Olympiad problems

2012 Today's Calculation Of Integral, 815

Prove that : $\left|\sum_{i=0}^n \left(1-\pi \sin \frac{i\pi}{4n}\cos \frac{i\pi}{4n}\right)\right|<1.$

2005 Today's Calculation Of Integral, 66

Tags: calculus computations , calculus , trigonometry , integration

Find the minimum value of $\int_0^{\frac{\pi}{2}} |\cos x -a|\sin x \ dx$

2002 National High School Mathematics League, 8

Tags: arithmetic sequence , binomial theorem , algebra , integration , calculus

Consider the expanded form of $\left(x+\frac{1}{2\sqrt[4]{x}}\right)^n$, put all items in number (from high power to low power). If the coefficients of the first three items are arithmetic sequence, then the number of items with an integral power is________.

1964 Spain Mathematical Olympiad, 2

Tags: calculus

The RTP tax is a function $f(x)$, where $x$ is the total of the annual profits (in pesetas). Knowing that: a) $f(x)$ is a continuous function b) The derivative $\frac{df(x)}{dx}$ on the interval $0 \leq 6000$ is constant and equals zero; in the interval $6000< x < P$ is constant and equals $1$; and when $x>P$ is constant and equal 0.14. c) $f(0)=0$ and $f(140000)=14000$. Determine the value of the amount $P$ (in pesetas) and represent graphically the function $y=f(x)$.

1969 AMC 12/AHSME, 35

Tags: analytic geometry , limit , calculus

Let $L(m)$ be the $x$-coordinate of the left end point of the intersection of the graphs of $y=x^2-6$ and $y=m$, where $-6<m<6$. Let $r=[L(-m)-L(m)]/m$. Then, as $m$ is made arbitrarily close to zero, the value of $r$ is: $\textbf{(A) }\text{arbitrarily close to zero}\qquad \textbf{(B) }\text{arbitrarily close to }\tfrac1{\sqrt6}\qquad$ $\textbf{(C) }\text{arbitrarily close to }\tfrac2{\sqrt6}\qquad\,\,\, \textbf{(D) }\text{arbitrarily large}\qquad$ $\textbf{(E) }\text{undetermined}$

1996 Romania National Olympiad, 4

Tags: real analysis , integral , monotone functions , limit , function , calculus

Let $f:[0,1) \to \mathbb{R}$ be a monotonic function. Prove that the limits [center]$\lim_{x \nearrow 1} \int_0^x f(t) \mathrm{d}t$ and $\lim_{n \to \infty} \frac{1}{n} \left[ f(0) + f \left(\frac{1}{n}\right) + \ldots + f \left( \frac{n-1}{n} \right) \right]$[/center] exist and are equal.

2012 Today's Calculation Of Integral, 843

Tags: special factorizations , function , integration , calculus computations , inequalities , calculus

Let $f(x)$ be a continuous function such that $\int_0^1 f(x)\ dx=1.$ Find $f(x)$ for which $\int_0^1 (x^2+x+1)f(x)^2dx$ is minimized.

2010 Today's Calculation Of Integral, 612

Tags: conic , parabola , quadratic , integration , calculus , calculus computations

For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality. \[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]

2000 District Olympiad (Hunedoara), 3

Tags: sequence , limit , real analysis , calculus

Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies $$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$ Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $

1995 AIME Problems, 10

Tags: prime number , number theory , integration , calculus

What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?

2013 Today's Calculation Of Integral, 879

Tags: integral , calculus computations , trigonometry , integration , calculus

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

2013 Today's Calculation Of Integral, 870

Tags: conic , hyperbola , ellipse , geometry , calculus , inequalities , integration

Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$ (1) Find all points of intersection of $E$ and $H$. (2) Find the area of the region expressed by the system of inequality \[\left\{ \begin{array}{ll} 3x^2+y^2\leq 3 &\quad \\ xy\geq \frac 34 , &\quad \end{array} \right.\]

2009 Today's Calculation Of Integral, 478

Tags: trigonometry , integration , calculus computations , calculus

Evaluate $ \int_0^{\frac{\pi}{4}} \{(x\sqrt{\sin x}\plus{}2\sqrt{\cos x})\sqrt{\tan x}\plus{}(x\sqrt{\cos x}\minus{}2\sqrt{\sin x})\sqrt{\cot x}\}\ dx.$

2012 Today's Calculation Of Integral, 829

Tags: logarithm , integration , calculus computations , calculus

Let $a$ be a positive constant. Find the value of $\ln a$ such that \[\frac{\int_1^e \ln (ax)\ dx}{\int_1^e x\ dx}=\int_1^e \frac{\ln (ax)}{x}\ dx.\]

1994 Irish Math Olympiad, 1

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

Let $ x,y$ be positive integers with $ y>3$ and $ x^2\plus{}y^4\equal{}2((x\minus{}6)^2\plus{}(y\plus{}1)^2).$ Prove that: $ x^2\plus{}y^4\equal{}1994.$

1997 Romania National Olympiad, 2

Tags: real analysis , calculus , integration , function

Prove that: $\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2$ Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural) :D

2006 Moldova National Olympiad, 12.2

Tags: limit , floor function , integration , calculus , algebra , function

Let $a, b, n \in \mathbb{N}$, with $a, b \geq 2.$ Also, let $I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx $ and $I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.$ Find $\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.$

2009 Today's Calculation Of Integral, 483

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

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}}.$