton University Math Competition, 4

Tags: quadratic , quadratic formula , algebra , integration , calculus

Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$. (Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)

2003 Flanders Math Olympiad, 4

Tags: ratio , parameterization , least common multiple , analytic geometry , number theory , integration , calculus

Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with $r=2\sqrt2$ goes through 4 points) Prove that $\forall n\in \mathbb{N}, \exists r$ so that the circle with midpoint 0,0 and radius $r$ goes through at least $n$ points.

1999 Mexico National Olympiad, 6

Tags: integration , calculus , combinatorics

A polygon has each side integral and each pair of adjacent sides perpendicular (it is not necessarily convex). Show that if it can be covered by non-overlapping $2 x 1$ dominos, then at least one of its sides has even length.

1989 Putnam, A6

Tags: power series , calculus

Let $\alpha=1+a_1x+a_2x^2+\ldots$ be a formal power series with coefficients in the field of two elements. Let $$a_n=\begin{cases}1&\text{if every block of zeroes in the binary expansion of }n\text{ has an even number of zeroes}\\0&\text{otherwise}\end{cases}$$(For example, $a_{36}=1$ since $36=100100_2$) Prove that $\alpha^3+x\alpha+1=0$.

2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 2

Tags: trigonometry , inequalities , calculus

Let $f(x,\ y)=\frac{x+y}{(x^2+1)(y^2+1)}.$ (1) Find the maximum value of $f(x,\ y)$ for $0\leq x\leq 1,\ 0\leq y\leq 1.$ (2) Find the maximum value of $f(x,\ y),\ \forall{x,\ y}\in{\mathbb{R}}.$

1997 AMC 12/AHSME, 21

Tags: function , matrix , logarithm , integration , linear algebra , calculus

For any positive integer $ n$, let \[f(n) \equal{} \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\ 0, & \text{otherwise.} \end{cases}\] What is $ \sum_{n \equal{} 1}^{1997}{f(n)}$? $ \textbf{(A)}\ \log_8{2047}\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ \frac {55}{3}\qquad \textbf{(D)}\ \frac {58}{3}\qquad \textbf{(E)}\ 585$

2021 CIIM, 1

Tags: geometry , calculus , analytic geometry

For every $0 < \alpha < 1$, let $R(\alpha)$ be the region in $\mathbb{R}^2$ whose boundary is the convex pentagon of vertices $(0,1-\alpha), (\alpha, 0), (1, 0), (1,1)$ and $(0, 1)$. Let $R$ be the set of points that belong simultaneously to each of the regions $R(\alpha)$ with $0 < \alpha < 1$, that is, $R =\bigcap_{0<\alpha<1} R(\alpha)$. Determine the area of $R$.

1940 Putnam, A3

Tags: integration , calculus , function

Let $a$ be a real number. Find all real-valued functions $f$ such that $$\int f(x)^{a} dx=\left( \int f(x) dx \right)^{a}$$ when constants of integration are suitably chosen.

2021 JHMT HS, 4

Tags: calculus , functional equation , function , algebra

There is a unique differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ satisfying $f(x) + (f(x))^3 = x + x^7$ for all real $x.$ The derivative of $f(x)$ at $x = 2$ can be expressed as a common fraction $a/b.$ Compute $a + b.$

2011 Today's Calculation Of Integral, 721

Tags: integration , function , calculus computations , calculus

For constant $a$, find the differentiable function $f(x)$ satisfying $\int_0^x (e^{-x}-ae^{-t})f(t)dt=0$.

2008 AMC 12/AHSME, 18

Tags: calculus , 3d geometry , pyramid , area of a triangle , tetrahedron , analytic geometry , geometry

Triangle $ ABC$, with sides of length $ 5$, $ 6$, and $ 7$, has one vertex on the positive $ x$-axis, one on the positive $ y$-axis, and one on the positive $ z$-axis. Let $ O$ be the origin. What is the volume of tetrahedron $ OABC$? $ \textbf{(A)}\ \sqrt{85} \qquad \textbf{(B)}\ \sqrt{90} \qquad \textbf{(C)}\ \sqrt{95} \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ \sqrt{105}$

