Olympiad problems

2013 Today's Calculation Of Integral, 875

Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$

1975 Miklós Schweitzer, 7

Tags: calculus , real analysis , analytic geometry , function , derivative , graphing lines , slope

Let $ a<a'<b<b'$ be real numbers and let the real function $ f$ be continuous on the interval $ [a,b']$ and differentiable in its interior. Prove that there exist $ c \in (a,b), c'\in (a',b')$ such that \[ f(b)\minus{}f(a)\equal{}f'(c)(b\minus{}a),\] \[ f(b')\minus{}f(a')\equal{}f'(c')(b'\minus{}a'),\] and $ c<c'$. [i]B. Szokefalvi Nagy[/i]

2012 Turkey Team Selection Test, 1

Tags: function , algebra , integration , calculus , limit

Let $S_r(n)=1^r+2^r+\cdots+n^r$ where $n$ is a positive integer and $r$ is a rational number. If $S_a(n)=(S_b(n))^c$ for all positive integers $n$ where $a, b$ are positive rationals and $c$ is positive integer then we call $(a,b,c)$ as [i]nice triple.[/i] Find all nice triples.

1983 Putnam, A2

Tags: calculus , analysis

The shorthand of a clock has the length 3, the longhand has the length 4. Determine the distance between the endpoints of the hands at the time, where their distance increases the most.

1978 AMC 12/AHSME, 11

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

If $r$ is positive and the line whose equation is $x + y = r$ is tangen to the circle whose equation is $x^2 + y ^2 = r$, then $r$ equals $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{2}\qquad \textbf{(E) }2\sqrt{2}$

2021 JHMT HS, 5

Tags: geometry , optimization , calculus

For real numbers $x,$ let $T_x$ be the triangle with vertices $(5, 5^3),$ $(8, 8^3),$ and $(x, x^3)$ in $\mathbb{R}^2.$ Over all $x$ in the interval $[5, 8],$ the area of the triangle $T_x$ is maximized at $x = \sqrt{n},$ for some positive integer $n.$ Compute $n.$

2012 Today's Calculation Of Integral, 838

Tags: calculus , integration , calculus computations , inequalities

Prove that : $\frac{e-1}{e}<\int_0^1 e^{-x^2}dx<\frac{\pi}{4}.$

2024 CMI B.Sc. Entrance Exam, 1

Tags: geometry , integral calculus , calculus

(a) Sketch qualitativly the region with maximum area such that it lies in the first quadrant and is bound by $y=x^2-x^3$ and $y=kx$ where $k$ is a constent. The region must not have any other lines closing it. Note: $kx$ lies above $x^2-x^3$ (b) Find an expression for the volume of the solid obtained by spinning this region about the $y$ axis.

2012 Today's Calculation Of Integral, 781

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

Let $l,\ m$ be the tangent lines passing through the point $A(a,\ a-1)$ on the line $y=x-1$ and touch the parabola $y=x^2$. Note that the slope of $l$ is greater than that of $m$. (1) Exress the slope of $l$ in terms of $a$. (2) Denote $P,\ Q$ be the points of tangency of the lines $l,\ m$ and the parabola $y=x^2$. Find the minimum area of the part bounded by the line segment $PQ$ and the parabola $y=x^2$. (3) Find the minimum distance between the parabola $y=x^2$ and the line $y=x-1$.

2008 Estonia Team Selection Test, 3

Tags: function , calculus , inequalities , algebra

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

2009 Today's Calculation Of Integral, 436

Tags: calculus , calculus computations , integration , geometry

Find the minimum area bounded by the graphs of $ y\equal{}x^2$ and $ y\equal{}kx(x^2\minus{}k)\ (k>0)$.

2006 Harvard-MIT Mathematics Tournament, 2

Tags: calculus , trigonometry , limit

Compute $\displaystyle\lim_{x\to 0}\dfrac{e^{x\cos x}-1-x}{\sin(x^2)}.$

2007 Today's Calculation Of Integral, 197

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

Let $|a|<\frac{\pi}{2}.$ Evaluate the following definite integral. \[\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}\]

2004 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , geometry

A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at $1$ meter per second. Suddenly someone turns on the turntable; it spins at $30$ rpm. Consider the set $S$ of points the mouse can reach in his car within $1$ second after the turntable is set in motion. What is the area of $S$, in square meters?

2008 Mathcenter Contest, 6

Tags: function , functional equation , algebra , calculus

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[ f(x^2+y^2+2f(xy)) = (f(x+y))^2. \] for all $x,y \in \mathbb{R}$.

2012 Today's Calculation Of Integral, 779

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

Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane. When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.

2008 Harvard-MIT Mathematics Tournament, 6

Tags: calculus , limit , calculus computations , ratio

Determine the value of $ \lim_{n\rightarrow\infty}\sum_{k \equal{} 0}^n\binom{n}{k}^{ \minus{} 1}$.

2013 Today's Calculation Of Integral, 867

Tags: calculus , derivative , integration , function , quadratic , linear equation , algebra

Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

2004 Romania National Olympiad, 1

Tags: functional equation , calculus , derivative , function , integration , limit , algebra

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have \[ n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = n f(x) + \frac12 . \] [i]Mihai Piticari[/i]

1980 IMO, 8

Tags: number theory , calculus , integration

Prove that if $(a,b,c,d)$ are positive integers such that $(a+2^{\frac13}b+2^{\frac23}c)^2=d$ then $d$ is a perfect square (i.e is the square of a positive integer).

2018 ISI Entrance Examination, 3

Tags: function , calculus , continuous

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that for all $x\in\mathbb{R}$ and for all $t\geqslant 0$, $$f(x)=f(e^tx)$$ Show that $f$ is a constant function.

2009 Harvard-MIT Mathematics Tournament, 7

Tags: calculus , geometry

A line in the plane is called [i]strange[/i] if it passes through $(a,0)$ and $(0,10-a)$ for some $a$ in the interval $[0,10]$. A point in the plane is called [i]charming[/i] if it lies in the first quadrant and also lies [b]below[/b] some strange line. What is the area of the set of all charming points?

2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 3

Tags: calculus , trigonometry , integration , real analysis

Let $a$ be a positive real number. Evaluate $I=\int_0^{+\infty} \frac{\sin x\cos x}{x(x^2+a^2)}dx.$

1995 Flanders Math Olympiad, 2

Tags: function , calculus , algebra , quadratic , integration

How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$?

1990 Flanders Math Olympiad, 4

Tags: function , calculus , limit , calculus computations

Let $f:\mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0$ be a strictly decreasing function. (a) Be $a_n$ a sequence of strictly positive reals so that $\forall k \in \mathbb{N}_0:k\cdot f(a_k)\geq (k+1)\cdot f(a_{k+1})$ Prove that $a_n$ is ascending, that $\displaystyle\lim_{k\rightarrow +\infty} f(a_k)$ = 0and that $\displaystyle\lim_{k\rightarrow +\infty} a_k =+\infty$ (b) Prove that there exist such a sequence ($a_n$) in $\mathbb{R}^+_0$ if you know $\displaystyle\lim_{x\rightarrow +\infty} f(x)=0$.