Olympiad problems

2005 Today's Calculation Of Integral, 60

Let $a_n=\int_0^{\frac{\pi}{2}} \sin 2t\ (1-\sin t)^{\frac{n-1}{2}}dt\ (n=1,2,\cdots)$ Evaluate \[\sum_{n=1}^{\infty} (n+1)(a_n-a_{n+1})\]

2010 Today's Calculation Of Integral, 584

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

Find $ \lim_{x\rightarrow \infty} \left(\int_0^x \sqrt{1\plus{}e^{2t}}\ dt\minus{}e^x\right)$.

2008 China Western Mathematical Olympiad, 4

Tags: function , trigonometry , geometry , analytic geometry , vector , derivative , calculus

Let P be an interior point of a regular n-gon $ A_1 A_2 ...A_n$, the lines $ A_i P$ meet the regular n-gon at another point $ B_i$, where $ i\equal{}1,2,...,n$. Prove that sums of all $ PA_i\geq$ sum of all $ PB_i$.

2023 CMIMC Integration Bee, 12

Tags: integration , calculus

\[\lim_{n\to\infty} n^2 \int_0^1 x^n e^{-x}\log(x)\,\mathrm dx\] [i]Proposed by Connor Gordon and Vlad Oleksenko[/i]

2007 Today's Calculation Of Integral, 224

Tags: calculus , integration , trigonometry , calculus computations

Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.

1967 IMO Longlists, 28

Tags: trigonometry , calculus , parameterization , trigonometric identities , algebra

Find values of the parameter $u$ for which the expression \[y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}\] does not depend on $x.$

2003 China Team Selection Test, 1

Tags: trigonometry , calculus , geometry

Let $ ABCD$ be a quadrilateral which has an incircle centered at $ O$. Prove that \[ OA\cdot OC\plus{}OB\cdot OD\equal{}\sqrt{AB\cdot BC\cdot CD\cdot DA}\]

MIPT Undergraduate Contest 2019, 1.5 & 2.5

Tags: trigonometry , calculus , inequalities , real analysis

Prove the inequality $$\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2$$ for any sequence of real numbers $x_0, x_1, ..., x_n$ for which $x_0 = x_n = 0.$

2024 ISI Entrance UGB, P5

Tags: limit , lhopital , polynomial limit , calculus

Let $P(x)$ be a polynomial with real coefficients. Let $\alpha_1 , \dots , \alpha_k$ be the distinct real roots of $P(x)=0$. If $P'$ is the derivative of $P$, show that for each $i=1,\dots , k$ \[\lim_{x\to \alpha_i} \frac{(x-\alpha_i)P'(x)}{P(x)} = r_i, \] for some positive integer $r_i$.

2003 SNSB Admission, 5

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

Let be an holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ having the property that $ |f(z)|\le e^{|\text{Im}(z)|} , $ for all complex numbers $ z. $ Prove that the restriction of any of its derivatives (of any order) to the real numbers is everywhere dominated by $ 1. $

2018 Brazil Undergrad MO, 19

Tags: calculus , complex number

What is the largest amount of complex $ z $ solutions a system can have? $ | z-1 || z + 1 | = 1 $ $ Im (z) = b? $ (where $ b $ is a real constant)

2000 Romania National Olympiad, 2

Tags: real analysis , integration , integral , calculus , derivative

For any partition $ P $ of $ [0,1] $ , consider the set $$ \mathcal{A}(P)=\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f’\bigg|_{[0,1]}\right.\wedge\int_0^1 |f(x)|dx =1\wedge \left( y\in P\implies f (y ) =0\right)\right\} . $$ Prove that there exists a partition $ P_0 $ of $ [0,1] $ such that $$ g\in \mathcal{A}\left( P_0\right)\implies \sup_{x\in [0,1]} \big| g’(x)\big| >4\cdot \# P. $$ Here, $ \# D $ denotes the natural number $ d $ such that $ 0=x_0<x_1<\cdots <x_d=1 $ is a partition $ D $ of $ [0,1] . $

2011 Today's Calculation Of Integral, 680

Tags: integration , calculus , calculus computations

Let $a>0$. Evaluate $\int_0^a x^2\left(1-\frac{x}{a}\right)^adx$. [i]2011 Keio University entrance exam/Science and Technology[/i]

2023 CMIMC Integration Bee, 15

Tags: calculus , integration

\[\int_0^\infty \left(1-e^{-\pi/x^2}\right)^2\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

ICMC 6, 2

Tags: calculus , function , derivative

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f'(x) > f(x)>0$ for all real numbers $x$. Show that $f(8) > 2022f(0)$. [i]Proposed by Ethan Tan[/i]

2002 Iran Team Selection Test, 9

Tags: function , number theory , integration , calculus , inequalities

$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?

2006 VTRMC, Problem 4

Tags: calculus , differential equation

We want to find functions $p(t)$, $q(t)$, $f(t)$ such that (a) $p$ and $q$ are continuous functions on the open interval $(0,\pi)$. (b) $f$ is an infinitely differentiable nonzero function on the whole real line $(-\infty,\infty)$ such that $f(0)=f'(0)=f''(0)$. (c) $y=\sin t$ and $y=f(t)$ are solutions of the differential equation $y''+p(t)y'+q(t)y=0$ on $(0,\pi)$. Is this possible? Either prove this is not possible, or show this is possible by providing an explicit example of such $f,p,q$.

2025 Bulgarian Spring Mathematical Competition, 12.1

Tags: derivative , inequalities , calculus , algebra

In terms of the real numbers $a$ and $b$ determine the minimum value of $$ \sqrt{(x+a)^2+1}+\sqrt{(x+1-a)^2+1}+\sqrt{(x+b)^2+1}+\sqrt{(x+1-b)^2+1}$$ as well as all values of $x$ which attain it.

2012 IFYM, Sozopol, 4

Tags: number theory , calculus , integration , polynomial , gauss , prime number , algebra

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

2020 LIMIT Category 2, 19

Tags: calculus , probability , limti

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 $1.2$

2006 All-Russian Olympiad, 1

Tags: calculus , inequalities , derivative , function , trigonometry , real analysis

Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0<x<\frac{\pi}{2}$.

2007 Today's Calculation Of Integral, 256

Tags: calculus , calculus computations , integration , trigonometry

Find the value of $ a$ for which $ \int_0^{\pi} \{ax(\pi ^ 2 \minus{} x^2) \minus{} \sin x\}^2dx$ is minimized.

2010 Today's Calculation Of Integral, 524

Tags: function , calculus computations , calculus , integration

Evaluate the following definite integral. \[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]

2014 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , function , derivative , logarithm

Fix a positive real number $c>1$ and positive integer $n$. Initially, a blackboard contains the numbers $1,c,\ldots, c^{n-1}$. Every minute, Bob chooses two numbers $a,b$ on the board and replaces them with $ca+c^2b$. Prove that after $n-1$ minutes, the blackboard contains a single number no less than \[\left(\dfrac{c^{n/L}-1}{c^{1/L}-1}\right)^L,\] where $\phi=\tfrac{1+\sqrt 5}2$ and $L=1+\log_\phi(c)$.

2011 Today's Calculation Of Integral, 743

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

Evaluate $\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.$