2020 Jozsef Wildt International Math Competition, W52

Tags: calculus , derivative , inequalities

If $f\in C^{(3)}([0,1])$ such that $f(0)=f(1)=f'(0)=0$ and $|f'''(x)|\le1,(\forall)x\in[0,1]$, show that: a) $$|f(x)|\le\frac{x(1-x)}{\sqrt3}\cdot\left(\int^x_0\frac{f(t)}{t(1-t)}dt\right)^{1/2},(\forall)x\in[0,1]$$ b) $$|f'(x)|\le\frac{1-2x}{\sqrt3}\cdot\left(\int^x_0\frac{|f(t)|}{t(1-t)}dt\right)^{1/2},(\forall)x\in\left[0,\frac12\right]$$ c) $$\int^1_0(1-x)^2\cdot\frac{|f(x)|}xdx\ge9\int^1_0\left(\frac{f(x)}x\right)^2dx$$ [i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]

2013 Turkey Junior National Olympiad, 1

Tags: function , algebra , calculus , polynomial , vieta , inequalities , analytic geometry

Let $x, y, z$ be real numbers satisfying $x+y+z=0$ and $x^2+y^2+z^2=6$. Find the maximum value of \[ |(x-y)(y-z)(z-x) | \]

2009 Vietnam National Olympiad, 4

Tags: function , polynomial , logarithm , integration , induction , calculus , algebra

Let $ a$, $ b$, $ c$ be three real numbers. For each positive integer number $ n$, $ a^n \plus{} b^n \plus{} c^n$ is an integer number. Prove that there exist three integers $ p$, $ q$, $ r$ such that $ a$, $ b$, $ c$ are the roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} r \equal{} 0$.

1992 Cono Sur Olympiad, 1

Tags: calculus , integration , number theory

Prove that there aren't any positive integrer numbers $x,y,z$ such that $x^2+y^2=3z^2$.

2012 Putnam, 6

Tags: rectangle , calculus , function , integration , real analysis

Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2.$ Suppose that, for every rectangular region $R$ of area $1,$ the double integral of $f(x,y)$ over $R$ equals $0.$ Must $f(x,y)$ be identically $0?$

2003 SNSB Admission, 1

Tags: function , complex analysis , derivative , complex number , calculus , inequalities

Show that if a holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ has the property that the modulus of any of its derivatives (of any order) is everywhere dominated by $ 1, $ then $ |f(z)|\le e^{|\text{Im} (z)|} , $ for all complex numbers $ z. $

2007 Harvard-MIT Mathematics Tournament, 2

Tags: calculus , polynomial , algebra

Determine the real number $a$ having the property that $f(a)=a$ is a relative minimum of $f(x)=x^4-x^3-x^2+ax+1$.

2007 Harvard-MIT Mathematics Tournament, 4

Tags: function , parabola , calculus , conic

Find the real number $\alpha$ such that the curve $f(x)=e^x$ is tangent to the curve $g(x)=\alpha x^2$.

Today's calculation of integrals, 869

Tags: calculus , integration , trigonometry , calculus computations

Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$ Answer the questions below. (1) Find $I_n.$ (2) Find $\sum_{n=1}^{\infty} I_n.$

2023 IMC, 7

Tags: function , calculus

Let $V$ be the set of all continuous functions $f\colon [0,1]\to \mathbb{R}$, differentiable on $(0,1)$, with the property that $f(0)=0$ and $f(1)=1$. Determine all $\alpha \in \mathbb{R}$ such that for every $f\in V$, there exists some $\xi \in (0,1)$ such that \[f(\xi)+\alpha = f'(\xi)\]

2000 Moldova National Olympiad, Problem 6

Tags: calculus , integration

Show that there is a positive number $p$ such that $\int^\pi_0x^p\sin xdx=\sqrt[10]{2000}$.

1965 Putnam, B1

Tags: limit , integration , calculus , trigonometry

Evaluate $ \lim_{n\to\infty} \int_0^1 \int_0^1 \cdots \int_0^1 \cos ^ 2 \left\{\frac{\pi}{2n}(x_1\plus{}x_2\plus{}\cdots \plus{}x_n)\right\} dx_1dx_2\cdots dx_n.$