Olympiad problems

2014 Online Math Open Problems, 25

Tags: derivative , calculus , function , factorial , induction , integration

If \[ \sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq \] for relatively prime positive integers $p,q$, find $p+q$. [i]Proposed by Michael Kural[/i]

2007 Today's Calculation Of Integral, 250

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

For a positive constant number $ p$, find $ \lim_{n\to\infty} \frac {1}{n^{p \plus{} 1}}\sum_{k \equal{} 0}^{n \minus{} 1} \int_{2k\pi}^{(2k \plus{} 1)\pi} x^p\sin ^ 3 x\cos ^ 2x\ dx.$

2006 Romania National Olympiad, 4

Tags: least common multiple , number theory , integration , relatively prime , greatest common divisor , calculus

Let $A$ be a set of positive integers with at least 2 elements. It is given that for any numbers $a>b$, $a,b \in A$ we have $\frac{ [a,b] }{ a- b } \in A$, where by $[a,b]$ we have denoted the least common multiple of $a$ and $b$. Prove that the set $A$ has [i]exactly[/i] two elements. [i]Marius Gherghu, Slatina[/i]

2012 Today's Calculation Of Integral, 855

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

Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$. For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

2021 CIIM, 6

Tags: function , calculus , integration , real analysis

Let $0 \le a < b$ be real numbers. Prove that there is no continuous function $f : [a, b] \to \mathbb{R}$ such that \[ \int_a^b f(x)x^{2n} \mathrm dx>0 \quad \text{and} \quad \int_a^b f(x)x^{2n+1} \mathrm dx <0 \] for every integer $n \ge 0$.

2005 Today's Calculation Of Integral, 42

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

Let $0<t<\frac{\pi}{2}$. Evaluate \[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]

2008 Putnam, B2

Tags: calculus , integration , limit , factorial , logarithm

Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$

1987 AIME Problems, 11

Tags: algebra , algorithm , integration , calculus , inequalities , quadratic

Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.

1994 China Team Selection Test, 2

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

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.

2019 Simon Marais Mathematical Competition, B1

Tags: integration , calculus

Determine all pairs $(a,b)$ of real numbers with $a\leqslant b$ that maximise the integral $$\int_a^b e^{\cos (x)}(380-x-x^2) \mathrm{d} x.$$

1996 IMC, 2

Tags: calculus , integration , real analysis

Evaluate the definite integral $$\int_{-\pi}^{\pi}\frac{\sin nx}{(1+2^{x})\sin x} dx,$$ where $n$ is a natural number.

2008 Romania National Olympiad, 1

Tags: function , calculus , integration , real analysis , derivative , limit

Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have \[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.

2019 Ramnicean Hope, 2

Tags: integration , definite integral , quadratic , calculus

Calculate $ \int_1^4 \frac{\ln x}{(1+x)(4+x)} dx . $ [i]Ovidiu Țâțan[/i]

2013 India Regional Mathematical Olympiad, 6

Tags: integration , arithmetic sequence , number theory , calculus

Let $n \ge 4$ be a natural number. Let $A_1A_2 \cdots A_n$ be a regular polygon and $X = \{ 1,2,3....,n \} $. A subset $\{ i_1, i_2,\cdots, i_k \} $ of $X$, with $k \ge 3$ and $i_1 < i_2 < \cdots < i_k$, is called a good subset if the angles of the polygon $A_{i_1}A_{i_2}\cdots A_{i_k}$ , when arranged in the increasing order, are in an arithmetic progression. If $n$ is a prime, show that a proper good subset of $X$ contains exactly four elements.

2012 Today's Calculation Of Integral, 820

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

Let $P_k$ be a point whose $x$-coordinate is $1+\frac{k}{n}\ (k=1,\ 2,\ \cdots,\ n)$ on the curve $y=\ln x$. For $A(1,\ 0)$, find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n} \overline{AP_k}^2.$

2007 Today's Calculation Of Integral, 207

Tags: logarithm , calculus , integration , calculus computations

Evaluate the following definite integral. \[\int_{e^{e}}^{e^{e+1}}\left\{\frac{1}{\ln x \cdot\ln (\ln x)}+\ln (\ln (\ln x))\right\}dx\]

1998 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , geometry , asymptote , integration

Find the area of the region bounded by the graphs $y=x^2$, $y=x$, and $x=2$.

1982 Vietnam National Olympiad, 1

Tags: trigonometry , polynomial , algebra , integration , vieta , calculus

Determine a quadric polynomial with intergral coefficients whose roots are $\cos 72^{\circ}$ and $\cos 144^{\circ}.$

2018 Korea USCM, 3

Tags: calculus , integration , function

$\Phi$ is a function defined on collection of bounded measurable subsets of $\mathbb{R}$ defined as $$\Phi(S) = \iint_S (1-5x^2 + 4xy-5y^2 ) dx dy$$ Find the maximum value of $\Phi$.

2004 USA Team Selection Test, 3

Tags: inequalities , geometry , floor function , perimeter , vector , integration , calculus

Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?

PEN D Problems, 13

Tags: modular arithmetic , algebra , calculus , polynomial , congruence , integration

Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.

2012 Today's Calculation Of Integral, 777

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

Given two points $P,\ Q$ on the parabola $C: y=x^2-x-2$ in the $xy$ plane. Note that the $x$ coodinate of $P$ is less than that of $Q$. (a) If the origin $O$ is the midpoint of the lines　egment $PQ$, then find the equation of the line $PQ$. (b) If the origin $O$ divides internally the line segment $PQ$ by 2:1, then find the equation of $PQ$. (c) If the origin $O$ divides internally the line segment $PQ$ by 2:1, find the area of the figure bounded by the parabola $C$ and the line $PQ$.

1998 Hungary-Israel Binational, 3

Tags: number theory , integration , calculus

Let $ a, b, c, m, n$ be positive integers. Consider the trinomial $ f (x) = ax^{2}+bx+c$. Show that there exist $ n$ consecutive natural numbers $ a_{1}, a_{2}, . . . , a_{n}$ such that each of the numbers $ f (a_{1}), f (a_{2}), . . . , f (a_{n})$ has at least $ m$ different prime factors.

1989 IMO Longlists, 20

Tags: rectangle , number theory , integration , geometry , calculus

Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions: [b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$ [b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint. [b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length. Prove that $ R$ has at least one side of integral length. [i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.

2014 Contests, 903

Tags: real analysis , calculus , integration , function , geometric series , calculus computations

Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$. Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$