Olympiad problems

2005 Today's Calculation Of Integral, 50

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

Let $a,b$ be real numbers such that $a<b$. Evaluate \[\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}\].

Today's calculation of integrals, 768

Tags: limit , symmetry , geometry , integration , 3d geometry , calculus , sphere

Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying \[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\] in $xyz$-space. (1) Find $V(r)$. (2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$ (3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$

2012 Pre - Vietnam Mathematical Olympiad, 2

Tags: limit , topology , algebra

Compute $\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\left( {2 + \sqrt 3 } \right)}^n}} \right\}$

2008 Iran Team Selection Test, 8

Tags: algebra , relatively prime , limit , polynomial , number theory , function , search

Find all polynomials $ p$ of one variable with integer coefficients such that if $ a$ and $ b$ are natural numbers such that $ a \plus{} b$ is a perfect square, then $ p\left(a\right) \plus{} p\left(b\right)$ is also a perfect square.

2005 Today's Calculation Of Integral, 84

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

Evaluate \[\lim_{n\to\infty} n\int_0^\pi e^{-nx} \sin ^ 2 nx\ dx\]

2012 Today's Calculation Of Integral, 857

Tags: geometric transformation , calculus , integration , trigonometry , limit , rotation , geometry

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

2020 LIMIT Category 1, 7

Tags: limit , algebra , vieta , polynomial

Let $P(x)=x^6-x^5-x^3-x^2-x$ and $a,b,c$ and $d$ be the roots of the equation $x^4-x^3-x^2-1=0$, then determine the value of $P(a)+P(b)+P(c)+P(d)$ (A)$5$ (B)$6$ (C)$7$ (D)$8$

2005 Croatia National Olympiad, 1

Tags: algebra , limit

A sequence $(a_{n})$ is deﬁned by $a_{1}= 1$ and $a_{n}= a_{1}a_{2}...a_{n-1}+1$ for $n \geq 2.$ Find the smallest real number $M$ such that $\sum_{n=1}^{m}\frac{1}{a_{n}}<M\; \forall m\in\mathbb{N}$.

2013 Today's Calculation Of Integral, 880

Tags: geometry , geometric transformation , trigonometry , integration , calculus , limit , rotation

For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows. (1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$. (2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$ (3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.

2000 Putnam, 3

Tags: algebra , limit , calculus , trigonometry , derivative , polynomial

Let $f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)$, where each $a_j$ is areal and $a_N$ is not equal to $0$. Let $N_k$ denote the number of zeroes (including multiplicites) of $\dfrac{d^k f}{dt^k}$. Prove that \[ N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N. \] [color=green][Only zeroes in [0, 1) should be counted.][/color]

2010 Romania National Olympiad, 4

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

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$, and \[I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].\] Prove that a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$, for any $h\in [0,1]$. b) the sequence $(a_n)_{n\ge 1}$, defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$, is convergent if and only if $f(0)=0$. [i]Calin Popescu[/i]

2012 Today's Calculation Of Integral, 803

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

Answer the following questions: (1) Evaluate $\int_{-1}^1 (1-x^2)e^{-2x}dx.$ (2) Find $\lim_{n\to\infty} \left\{\frac{(2n)!}{n!n^n}\right\}^{\frac{1}{n}}.$

2013 District Olympiad, 1

Tags: limit , calculus , inequalities , calculus computations

Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate $\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$

2016 Korea USCM, 1

Tags: limit , logarithm , real analysis , riemann sum

Find the following limit. \[\lim_{n\to\infty} \frac{1}{n} \log \left(\sum_{k=2}^{2^n} k^{1/n^2} \right)\]

2004 IMC, 1

Tags: polynomial , linear algebra , matrix , limit , algebra

Let $A$ be a real $4\times 2$ matrix and $B$ be a real $2\times 4$ matrix such that \[ AB = \left(% \begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ \end{array}% \right). \] Find $BA$.

2010 ISI B.Math Entrance Exam, 5

Tags: calculus , limit , calculus computations

Let $a_1>a_2>.....>a_r$ be positive real numbers . Compute $\lim_{n\to \infty} (a_1^n+a_2^n+.....+a_r^n)^{\frac{1}{n}}$

1998 VJIMC, Problem 3

Tags: function , real analysis , continuity , limit

Give an example of a sequence of continuous functions on $\mathbb R$ converging pointwise to $0$ which is not uniformly convergent on any nonempty open set.

2010 Today's Calculation Of Integral, 625

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

Find $\lim_{t\rightarrow 0}\frac{1}{t^3}\int_0^{t^2} e^{-x}\sin \frac{x}{t}\ dx\ (t\neq 0).$ [i]2010 Kumamoto University entrance exam/Medicine[/i]

2002 District Olympiad, 1

Tags: limit , induction , real analysis

a) Evaluate \[\lim_{n\to \infty} \underbrace{\sqrt{a+\sqrt{a+\ldots+\sqrt{a+\sqrt{b}}}}}_{n\ \text{square roots}}\] with $a,b>0$. b)Let $(a_n)_{n\ge 1}$ and $(x_n)_{n\ge 1}$ such that $a_n>0$ and \[x_n=\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}},\ \forall n\in \mathbb{N}^*\] Prove that: 1) $(x_n)_{n\ge 1}$ is bounded if and only if $(a_n)_{n\ge 1}$ is bounded. 2) $(x_n)_{n\ge 1}$ is convergent if and only if $(a_n)_{n\ge 1}$ is convergent. [i]Valentin Matrosenco[/i]

1993 Flanders Math Olympiad, 4

Tags: limit , trigonometry

Define the sequence $oa_n$ as follows: $oa_0=1, oa_n= oa_{n-1} \cdot cos\left( \dfrac{\pi}{2^{n+1}} \right)$. Find $\lim\limits_{n\rightarrow+\infty} oa_n$.

2002 AMC 10, 1

Tags: ratio , function , limit

The ratio $ \dfrac{10^{2000}\plus{}10^{2002}}{10^{2001}\plus{}10^{2001}}$ is closest to which of the following numbers? $ \text{(A)}\ 0.1\qquad \text{(B)}\ 0.2\qquad \text{(C)}\ 1\qquad \text{(D)}\ 5\qquad \text{(E)}\ 10$

1986 Iran MO (2nd round), 2

Tags: limit , function , calculus , derivative , algebra

[b](a)[/b] Sketch the diagram of the function $f$ if \[f(x)=4x(1-|x|) , \quad |x| \leq 1.\] [b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$ [b](c)[/b] Let $g$ be a function such that \[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\] Is the function $g$ continuous in the point $x=0 \ ?$ [b](d)[/b] Sketch the diagram of $g.$