Olympiad problems

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} . $

2003 Turkey Team Selection Test, 2

Tags: limit , geometry , inequalities

Let $K$ be the intersection of the diagonals of a convex quadrilateral $ABCD$. Let $L\in [AD]$, $M \in [AC]$, $N \in [BC]$ such that $KL\parallel AB$, $LM\parallel DC$, $MN\parallel AB$. Show that \[\dfrac{Area(KLMN)}{Area(ABCD)} < \dfrac {8}{27}.\]

2012 Today's Calculation Of Integral, 792

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

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

1994 IMC, 6

Tags: real analysis , limit

Find $$\lim_{N\to\infty}\frac{\ln^2 N}{N} \sum_{k=2}^{N-2} \frac{1}{\ln k \cdot \ln (N-k)}$$

2009 Today's Calculation Of Integral, 440

Tags: integration , limit , calculus , calculus computations , real analysis

For $ a>1$, find $ \lim_{n\to\infty} \int_0^a \frac{e^x}{1\plus{}x^n}dx.$

2007 Nicolae Coculescu, 3

Tags: real analysis , seuqence , limit

Let be the sequence $ \left( a_n \right)_{n\ge 0} $ of positive real numbers defined by $$ a_n=1+\frac{a_{n-1}}{n} ,\quad\forall n\ge 1. $$ Calculate $ \lim_{n\to\infty } a_n ^n . $ [i]Florian Dumitrel[/i]

1998 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , trigonometry , logarithm , limit

Evaluate $\displaystyle\lim_{x\to 1}x^{\dfrac{x}{\sin(1-x)}}$.

1968 Miklós Schweitzer, 9

Tags: limit , real analysis , function

Let $ f(x)$ be a real function such that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f(x)}{e^x}\equal{}1\] and $ |f''(x)|\leq c|f'(x)|$ for all sufficiently large $ x$. Prove that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f'(x)}{e^x}\equal{}1.\] [i]P. Erdos[/i]

2012 South africa National Olympiad, 6

Tags: algebra , limit , function , inequalities

Find all functions $f:\mathbb{N}\to\mathbb{R}$ such that $f(km)+f(kn)-f(k)f(mn)\ge 1$ for all $k,m,n\in\mathbb{N}$.

1983 Putnam, A6

Tags: limit , calculus , real analysis , integration

Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

2010 ISI B.Math Entrance Exam, 8

Tags: function , calculus , calculus computations , limit

Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that $\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.

1982 IMO Longlists, 29

Tags: limit , algebra , function

Let $f : \mathbb R \to \mathbb R$ be a continuous function. Suppose that the restriction of $f$ to the set of irrational numbers is injective. What can we say about $f$? Answer the analogous question if $f$ is restricted to rationals.

2004 Putnam, B5

Tags: real analysis , logarithm , limit , integration , calculus

Evaluate $\lim_{x\to 1^-}\prod_{n=0}^{\infty}\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.

2013 IPhOO, 9

Tags: buoyancy , 3d geometry , sphere , limit , geometry

Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's launched up. Find $\lim_{r\to0} \left(f(r)\right) $. Express your answer is meters and round to the nearest integer. Assume the density of water is 1000 $\text{kg/m}^3$. [i](B. Dejean, 6 points)[/i]

1995 Putnam, 6

Tags: ceiling function , limit , floor function

For any $a>0$,set $\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}$. Show that there are no three positive reals $a,b,c$ such that \[ \mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset \] \[ \mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N} \]

2010 Today's Calculation Of Integral, 656

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

Find $\lim_{n\to\infty} n\int_0^{\frac{\pi}{2}} \frac{1}{(1+\cos x)^n}dx\ (n=1,\ 2,\ \cdots).$

2010 Gheorghe Vranceanu, 2

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

Let be a natural number $ n, $ a nonzero number $ \alpha, \quad n $ numbers $ a_1,a_2,\ldots ,a_n $ and $ n+1 $ functions $ f_0,f_1,f_2,\ldots ,f_n $ such that $ f_0=\alpha $ and the rest are defined recursively as $$ f_k (x)=a_k+\int_0^x f_{k-1} (x)dx . $$ Prove that if all these functions are everywhere nonnegative, then the sum of all these functions is everywhere nonnegative.

2014 BMT Spring, 5

Tags: real analysis , limit

Determine $$\lim_{x\to\infty}\frac{\sqrt{x+2014}}{\sqrt x+\sqrt{x+2014}}$$

2020 LIMIT Category 2, 8

Tags: set , probability , limit

Let $S$ be a finite set of size $s\geq 1$ defined with a uniform probability $\mathbb{P}$( i.e. for any subset $X\subset S$ of size $x$, $\mathbb{P}(x)=\frac{x}{s}$). Suppose $A$ and $B$ are subsets of $S$. They are said to be independent iff $\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(A\cap B)$. Which if these is sufficient for independence? (A)$|A\cup B|=|A|+|B|$ (B)$|A\cap B|=|A|+|B|$ (C)$|A\cup B|=|A|\cdot |B|$ (D)$|A\cap B|=|A|\cdot |B|$

2010 Today's Calculation Of Integral, 649

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

Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$, $I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$ Find $\lim_{n\to\infty} I_n.$ [i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]

2019 ISI Entrance Examination, 6

Tags: limit , sequence , real analysis

For all natural numbers $n$, let $$A_n=\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}\quad\text{(n many radicals)}$$ [b](a)[/b] Show that for $n\geqslant 2$, $$A_n=2\sin\frac{\pi}{2^{n+1}}$$ [b](b)[/b] Hence or otherwise, evaluate the limit $$\lim_{n\to\infty} 2^nA_n$$

1966 Swedish Mathematical Competition, 5

Tags: combinatorial geometry , lattice , limit

Let $f(r)$ be the number of lattice points inside the circle radius $r$, center the origin. Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$, put $g(r) = f(r) - kr^2$. Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$?

2000 VJIMC, Problem 3

Tags: limit , real analysis

Let $a_1,a_2,\ldots$ be a bounded sequence of reals. Is it true that the fact $$\lim_{N\to\infty}\frac1N\sum_{n=1}^Na_n=b\enspace\text{ and }\enspace\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac{a_n}n=c$$implies $b=c$?

1984 AIME Problems, 15

Tags: algorithm , algebra , polynomial , limit

Determine $w^2+x^2+y^2+z^2$ if \[ \begin{array}{l} \displaystyle \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 \\ \displaystyle \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 \\ \displaystyle \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 \\ \displaystyle \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 \\ \end{array} \]

Today's calculation of integrals, 882

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

Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.