Olympiad problems

2023 ISI Entrance UGB, 2

Let $a_0 = \frac{1}{2}$ and $a_n$ be defined inductively by \[a_n = \sqrt{\frac{1+a_{n-1}}{2}} \text{, $n \ge 1$.} \] [list=a] [*] Show that for $n = 0,1,2, \ldots,$ \[a_n = \cos(\theta_n) \text{ for some $0 < \theta_n < \frac{\pi}{2}$, }\] and determine $\theta_n$. [*] Using (a) or otherwise, calculate \[ \lim_{n \to \infty} 4^n (1 - a_n).\] [/list]

2009 Today's Calculation Of Integral, 516

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

Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$. (1) Find the local minimum value of $ f(x)$. (2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$.

2007 Today's Calculation Of Integral, 254

Tags: calculus , integration , calculus computations , logarithm

Evaluate $ \int_e^{e^2} \frac {(\ln x)^7\minus{}7!}{(\ln x)^8}\ dx.$ Sorry, I have deleted my first post because that was wrong. kunny

2011 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , integration

Find all integers $x$ such that $2x^2+x-6$ is a positive integral power of a prime positive integer.

2010 Today's Calculation Of Integral, 591

Tags: calculus , integration , inequalities , calculus computations

Let $ a,\ b,\ c$ be real numbers such that $ a\geq b\geq c\geq 1$. Prove the following inequality: \[ \int_0^1 \{(1\minus{}ax)^3\plus{}(1\minus{}bx)^3\plus{}(1\minus{}cx)^3\minus{}3x\}\ dx\geq ab\plus{}bc\plus{}ca\minus{}\frac 32(a\plus{}b\plus{}c)\minus{}\frac 34abc.\]

2005 Today's Calculation Of Integral, 17

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

Calculate the following indefinite integrals. [1] $\int \frac{dx}{e^x-e^{-x}}$ [2] $\int e^{-ax}\cos 2x dx\ (a\neq 0)$ [3] $\int (3^x-2)^2 dx$ [4] $\int \frac{x^4+2x^3+3x^2+1}{(x+2)^5}dx$ [5] $\int \frac{dx}{1-\cos x}dx$

2005 Today's Calculation Of Integral, 27

Tags: calculus , integration , trigonometry , quadratic , algebra , calculus computations

Let $f(x)=t\sin x+(1-t)\cos x\ (0\leqq t\leqq 1)$. Find the maximum and minimum value of the following $P(t)$. \[P(t)=\left\{\int_0^{\frac{\pi}{2}} e^x f(x) dx \right\}\left\{\int_0^{\frac{\pi}{2}} e^{-x} f(x)dx \right\}\]

1999 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , integration , trigonometry , function

Find \[\int_{-4\pi\sqrt{2}}^{4\pi\sqrt{2}}\left(\dfrac{\sin x}{1+x^4}+1\right)dx.\]

2008 IMS, 3

Tags: geometry , parabola , ratio , function , calculus , derivative , conic

Let $ A,B$ be different points on a parabola. Prove that we can find $ P_1,P_2,\dots,P_{n}$ between $ A,B$ on the parabola such that area of the convex polygon $ AP_1P_2\dots P_nB$ is maximum. In this case prove that the ratio of $ S(AP_1P_2\dots P_nB)$ to the sector between $ A$ and $ B$ doesn't depend on $ A$ and $ B$, and only depends on $ n$.

2011 Gheorghe Vranceanu, 2

Tags: limit , squeezing , am-gm , algebra , caluclus , calculus

$ a>0,\quad\lim_{n\to\infty }\sum_{i=1}^n \frac{1}{n+a^i} $

2015 AMC 10, 25

Tags: probability , integration , geometry , trigonometry , calculus , rectangle , geometric probability

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

2012 Today's Calculation Of Integral, 810

Tags: calculus , integration , function , calculus computations

Given the functions $f(x)=xe^{x}+2x\int_0^2 |g(t)|dt-1,\ g(x)=x^2-x\int_0^1 f(t)dt$, evaluate $\int_0^2 |g(t)|dt.$

2020 LIMIT Category 2, 19

Tags: probability , limti , calculus

Consider an unbiased coin which is tossed infinitely many times. Let $A_n$ be the event that no two successive heads occur in the first $n$ tosses of this experiment. Then which of the following is incorrect : (A) $\lim_{n \to \infty} P(A_n)=0$ (B) $\lim_{n \to \infty}3^n P(A_n)=0$ (C) $2^nP(A_n) +2^{n+1}P(A_{n+1})=2^{n+2}P(A_{n+2}$ (D) $\lim_{n \to \infty} \frac{P(A_n)}{P(A_{n+1})}$ is lesser than $1.2$

2012 Today's Calculation Of Integral, 835

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

Evaluate the following definite integrals. (a) $\int_1^2 \frac{x-1}{x^2-2x+2}\ dx$ (b) $\int_0^1 \frac{e^{4x}}{e^{2x}+2}\ dx$ (c) $\int_1^e x\ln \sqrt{x}\ dx$ (d) $\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx$

2005 Today's Calculation Of Integral, 22

Tags: calculus , integration , induction , trigonometry , calculus computations

Evaluate \[\int_0^1 (1-x^2)^n dx\ (n=0,1,2,\cdots)\]

2007 Today's Calculation Of Integral, 238

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

Find $ \lim_{a\to\infty} \frac {1}{a^2}\int_0^a \log (1 \plus{} e^x)\ dx.$

1991 Arnold's Trivium, 18

Tags: integration , linear algebra , matrix , quadratic , gauss , calculus

Calculate \[\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n\]

2010 Today's Calculation Of Integral, 586

Tags: calculus , integration , algebra , polynomial , trigonometry , limit , calculus computations

Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.

1979 IMO Longlists, 72

Tags: polynomial , calculus , integration , algebra

Let $f (x)$ be a polynomial with integer coefficients. Prove that if $f (x)= 1979$ for four different integer values of $x$, then $f (x)$ cannot be equal to $2\times 1979$ for any integral value of $x$.

2009 Stanford Mathematics Tournament, 5

Tags: calculus , derivative

Find the minimum possible value of $2x^2+2xy+4y+5y^2-x$ for real numbers $x$ and $y$.

2010 Contests, 522

Tags: calculus , integration , limit , function , geometry , derivative , logarithm

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

2006 Harvard-MIT Mathematics Tournament, 1

Tags: calculus , derivative , algebra , polynomial

A nonzero polynomial $f(x)$ with real coefficients has the property that $f(x)=f^\prime(x)f^{\prime\prime}(x)$. What is the leading coefficient of $f(x)$?

1990 National High School Mathematics League, 10

Tags: calculus , integration

Define $f(n):$ the number of integral points of line segment $OA_n$ ($O$ and $A_n$ not included), where $A_n(n,n+3)$. Then, $f(1)+f(2)+\cdots+f(1990)=$________.

2017 District Olympiad, 1

Tags: function , integral , inequalities , calculus

Let $ f,g:[0,1]\longrightarrow{R} $ be two continuous functions such that $ f(x)g(x)\ge 4x^2, $ for all $ x\in [0,1] . $ Prove that $$ \left| \int_0^1 f(x)dx \right| \ge 1\text{ or } \left| \int_0^1 g(x)dx \right| \ge 1. $$

2010 Today's Calculation Of Integral, 588

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^{\frac{\pi}{2}} e^{xe^x}\{(x\plus{}1)e^x(\cos x\plus{}\sin x)\plus{}\cos x\minus{}\sin x\}dx$.