Olympiad problems

2010 Contests, 1

suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)

2007 ISI B.Math Entrance Exam, 3

Tags: algebra , polynomial , limit , complex number

For a natural number $n>1$ , consider the $n-1$ points on the unit circle $e^{\frac{2\pi ik}{n}}\ (k=1,2,...,n-1) $ . Show that the product of the distances of these points from $1$ is $n$.

2019 Jozsef Wildt International Math Competition, W. 7

Tags: limit , integration , summation , harmonic numbers , sequence

If $$\Omega_n=\sum \limits_{k=1}^n \left(\int \limits_{-\frac{1}{k}}^{\frac{1}{k}}(2x^{10} + 3x^8 + 1)\cos^{-1}(kx)dx\right)$$Then find $$\Omega=\lim \limits_{n\to \infty}\left(\Omega_n-\pi H_n\right)$$

2019 District Olympiad, 1

Tags: limit , convergence , calculus , sequence

Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n.$$

2024 District Olympiad, P4

Tags: real analysis , limit

Consider the functions $f,g:\mathbb{R}\to\mathbb{R}$ such that $f{}$ is continous. For any real numbers $a<b<c$ there exists a sequence $(x_n)_{n\geqslant 1}$ which converges to $b{}$ and for which the limit of $g(x_n)$ as $n{}$ tends to infinity exists and satisfies \[f(a)<\lim_{n\to\infty}g(x_n)<f(c).\][list=a] [*]Give an example of a pair of such functions $f,g$ for which $g{}$ is discontinous at every point. [*]Prove that if $g{}$ is monotonous, then $f=g.$ [/list]

2014 Math Prize For Girls Problems, 19

Tags: limit , function , probability , expected value , ceiling function

Let $n$ be a positive integer. Let $(a, b, c)$ be a random ordered triple of nonnegative integers such that $a + b + c = n$, chosen uniformly at random from among all such triples. Let $M_n$ be the expected value (average value) of the largest of $a$, $b$, and $c$. As $n$ approaches infinity, what value does $\frac{M_n}{n}$ approach?

1968 Miklós Schweitzer, 9

Tags: function , limit , real analysis

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]

2007 Today's Calculation Of Integral, 235

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

Show that a function $ f(x)\equal{}\int_{\minus{}1}^1 (1\minus{}|\ t\ |)\cos (xt)\ dt$ is continuous at $ x\equal{}0$.

2009 Today's Calculation Of Integral, 482

Tags: calculus , integration , limit , geometry , rectangle , inequalities , logarithm

Let $ n$ be natural number. Find the limit value of ${ \lim_{n\to\infty} \frac{1}{n}(\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}}+\cdots\cdots +\frac{n}{\sqrt{n^2+1}}).$

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.$

2019 Jozsef Wildt International Math Competition, W. 43

Tags: limit , integration , polynomial , algebra , sequence

Consider the sequence of polynomials $P_0(x) = 2$, $P_1(x) = x$ and $P_n(x) = xP_{n-1}(x) - P_{n-2}(x)$ for $n \geq 2$. Let $x_n$ be the greatest zero of $P_n$ in the the interval $|x| \leq 2$. Show that $$\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}$$

2009 Miklós Schweitzer, 3

Tags: search , limit , set theory , combinatorics

Prove that there exist positive constants $ c$ and $ n_0$ with the following property. If $ A$ is a finite set of integers, $ |A| \equal{} n > n_0$, then \[ |A \minus{} A| \minus{} |A \plus{} A| \leq n^2 \minus{} c n^{8/5}.\]

1996 Romania National Olympiad, 4

Tags: limit , function , real analysis , calculus , monotone functions , integral

Let $f:[0,1) \to \mathbb{R}$ be a monotonic function. Prove that the limits [center]$\lim_{x \nearrow 1} \int_0^x f(t) \mathrm{d}t$ and $\lim_{n \to \infty} \frac{1}{n} \left[ f(0) + f \left(\frac{1}{n}\right) + \ldots + f \left( \frac{n-1}{n} \right) \right]$[/center] exist and are equal.

2006 IMS, 4

Tags: advanced fields , function , limit , topology

Assume that $X$ is a seperable metric space. Prove that if $f: X\longrightarrow\mathbb R$ is a function that $\lim_{x\rightarrow a}f(x)$ exists for each $a\in\mathbb R$. Prove that set of points in which $f$ is not continuous is countable.

2020 LIMIT Category 1, 1

Tags: irrational , limit

If $a$ is a rational number and $b$ is an irrational number such that $ab$ is rational, then which of the following is false? (A)$ab^2$ is irrational (B)$a^2b$ is rational (C)$\sqrt{ab}$ is rational (D)$a+b$ is irrational

2000 JBMO ShortLists, 11

Tags: function , floor function , limit , algebra

Prove that for any integer $n$ one can find integers $a$ and $b$ such that \[n=\left[ a\sqrt{2}\right]+\left[ b\sqrt{3}\right] \]

2007 Harvard-MIT Mathematics Tournament, 1

Tags: calculus , limit

Compute: \[\lim_{x\to 0}\text{ }\dfrac{x^2}{1-\cos(x)}\]

1989 IMO Longlists, 5

Tags: inequalities , induction , trigonometry , limit , algebra

The sequences $ a_0, a_1, \ldots$ and $ b_0, b_1, \ldots$ are defined for $ n \equal{} 0, 1, 2, \ldots$ by the equalities \[ a_0 \equal{} \frac {\sqrt {2}}{2}, \quad a_{n \plus{} 1} \equal{} \frac {\sqrt {2}}{2} \cdot \sqrt {1 \minus{} \sqrt {1 \minus{} a^2_n}} \] and \[ b_0 \equal{} 1, \quad b_{n \plus{} 1} \equal{} \frac {\sqrt {1 \plus{} b^2_n} \minus{} 1}{b_n} \] Prove the inequalities for every $ n \equal{} 0, 1, 2, \ldots$ \[ 2^{n \plus{} 2} a_n < \pi < 2^{n \plus{} 2} b_n. \]

2010 Today's Calculation Of Integral, 522

Tags: calculus , integration , limit , function , geometry , derivative , logarithm

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

2004 Nicolae Coculescu, 1

Tags: limit , series , harmonic series , calculus , sequence

Calculate $ \lim_{n\to\infty } \left( e^{1+1/2+1/3+\cdots +1/n+1/(n+1)} -e^{1+1/2+1/3+\cdots +1/n} \right) . $

1982 IMO Longlists, 29

Tags: function , limit , algebra

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.

2012 Today's Calculation Of Integral, 857

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

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.