Olympiad problems

2006 Putnam, A6

Tags: probability , calculus , integration , trigonometry , geometry

Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral.

2006 South africa National Olympiad, 2

Tags: calculus , derivative , trigonometry , geometry

Triangle $ABC$ has $BC=1$ and $AC=2$. What is the maximum possible value of $\hat{A}$.

1967 Miklós Schweitzer, 9

Tags: calculus , derivative , symmetry , advanced fields

Let $ F$ be a surface of nonzero curvature that can be represented around one of its points $ P$ by a power series and is symmetric around the normal planes parallel to the principal directions at $ P$. Show that the derivative with respect to the arc length of the curvature of an arbitrary normal section at $ P$ vanishes at $ P$. Is it possible to replace the above symmetry condition by a weaker one? [i]A. Moor[/i]

2001 China Team Selection Test, 3

Tags: algebra , function , calculus , polynomial , minimization

Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.

2009 Today's Calculation Of Integral, 450

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

Let $ a,\ b$ be postive real numbers. Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{n}{(k\plus{}an)(k\plus{}bn)}.$

1999 USAMTS Problems, 4

Tags: analytic geometry , calculus , integration , geometry , perimeter

We say a triangle in the coordinate plane is [i]integral[/i] if its three vertices have integer coordinates and if its three sides have integer lengths. (a) Find an integral triangle with perimeter of $42$. (b) Is there an integral triangle with perimeter of $43$?

2012 Today's Calculation Of Integral, 852

Tags: calculus , integration , trigonometry , algebra , polynomial , absolute value , calculus computations

Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows. (1) $g_n(x)=(1+x)^n$ (2) $g_n(x)=\sin n\pi x$ (3) $g_n(x)=e^{nx}$

2021 Nigerian Senior MO Round 3, 5

Tags: algebra , calculus , derivative

Let $f(x)=\frac{P(x)}{Q(x)}$. Where $P(x), Q(x)$ are two non constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})$ for all infinitely many values of $x$. a. Show that $deg(P) <deg(Q).$ b. Show that $P'(1)=2Q'(1)- deg(Q). Q(1)$ Here $P'(x)$ denotes the derivatives of $P(x)$ as usual

2005 Today's Calculation Of Integral, 12

Tags: calculus , integration , trigonometry , calculus computations

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

2014 Contests, 2

Tags: calculus , integration , arithmetic sequence , algebra

The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.

2022 Brazil Undergrad MO, 1

Tags: function , functional analysis , calculus

Let $0<a<1$. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ continuous at $x = 0$ such that $f(x) + f(ax) = x,\, \forall x \in \mathbb{R}$

PEN K Problems, 26

Tags: functional equation , function , calculus , integration , induction

The function $f: \mathbb{N}\to\mathbb{N}_{0}$ satisfies for all $m,n\in\mathbb{N}$: \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine $f(1982)$.

1992 IMO Longlists, 74

Tags: pigeonhole principle , algebra , calculus , limit , logarithm

Let $S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.$ Show that every real number $x \geq 0$ is an accumulation point of $S.$

1983 IMO Longlists, 12

Tags: geometry , geometric transformation , rotation , calculus , function , totient function , number theory

The number $0$ or $1$ is to be assigned to each of the $n$ vertices of a regular polygon. In how many different ways can this be done (if we consider two assignments that can be obtained one from the other through rotation in the plane of the polygon to be identical)?

2009 Vietnam National Olympiad, 4

Tags: algebra , polynomial , induction , calculus , integration , function , logarithm

Let $ a$, $ b$, $ c$ be three real numbers. For each positive integer number $ n$, $ a^n \plus{} b^n \plus{} c^n$ is an integer number. Prove that there exist three integers $ p$, $ q$, $ r$ such that $ a$, $ b$, $ c$ are the roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} r \equal{} 0$.

2013 Putnam, 4

Tags: function , integration , inequalities , calculus

For any continuous real-valued function $f$ defined on the interval $[0,1],$ let \[\mu(f)=\int_0^1f(x)\,dx,\text{Var}(f)=\int_0^1(f(x)-\mu(f))^2\,dx, M(f)=\max_{0\le x\le 1}|f(x)|.\] Show that if $f$ and $g$ are continuous real-valued functions defined on the interval $[0,1],$ then \[\text{Var}(fg)\le 2\text{Var}(f)M(g)^2+2\text{Var}(g)M(f)^2.\]

2005 Today's Calculation Of Integral, 7

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

Calculate the following indefinite integrals. [1] $\int \sqrt{x}(\sqrt{x}+1)^2 dx$ [2] $\int (e^x+2e^{x+1}-3e^{x+2})dx$ [3] $\int (\sin ^2 x+\cos x)\sin x dx$ [4] $\int x\sqrt{2-x} dx$ [5] $\int x\ln x dx$

2009 Harvard-MIT Mathematics Tournament, 2

Tags: calculus , trigonometry , function , integration

The differentiable function $F:\mathbb{R}\to\mathbb{R}$ satisfies $F(0)=-1$ and \[\dfrac{d}{dx}F(x)=\sin (\sin (\sin (\sin(x))))\cdot \cos( \sin (\sin (x))) \cdot \cos (\sin(x))\cdot\cos(x).\] Find $F(x)$ as a function of $x$.

1999 Tuymaada Olympiad, 2

Tags: algebra , polynomial , function , calculus , derivative

Can the graphs of a polynomial of degree 20 and the function $\displaystyle y={1\over x^{40}}$ have exactly 30 points of intersection? [i]Proposed by K. Kokhas[/i]

2011 Harvard-MIT Mathematics Tournament, 1

Tags: hmmt , calculus , integration , angle bisector , geometry

Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.

2022 CMIMC Integration Bee, 14

Tags: calculus , integration

\[\int_2^\infty \frac{\pi(x)}{x^3 - x}\,dx\] [i]Proposed by Vlad Oleksenko[/i]

2005 Bundeswettbewerb Mathematik, 2

Tags: calculus , number theory

Let $a$ be such an integer, that $3a$ can be written in the form $x^2 + 2y^2$, with integers $x$ and $y$. Prove that the number $a$ can also be written in this form. [b]Additional problems:[/b] [b]a)[/b] Find a general (necessary and sufficent) criterion for an integer $n$ to be of that form. [b]b)[/b] In how many ways can the integer $n$ be represented in that way?

1992 Brazil National Olympiad, 1

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

The equation $x^3+px+q=0$ has three distinct real roots. Show that $p<0$

2013 Today's Calculation Of Integral, 860

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

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

1969 Miklós Schweitzer, 5

Tags: function , integration , calculus , real analysis

Find all continuous real functions $ f,g$ and $ h$ defined on the set of positive real numbers and satisfying the relation \[ f(x\plus{}y)\plus{}g(xy)\equal{}h(x)\plus{}h(y)\] for all $ x>0$ and $ y>0$. [i]Z. Daroczy[/i]