Olympiad problems

1999 VJIMC, Problem 2

Let $a,b\in\mathbb R$, $a\le b$. Assume that $f:[a,b]\to[a,b]$ satisfies $f(x)-f(y)\le|x-y|$ for every $x,y\in[a,b]$. Choose an $x_1\in[a,b]$ and define $$x_{n+1}=\frac{x_n+f(x_n)}2,\qquad n=1,2,3,\ldots.$$Show that $\{x_n\}^\infty_{n=1}$ converges to some fixed point of $f$.

2006 IberoAmerican Olympiad For University Students, 4

Tags: calculus , integration , algebra , polynomial , function , real analysis

Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval \[a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b\] such that \[\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots\] for all polynomials $f$ with real coefficients and degree less than $n$.

2019 ISI Entrance Examination, 8

Tags: distance , coordinate geometry , real analysis

Consider the following subsets of the plane:$$C_1=\Big\{(x,y)~:~x>0~,~y=\frac1x\Big\} $$ and $$C_2=\Big\{(x,y)~:~x<0~,~y=-1+\frac1x\Big\}$$ Given any two points $P=(x,y)$ and $Q=(u,v)$ of the plane, their distance $d(P,Q)$ is defined by $$d(P,Q)=\sqrt{(x-u)^2+(y-v)^2}$$ Show that there exists a unique choice of points $P_0\in C_1$ and $Q_0\in C_2$ such that $$d(P_0,Q_0)\leqslant d(P,Q)\quad\forall ~P\in C_1~\text{and}~Q\in C_2.$$

2008 Harvard-MIT Mathematics Tournament, 9

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

([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty} n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)} \left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.

2005 Gheorghe Vranceanu, 3

Tags: limit , euler s number , real analysis , calculus

$ \lim_{n\to\infty }\left( \frac{1}{e}\sum_{i=0}^n \frac{1}{i!} \right)^{n!} $

2019 LIMIT Category C, Problem 12

Tags: real analysis

$\lim_{x\to0}x\left\lfloor\frac1x\right\rfloor=?$

2019 Ramnicean Hope, 1

Tags: limit , factorial , trigonometric limit , real analysis

Calculate $ \lim_{n\to\infty }\left(\lim_{x\to 0} \left( -\frac{n}{x}+1+\frac{1}{x}\sum_{r=2}^{n+1}\sqrt[r!]{1+\sin rx}\right)\right) . $ [i]Constantin Rusu[/i]

1983 Miklós Schweitzer, 5

Tags: function , integration , limit , real analysis

Let $ g : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $ x+g(x)$ is strictly monotone (increasing or decreasing), and let $ u : [0,\infty) \rightarrow \mathbb{R}$ be a bounded and continuous function such that \[ u(t)+ \int_{t-1}^tg(u(s))ds\] is constant on $ [1,\infty)$. Prove that the limit $ \lim_{t\rightarrow \infty} u(t)$ exists. [i]T. Krisztin[/i]

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 5

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

Let $a,\ b>0$ be real numbers, $n\geq 2$ be integers. Evaluate $I_n=\int_{-\infty}^{\infty} \frac{exp(ia(x-ib))}{(x-ib)^n}dx.$

2022 Taiwan TST Round 3, A

Tags: algebra , real analysis , inequalities

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

2020 Jozsef Wildt International Math Competition, W21

Tags: limit , real analysis

Evaluate $$\lim_{n\to\infty}\left(\frac{1+\frac13+\ldots+\frac1{2n+1}}{\ln\sqrt n}\right)^{\ln\sqrt n}$$ [i]Proposed by Ángel Plaza[/i]

2007 Nicolae Păun, 4

Tags: function , topology , darboux s property , real analysis , pathological , monotone

Construct a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following properties: $ \text{(i)} f $ is not monotonic on any real interval. $ \text{(ii)} f $ has Darboux property (intermediate value property) on any real interval. $ \text{(iii)} f(x)\leqslant f\left( x+1/n \right) ,\quad \forall x\in\mathbb{R} ,\quad \forall n\in\mathbb{N} $ [i]Alexandru Cioba[/i]

1962 Miklós Schweitzer, 9

Tags: geometry , 3d geometry , prism , real analysis

Find the minimum possible sum of lengths of edges of a prism all of whose edges are tangent of a unit sphere. [Muller-Pfeiffer].

2018 District Olympiad, 4

Tags: function , continuity , real analysis

Let $a < b$ be real numbers and let $f : (a, b) \to \mathbb{R}$ be a function such that the functions $g : (a, b) \to \mathbb{R}$, $g(x) = (x - a) f(x)$ and $h : (a, b) \to \mathbb{R}$, $h(x) = (x - b) f(x)$ are increasing. Show that the function $f$ is continuous on $(a, b)$.

