Olympiad problems

2008 IMS, 3

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

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$.

2020 Jozsef Wildt International Math Competition, W2

Tags: limit , real analysis , calculus

Let $\left(a_n\right)_{n\geq1}$ be a sequence of nonnegative real numbers which converges to $a \in \mathbb{R}$. [list=1] [*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+a_nx^n \right)^ndx}$$ [*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+\frac{a_1x+a_3x^3+\cdots+a_{2n-1}x^{2n-1}}{n} \right)^ndx}$$ [/list]

2024 CMIMC Integration Bee, 4

Tags: calculus , integration

\[\int_0^1 (x^6+6x^5+15x^4+15x^2+6x+1)\mathrm dx\] [i]Proposed by Robert Trosten[/i]

2022 JHMT HS, 1

Tags: derivative , integration , calculus

Compute the value of \[ \frac{d}{dx}\int_{1}^{10} x^3\,dx. \]

2013 Today's Calculation Of Integral, 890

Tags: algorithm , calculus , integration , function , calculus computations

A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

2009 Today's Calculation Of Integral, 510

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

(1) Evaluate $ \int_0^{\frac{\pi}{2}} (x\cos x\plus{}\sin ^ 2 x)\sin x\ dx$. (2) For $ f(x)\equal{}\int_0^x e^t\sin (x\minus{}t)\ dt$, find $ f''(x)\plus{}f(x)$.

2004 Harvard-MIT Mathematics Tournament, 8

Tags: graphing lines , geometry , rotation , function , analytic geometry , geometric transformation , calculus

If $x$ and $y$ are real numbers with $(x+y)^4=x-y$, what is the maximum possible value of $y$?

1976 Miklós Schweitzer, 4

Tags: algebra , calculus , function , domain , ring theory , integration , polynomial

Let $ \mathbb{Z}$ be the ring of rational integers. Construct an integral domain $ I$ satisfying the following conditions: a)$ \mathbb{Z} \varsubsetneqq I$; b) no element of $ I \minus{} \mathbb{Z}$ (only in $ I$) is algebraic over $ \mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $ \mathbb{Z}$); c) $ I$ only has trivial endomorphisms. [i]E. Fried[/i]

2003 Alexandru Myller, 2

Tags: calculus , trigonometric integral , integration , trigonometry

Calculate $ \int_0^{2\pi }\prod_{i=1}^{2002} cos^i (it) dt. $ [i]Dorin Andrica[/i]

2005 Today's Calculation Of Integral, 21

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

[1] Tokyo Univ. of Science: $\int \frac{\ln x}{(x+1)^2}dx$ [2] Saitama Univ.: $\int \frac{5}{3\sin x+4\cos x}dx$ [3] Yokohama City Univ.: $\int_1^{\sqrt{3}} \frac{1}{\sqrt{x^2+1}}dx$ [4] Daido Institute of Technology: $\int_0^{\frac{\pi}{2}} \frac{\sin ^ 3 x}{\sin x +\cos x}dx$ [5] Gunma Univ.: $\int_0^{\frac{3\pi}{4}} \{(1+x)\sin x+(1-x)\cos x\}dx$

2009 Today's Calculation Of Integral, 406

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

Find $ \lim_{n\to\infty} \int_0^{\frac{\pi}{2}} x|\cos (2n\plus{}1)x|\ dx$.

1977 USAMO, 5

Tags: calculus , convex-concave inequalities , function , inequalities

If $ a,b,c,d,e$ are positive numbers bounded by $ p$ and $ q$, i.e, if they lie in $ [p,q], 0 < p$, prove that \[ (a \plus{} b \plus{} c \plus{} d \plus{} e)\left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \plus{} \frac {1}{d} \plus{} \frac {1}{e}\right) \le 25 \plus{} 6\left(\sqrt {\frac {p}{q}} \minus{} \sqrt {\frac {q}{p}}\right)^2\] and determine when there is equality.

2012 Online Math Open Problems, 40

Tags: calculus , algebra , derivative , trigonometry , circumcircle , geometry

Suppose $x,y,z$, and $w$ are positive reals such that \[ x^2 + y^2 - \frac{xy}{2} = w^2 + z^2 + \frac{wz}{2} = 36 \] \[ xz + yw = 30. \] Find the largest possible value of $(xy + wz)^2$. [i]Author: Alex Zhu[/i]

2007 Moldova National Olympiad, 12.8

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

Find all continuous functions $f\colon [0;1] \to R$ such that \[\int_{0}^{1}f(x)dx = 2\int_{0}^{1}(f(x^{4}))^{2}dx+\frac{2}{7}\]

2012 Hanoi Open Mathematics Competitions, 7

Tags: calculus , integration

[b]Q7.[/b] Find all integers $n$ such that $60+2n-n^2$ is a perfect square.

2005 Putnam, B4

Tags: calculus , function , integration , induction

For positive integers $ m$ and $ n$, let $ f\left(m,n\right)$ denote the number of $ n$-tuples $ \left(x_1,x_2,\dots,x_n\right)$ of integers such that $ \left|x_1\right| \plus{} \left|x_2\right| \plus{} \cdots \plus{} \left|x_n\right|\le m$. Show that $ f\left(m,n\right) \equal{} f\left(n,m\right)$.

2007 ISI B.Stat Entrance Exam, 7

Tags: trigonometry , prism , 3d geometry , calculus computations , calculus , geometry

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and find the height of this largest prism.

2011 Today's Calculation Of Integral, 764

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

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

2023 CMIMC Integration Bee, 5

Tags: integration , calculus

\[\int_1^\infty \frac{1}{x\sqrt{x^{2023}-1}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2010 Today's Calculation Of Integral, 540

Tags: calculus , calculus computations , logarithm , integration

Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx$.

2021 The Chinese Mathematics Competition, Problem 2

Tags: calculus

Let $z=z(x,y)$ be implicit function with two variables from $2sin(x+2y-3z)=x+2y-3z$. Find $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$.

2005 Today's Calculation Of Integral, 35

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

Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.

1998 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , graphing lines , slope , derivative , analytic geometry

Find the slopes of all lines passing through the origin and tangent to the curve $y^2=x^3+39x-35$.

2010 Putnam, A3

Tags: calculus , probability , derivative , function , complex analysis , vector

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

2013 ELMO Problems, 2

Tags: function , inequalities , calculus , derivative , 3-variable inequality , logarithm

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]