Olympiad problems

2002 District Olympiad, 4

Let be a continuous and periodic function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ of period $ 1. $ Show: [b]a)[/b] $ a\in\mathbb{R}\implies\int_a^{a+1} f(x)dx =\int_0^1 f(x) dx . $ [b]b)[/b] $ \lim_{n\to\infty} \int_0^1 f(x)f(nx) dx=\left( \int_0^1 f(x) dx \right)^2 . $ [i]C. Mortici[/i]

1985 IMO, 6

Tags: limit , polynomial , calculus , fixed point , sequence

For every real number $x_1$, construct the sequence $x_1,x_2,\ldots$ by setting: \[ x_{n+1}=x_n(x_n+{1\over n}). \] Prove that there exists exactly one value of $x_1$ which gives $0<x_n<x_{n+1}<1$ for all $n$.

Today's calculation of integrals, 857

Tags: calculus , integration , limit , trigonometry , geometry , geometric transformation , rotation

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

2011 Tokio University Entry Examination, 3

Tags: calculus , integration , analytic geometry , geometry , perimeter , limit , logarithm

Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by　distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$. (1) Find $u(t),\ v(t)$. (2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$. (3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$. [i]2011 Tokyo University entrance exam/Science, Problem 3[/i]

2015 Vietnam National Olympiad, 2

Tags: induction , calculus , derivative , inequalities

If $a,b,c$ are nonnegative real numbers, then \[{ 3(a^2+b^2+c^2) \geq (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2 \geq (a+b+c)^2.}\]

2003 SNSB Admission, 1

Tags: complex analysis , function , calculus , derivative , complex number , 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. $

1965 AMC 12/AHSME, 34

Tags: quadratic , calculus , function

For $ x \ge 0$ the smallest value of $ \frac {4x^2 \plus{} 8x \plus{} 13}{6(1 \plus{} x)}$ is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \frac {25}{12} \qquad \textbf{(D)}\ \frac {13}{6} \qquad \textbf{(E)}\ \frac {34}{5}$

2011 Today's Calculation Of Integral, 696

Tags: calculus , integration , algebra , polynomial , inequalities , function , linear algebra

Let $P(x),\ Q(x)$ be polynomials such that : \[\int_0^2 \{P(x)\}^2dx=14,\ \int_0^2 P(x)dx=4,\ \int_0^2 \{Q(x)\}^2dx=26,\ \int_0^2 Q(x)dx=2.\] Find the maximum and the minimum value of $\int_0^2 P(x)Q(x)dx$.

Today's calculation of integrals, 865

Tags: calculus , integration , geometry , geometric transformation , rotation , analytic geometry , rectangle

Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.

2015 Belarus Team Selection Test, 2

Tags: algebra , function , calculus

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

2011 Baltic Way, 4

Tags: calculus , inequalities

Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=4$. Prove the inequality \[\frac{a}{a^3+8}+\frac{b}{b^3+8}+\frac{c}{c^3+8}+\frac{d}{d^3+8}\le\frac{4}{9}\]

1999 Harvard-MIT Mathematics Tournament, 6

Tags: calculus , derivative , trigonometry

Evaluate $\dfrac{d}{dx}\left(\sin x - \dfrac{4}{3}\sin^3 x\right)$ when $x=15$.

2009 Today's Calculation Of Integral, 483

Tags: calculus , integration , limit , function , calculus computations , logarithm

Let $ n\geq 2$ be natural number. Answer the following questions. (1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$ (2) Prove the following inequality. $ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$ (3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$

2024 IMC, 2

Tags: real analysis , calculus , logarithm

For $n=1,2,\dots$ let \[S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}),\] where $\log$ denotes the natural logarithm. Find $\lim_{n \to \infty} S_n$.

1984 Vietnam National Olympiad, 1

Tags: calculus , integration , quadratic , number theory , algebra

$(a)$ Let $x, y$ be integers, not both zero. Find the minimum possible value of $|5x^2 + 11xy - 5y^2|$. $(b)$ Find all positive real numbers $t$ such that $\frac{9t}{10}=\frac{[t]}{t - [t]}$.

PEN H Problems, 54

Tags: calculus , integration , ratio , geometry , perimeter , quadratic

Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is $p^{m}$, where $p$ is a prime and $m$ is an integer, is $m+1$ if $p=2$ and $2m+1$ if $p \neq 2$.

1999 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , geometry , circumcircle

Let $A_n$ be the area outside a regular $n$-gon of side length $1$ but inside its circumscribed circle, let $B_n$ be the area inside the $n$-gon but outside its inscribed circle. Find the limit as $n$ tends to infinity of $\dfrac{A_n}{B_n}$.

2002 USA Team Selection Test, 6

Tags: calculus , quadratic , analytic geometry , algorithm , number theory

Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$.

2010 Today's Calculation Of Integral, 644

Tags: calculus , integration , trigonometry , inequalities , calculus computations , logarithm

For a constant $p$ such that $\int_1^p e^xdx=1$, prove that \[\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.\] Own

2005 Today's Calculation Of Integral, 18

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Calculate the following indefinite integrals. [1] $\int (\sin x+\cos x)^4 dx$ [2] $\int \frac{e^{2x}}{e^x+1}dx$ [3] $\int \sin ^ 4 xdx$ [4] $\int \sin 6x\cos 2xdx$ [5] $\int \frac{x^2}{\sqrt{(x+1)^3}}dx$

2024 CMIMC Integration Bee, 10

Tags: calculus , integration

\[\int_{-1}^1 \sqrt[3]{x}\log(1+e^x)\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2013 Online Math Open Problems, 25

Tags: modular arithmetic , calculus , integration

Positive integers $x,y,z \le 100$ satisfy \begin{align*} 1099x+901y+1110z &= 59800 \\ 109x+991y+101z &= 44556 \end{align*} Compute $10000x+100y+z$. [i]Evan Chen[/i]

2002 VJIMC, Problem 4

Tags: calculus , integration

Prove that $$\lim_{n\to\infty}n^2\left(\int^1_0\sqrt[n]{1+x^n}\text dx-1\right)=\frac{\pi^2}{12}.$$

II Soros Olympiad 1995 - 96 (Russia), 11.4

Tags: algebra , analysis , calculus

Consider the graph of the function $y = (1 -x^2)^3$. Find the set of points $M(x,y)$ through which you can draw at least $6$ lines touching this graph.

1997 AMC 12/AHSME, 13

Tags: calculus , integration

How many two-digit positive integers $ N$ have the property that the sum of $ N$ and the number obtained by reversing the order of the digits of $ N$ is a perfect square? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$