Olympiad problems

2011 Today's Calculation Of Integral, 750

Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$

2007 Today's Calculation Of Integral, 191

Tags: calculus computations , geometric series , calculus , integration , limit , geometry , ratio

(1) For integer $n=0,\ 1,\ 2,\ \cdots$ and positive number $a_{n},$ let $f_{n}(x)=a_{n}(x-n)(n+1-x).$ Find $a_{n}$ such that the curve $y=f_{n}(x)$ touches to the curve $y=e^{-x}.$ (2) For $f_{n}(x)$ defined in (1), denote the area of the figure bounded by $y=f_{0}(x), y=e^{-x}$ and the $y$-axis by $S_{0},$ for $n\geq 1,$ the area of the figure bounded by $y=f_{n-1}(x),\ y=f_{n}(x)$ and $y=e^{-x}$ by $S_{n}.$ Find $\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).$

2005 ISI B.Math Entrance Exam, 5

Tags: function , inequalities , graphing lines , slope , analytic geometry , calculus

Find the point in the closed unit disc $D=\{ (x,y) | x^2+y^2\le 1 \}$ at which the function $f(x,y)=x+y$ attains its maximum .

Today's calculation of integrals, 881

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

Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

2003 Moldova National Olympiad, 12.8

Tags: calculus computations , limit , function , calculus

Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by $F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]

2020 LIMIT Category 2, 16

Tags: limit , derivative , calculus

The $n^{th}$ derivative of a function $f(x)$ (if it exists) is denoted by $f^{(n)}(x) $. Let $f(x)=\frac{e^x}{x}$. Suppose $f$ is differentiable infinitely many times in $(0,\infty) $. Then find $\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}$

2008 Purple Comet Problems, 19

Tags: calculus , derivative , geometry

One side of a triangle has length $75$. Of the other two sides, the length of one is double the length of the other. What is the maximum possible area for this triangle

2002 Tuymaada Olympiad, 4

Tags: pigeonhole principle , integration , calculus , floor function , number theory

A real number $a$ is given. The sequence $n_{1}< n_{2}< n_{3}< ...$ consists of all the positive integral $n$ such that $\{na\}< \frac{1}{10}$. Prove that there are at most three different numbers among the numbers $n_{2}-n_{1}$, $n_{3}-n_{2}$, $n_{4}-n_{3}$, $\ldots$. [i]A corollary of a theorem from ergodic theory[/i]

2005 ISI B.Math Entrance Exam, 2

Tags: calculus computations , limit , induction , calculus

Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for all $n\ge 2$ . Define : $P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)$ Compute $\lim_{n\to \infty} P_n$

1991 Arnold's Trivium, 21

Tags: derivative , calculus , integration

Find the derivative of the solution of the equation $\ddot{x} = \dot{x}^2 + x^3$ with initial condition $x(0) = 0$, $\dot{x}(0) = A$ with respect to $A$ for $A = 0$.

2013 Stanford Mathematics Tournament, 3

Tags: limit , trigonometry , calculus , analytic geometry

Suppose $a$ and $b$ are real numbers such that \[\lim_{x\to 0}\frac{\sin^2 x}{e^{ax}-bx-1}=\frac{1}{2}.\] Determine all possible ordered pairs $(a, b)$.

2008 Harvard-MIT Mathematics Tournament, 19

Tags: derivative , 3d geometry , trigonometry , tetrahedron , calculus , geometry

Let $ ABCD$ be a regular tetrahedron, and let $ O$ be the centroid of triangle $ BCD$. Consider the point $ P$ on $ AO$ such that $ P$ minimizes $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$. Find $ \sin \angle PBO$.

1989 Balkan MO, 2

Tags: absolute value , combinatorial geometry , gauss , derivative , polynomial , algebra , calculus

Let $\overline{a_{n}a_{n-1}\ldots a_{1}a_{0}}$ be the decimal representation of a prime positive integer such that $n>1$ and $a_{n}>1$. Prove that the polynomial $P(x)=a_{n}x^{n}+\ldots +a_{1}x+a_{0}$ cannot be written as a product of two non-constant integer polynomials.

2004 Putnam, A6

Tags: calculus , integration , function

Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0\le x\le1,0\le y\le1.$ Show that $\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx$ $\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.$

2016 Israel Team Selection Test, 1

Tags: calculus , lagrange , inequalities

Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \geq 2$.

Today's calculation of integrals, 887

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

For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx. (1) Find $f(\sqrt{3})$ (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.

2009 Today's Calculation Of Integral, 439

Tags: calculus , analytic geometry , integration , logarithm , inequalities , calculus computations

Find the volume of the solid defined by the inequality $ x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2$. Note that you may not directively use double integral here for Japanese high school students who don't study it.

Today's calculation of integrals, 888

Tags: analytic geometry , calculus computations , trigonometry , integration , calculus , geometry

In the coordinate plane, given a circle $K: x^2+y^2=1,\ C: y=x^2-2$. Let $l$ be the tangent line of $K$ at $P(\cos \theta,\ \sin \theta)\ (\pi<\theta <2\pi).$ Find the minimum area of the part enclosed by $l$ and $C$.

2007 Today's Calculation Of Integral, 244

Tags: calculus computations , calculus , integration , function , geometry

A quartic funtion $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

2005 Postal Coaching, 24

Tags: integration , calculus , number theory

Find all nonnegative integers $x,y$ such that \[ 2 \cdot 3^{x} +1 = 7 \cdot 5^{y}. \]

2007 Today's Calculation Of Integral, 220

Tags: inequalities , calculus , integration , calculus computations

Prove that $ \frac{\pi}{2}\minus{}1<\int_{0}^{1}e^{\minus{}2x^{2}}\ dx$.

2012 Today's Calculation Of Integral, 771

Tags: conic , calculus , integration , analytic geometry , parabola , geometry , calculus computations

(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis. (2) For the range of $a$ found in the previous question, express the area bounded by the two tangent lines and the parabola $y=(x+a)^2$ in terms of $a$.

1995 IMC, 2

Tags: calculus , function , integral , real analysis , inequalities

Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$, we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.

2013 VJIMC, Problem 4

Tags: calculus , integration

Let $\mathcal F$ be the set of all continuous functions $f:[0,1]\to\mathbb R$ with the property $$\left|\int^x_0\frac{f(t)}{\sqrt{x-t}}\text dt\right|\le1\enspace\text{for all }x\in(0,1].$$Compute $\sup_{f\in\mathcal F}\left|\int^1_0f(x)\text dx\right|$.

1985 IMO Longlists, 63

Tags: algebra , inequalities , sequence , inequality , calculus , recurrence relation

Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]