Olympiad problems

2009 Today's Calculation Of Integral, 511

Suppose that $ f(x),\ g(x)$ are differential fuctions and their derivatives are continuous. Find $ f(x),\ g(x)$ such that $ f(x)\equal{}\frac 12\minus{}\int_0^x \{f'(t)\plus{}g(t)\}\ dt\ \ g(x)\equal{}\sin x\minus{}\int_0^{\pi} \{f(t)\minus{}g'(t)\}\ dt$.

2003 Costa Rica - Final Round, 3

Tags: calculus , function , inequalities , induction

If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds.

2005 Czech-Polish-Slovak Match, 1

Tags: function , derivative , analytic geometry , algebra , graphing lines , calculus , polynomial

Let $n$ be a given positive integer. Solve the system \[x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,\] \[x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}\] in the set of nonnegative real numbers.

2008 Harvard-MIT Mathematics Tournament, 7

Tags: function , ratio , quadratic , floor function , geometric series , integration , calculus

Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.

2011 Laurențiu Duican, 2

Tags: squeeze theorem , definite integral , limit , integration , calculus , real analysis

$ \lim_{n\to\infty } \int_{\pi }^{2\pi } \frac{|\sin (nx) +\cos (nx)|}{ x} dx ? $ [i]Gabriela Boeriu[/i]

2005 South East Mathematical Olympiad, 1

Tags: derivative , parabola , conic , calculus , function , algebra

Let $a \in \mathbb{R}$ be a parameter. (1) Prove that the curves of $y = x^2 + (a + 2)x - 2a + 1$ pass through a fixed point; also, the vertices of these parabolas all lie on the curve of a certain parabola. (2) If the function $x^2 + (a + 2)x - 2a + 1 = 0$ has two distinct real roots, find the value range of the larger root.

1998 Vietnam Team Selection Test, 1

Tags: polynomial , algebra , rational root theorem , calculus , integration , number theory

Find all integer polynomials $P(x)$, the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.

1994 Vietnam Team Selection Test, 2

Tags: integration , number theory , calculus

Consider the equation \[x^2 + y^2 + z^2 + t^2 - N \cdot x \cdot y \cdot z \cdot t - N = 0\] where $N$ is a given positive integer. a) Prove that for an infinite number of values of $N$, this equation has positive integral solutions (each such solution consists of four positive integers $x, y, z, t$), b) Let $N = 4 \cdot k \cdot (8 \cdot m + 7)$ where $k,m$ are no-negative integers. Prove that the considered equation has no positive integral solutions.

2000 India Regional Mathematical Olympiad, 3

Tags: inequalities , calculus , integration , induction

Suppose $\{ x_n \}_{n\geq 1}$ is a sequence of positive real numbers such that $x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots$, and for all $n$ \[ \frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 . \] Show that for all $k$ \[ \frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3. \]

Today's calculation of integrals, 862

Tags: parabola , calculus , graphing lines , conic , analytic geometry , integration , slope

Draw a tangent with positive slope to a parabola $y=x^2+1$. Find the $x$-coordinate such that the area of the figure bounded by the parabola, the tangent and the coordinate axisis is $\frac{11}{3}.$

2022 CMIMC Integration Bee, 8

Tags: integration , calculus

\[\int_{-\infty}^{0} \frac{1}{e^{-x}+2e^{x}+e^{3x}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

1950 Miklós Schweitzer, 10

Tags: derivative , advanced fields , calculus

Consider an arc of a planar curve such that the total curvature of the arc is less than $ \pi$. Suppose, further, that the curvature and its derivative with respect to the arc length exist at every point of the arc and the latter nowhere equals zero. Let the osculating circles belonging to the endpoints of the arc and one of these points be given. Determine the possible positions of the other endpoint.

2015 Vietnam National Olympiad, 2

Tags: derivative , inequalities , calculus , induction

If $a,b,c$ are nonnegative real numbers, then \[{ 3(a^2+b^2+c^2) \geq (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2 \geq (a+b+c)^2.}\]

2004 IMC, 6

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

For every complex number $z$ different from 0 and 1 we define the following function \[ f(z) := \sum \frac 1 { \log^4 z } \] where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$. b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have \[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]

2009 Today's Calculation Of Integral, 453

Tags: calculus computations , trigonometry , calculus , integration

Find the minimum value of $ \int_0^{\frac{\pi}{2}} |x\sin t\minus{}\cos t|\ dt\ (x>0).$

2007 Today's Calculation Of Integral, 216

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

Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n$. Find $ \lim_{n\to\infty}(a_{n}\minus{}\ln n)$.

Today's calculation of integrals, 874

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

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

2009 Princeton University Math Competition, 2

Tags: calculus , ceiling function , logarithm , induction , floor function , integration

It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?

2025 ISI Entrance UGB, 1

Tags: calculus , fixed point

Suppose $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and $| f'(x)| < \frac{1}{2}$ for all $x \in \mathbb{R}$. Show that for some $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.

2006 Petru Moroșan-Trident, 3

Tags: primitive , indefinite integral , calculus

Determine the primitives of: [b]1)[/b] $ (0,\pi /2)\ni x\mapsto\frac{x^2}{-x+\tan x} $ [b]2)[/b] $ 1<x\mapsto \frac{-1+\ln x}{x^2-\ln^2 x} $ [i]Ion Nedelcu[/i]

2011 Today's Calculation Of Integral, 762

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

Define a function $f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)$ by \[f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.\] (1) Let $f_n(x)=a_n\sin x+b_n\cos x.$ Express $a_{n+1},\ b_{n+1}$ in terms of $a_n,\ b_n.$ (2) Find $\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).$

2009 Today's Calculation Of Integral, 399

Tags: logarithm , calculus , integration , calculus computations

Evaluate $ \int_0^{\sqrt{2}\minus{}1} \frac{1\plus{}x^2}{1\minus{}x^2}\ln \left(\frac{1\plus{}x}{1\minus{}x}\right)\ dx$.

2000 Romania National Olympiad, 1

Tags: calculus , continuity , integral , limit , real analysis , integration

Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ [b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. [b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $

2006 Harvard-MIT Mathematics Tournament, 4

Tags: calculus

Compute $\displaystyle\sum_{k=1}^\infty \dfrac{k^4}{k!}$.

1982 Dutch Mathematical Olympiad, 4

Tags: integration , greatest common divisor , number theory , calculus

Determine $ \gcd (n^2\plus{}2,n^3\plus{}1)$ for $ n\equal{}9^{753}$.