2009 Miklós Schweitzer, 9

Tags: calculus , integration , inequalities , function , invariant , real analysis

Let $ P\subseteq \mathbb{R}^m$ be a non-empty compact convex set and $ f: P\rightarrow \mathbb{R}_{ \plus{} }$ be a concave function. Prove, that for every $ \xi\in \mathbb{R}^m$ \[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]

2007 Mathematics for Its Sake, 2

Tags: limit , arithmetic sequence , real analysis , arithmetic progression , sequence

Let $ \left( a_n \right)_{n\ge 1} $ be an arithmetic progression of positive real numbers, and $ m $ be a natural number. Calculate: [b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^{2m+2}} \sum_{1\le i<j\le n} a_i^ma_j^m $ [b]b)[/b] $ \lim_{n\to\infty } \frac{1}{a_n^{2m+2}} \sum_{1\le i<j\le n} a_i^ma_j^m $ [i]Dumitru Acu[/i]

2018 IMC, 10

Tags: real analysis

For $R>1$ let $\mathcal{D}_R =\{ (a,b)\in \mathbb{Z}^2: 0<a^2+b^2<R\}$. Compute $$\lim_{R\rightarrow \infty}{\sum_{(a,b)\in \mathcal{D}_R}{\frac{(-1)^{a+b}}{a^2+b^2}}}.$$ [i]Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro[/i]

2012 Romania National Olympiad, 1

Tags: function , limit , real analysis

[color=darkred]Let $f,g\colon [0,1]\to [0,1]$ be two functions such that $g$ is monotonic, surjective and $|f(x)-f(y)|\le |g(x)-g(y)|$ , for any $x,y\in [0,1]$ . [list] [b]a)[/b] Prove that $f$ is continuous and that there exists some $x_0\in [0,1]$ with $f(x_0)=g(x_0)$ . [b]b)[/b] Prove that the set $\{x\in [0,1]\, |\, f(x)=g(x)\}$ is a closed interval. [/list][/color]

2019 Ramnicean Hope, 2

Tags: function graphing , real analysis , infimum

Calculate $ \inf_{x> 0} \sqrt{(1+x)^2+4/x} . $ [i]Constantin Rusu[/i] and [i]Mihai Neagu[/i]

2014 Contests, 4

Tags: trigonometry , real analysis

Let $f,g$ are defined in $(a,b)$ such that $f(x),g(x)\in\mathcal{C}^2$ and non-decreasing in an interval $(a,b)$ . Also suppose $f^{\prime \prime}(x)=g(x),g^{\prime \prime}(x)=f(x)$. Also it is given that $f(x)g(x)$ is linear in $(a,b)$. Show that $f\equiv 0 \text{ and } g\equiv 0$ in $(a,b)$.

1995 IMC, 2

Tags: function , calculus , real analysis , integral , inequalities

Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$, we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.

2019 Miklós Schweitzer, 8

Tags: real analysis

Let $f: \mathbb{R} \to \mathbb{R}$ be a measurable function such that $f(x+t) - f(x)$ is locally integrable for every $t$ as a function of $x$. Prove that $f$ is locally integrable.

2003 VJIMC, Problem 3

Tags: limit , real analysis , sequence

Let $\{a_n\}^\infty_{n=0}$ be the sequence of real numbers satisfying $a_0=0$, $a_1=1$ and $$a_{n+2}=a_{n+1}+\frac{a_n}{2^n}$$for every $n\ge0$. Prove that $$\lim_{n\to\infty}a_n=1+\sum_{n=1}^\infty\frac1{2^{\frac{n(n-1)}2}\displaystyle\prod_{k=1}^n(2^k-1)}.$$

2011 Laurențiu Duican, 3

Tags: function , real analysis , continuity , fractional part

Let be two continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R} $ satisfying the following equations: $$ \lim_{x\to\infty } f(x) =\infty =\lim_{x\to\infty } g(x) $$ Prove that there exists a divergent sequence $ \left( k_n \right)_{n\ge 1} $ of nonnegative integers which has the property that each term (function) of the sequence of functions $ \left( h_{n} \right)_{n\ge 1} :[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ h_{n} (x) =f\left( k_n+g(x) -\left\lfloor g(x) \right\rfloor \right) , $$ doesn't have limit at $ \infty . $ [i]Romeo Ilie[/i]

2018 IMC, 4

Tags: differentiable function , real analysis

Find all differentiable functions $f:(0,\infty) \to \mathbb{R}$ such that $$f(b)-f(a)=(b-a)f’(\sqrt{ab}) \qquad \text{for all}\qquad a,b>0.$$ [i]Proposed by Orif Ibrogimov, National University of Uzbekistan[/i]