Olympiad problems

1996 China National Olympiad, 2

Tags: inequalities , trigonometry , calculus , derivative , integration

Let $n$ be a natural number. Suppose that $x_0=0$ and that $x_i>0$ for all $i\in\{1,2,\ldots ,n\}$. If $\sum_{i=1}^nx_i=1$ , prove that \[1\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2} \]

2010 CIIM, Problem 4

Tags: function , calculus , derivative

Let $f:[0,1] \to [0,1]$ a increasing continuous function, diferentiable in $(0,1)$ and with derivative smaller than 1 in every point. The sequence of sets $A_1,A_2,A_3,\dots$ is define as: $A_1 = f([0,1])$, and for $n \geq 2, A_n = f(A_{n-1}).$ Prove that $\displaystyle \lim_{n\to+\infty} d(A_n) = 0$, where $d(A)$ is the diameter of the set $A$. Note: The diameter of a set $X$ is define as $d(X) = \sup_{x,y\in X} |x-y|.$

2012 Math Prize For Girls Problems, 19

Tags: function , calculus , derivative , induction

Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by \[ a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr), \] where there are $n$ iterations of $L$. For example, \[ a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr). \] As $n$ approaches infinity, what value does $n a_n$ approach?

1970 IMO Longlists, 25

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

A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.

1996 Vietnam National Olympiad, 3

Tags: algebra , polynomial , calculus , derivative , inequalities

Prove that:$a+b+c+d \geq \frac{2}{3}(ab+bc+ca+ad+ac+bd)$ where $a;b;c;d$ are positive real numbers satisfying $2(ab+bc+cd+da+ac+bd)+abc+bcd+cda+dab=16$

2006 Romania Team Selection Test, 4

Tags: function , integration , calculus , derivative , inequalities

Let $p$, $q$ be two integers, $q\geq p\geq 0$. Let $n \geq 2$ be an integer and $a_0=0, a_1 \geq 0, a_2, \ldots, a_{n-1},a_n = 1$ be real numbers such that \[ a_{k} \leq \frac{ a_{k-1} + a_{k+1} } 2 , \ \forall \ k=1,2,\ldots, n-1 . \] Prove that \[ (p+1) \sum_{k=1}^{n-1} a_k^p \geq (q+1) \sum_{k=1}^{n-1} a_k^q . \]

2005 Today's Calculation Of Integral, 35

Tags: calculus , integration , derivative , function , calculus computations

Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.

2024 Brazil Undergrad MO, 2

Tags: real analysis , derivative , fixed point , algebra , polynomial

For each pair of integers $ j, k \geq 2 $, define the function $ f_{jk} : \mathbb{R} \to \mathbb{R} $ given by \[ f_{jk}(x) = 1 - (1 - x^j)^k. \] (a) Prove that for any integers $ j, k \geq 2 $, there exists a unique real number $ p_{jk} \in (0, 1) $ such that $ f_{jk}(p_{jk}) = p_{jk} $. Furthermore, defining $ \lambda_{jk} := f'_{jk}(p_{jk}) $, prove that $ \lambda_{jk} > 1 $. (b) Prove that $ p^j_{jk} = 1 - p_{kj} $ for any integers $ j, k \geq 2 $. (c) Prove that $ \lambda_{jk} = \lambda_{kj} $ for any integers $ j, k \geq 2 $.

2005 Korea National Olympiad, 4

Tags: function , induction , calculus , derivative , functional equation , algebra

Find all $f: \mathbb R \to\mathbb R$ such that for all real numbers $x$, $f(x) \geq 0$ and for all real numbers $x$ and $y$, \[ f(x+y)+f(x-y)-2f(x)-2y^2=0. \]

2011 Morocco National Olympiad, 1

Tags: calculus , derivative , inequalities

Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.

1992 Putnam, A4

Tags: function , calculus , derivative , limit , trigonometry , search

Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$

2024 Mongolian Mathematical Olympiad, 1

Tags: calculus , derivative , algebra , inequality , bruh

Let $P(x)$ and $Q(x)$ be polynomials with nonnegative coefficients. We denote by $P'(x)$ the derivative of $P(x)$. Suppose that $P(0)=Q(0)=0$ and $Q(1) \leq 1 \leq P'(0)$. $(1)$ Prove that $0 \leq Q(x) \leq x \leq P(x)$ for all $0 \leq x \leq 1$. $(2)$ Prove that $P(Q(x)) \leq Q(P(x))$ for all $0 \leq x \leq 1$. [i]Proposed by Otgonbayar Uuye.[/i]

2008 IMS, 3

Tags: geometry , parabola , ratio , function , calculus , derivative , conic

Let $ A,B$ be different points on a parabola. Prove that we can find $ P_1,P_2,\dots,P_{n}$ between $ A,B$ on the parabola such that area of the convex polygon $ AP_1P_2\dots P_nB$ is maximum. In this case prove that the ratio of $ S(AP_1P_2\dots P_nB)$ to the sector between $ A$ and $ B$ doesn't depend on $ A$ and $ B$, and only depends on $ n$.

2009 Stanford Mathematics Tournament, 5

Tags: calculus , derivative

Find the minimum possible value of $2x^2+2xy+4y+5y^2-x$ for real numbers $x$ and $y$.

2010 Contests, 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$.

2006 Harvard-MIT Mathematics Tournament, 1

Tags: calculus , derivative , algebra , polynomial

A nonzero polynomial $f(x)$ with real coefficients has the property that $f(x)=f^\prime(x)f^{\prime\prime}(x)$. What is the leading coefficient of $f(x)$?

2006 Petru Moroșan-Trident, 2

Tags: function , find all functions , real analysis , calculus , derivative

Find the twice-differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that have the property that $$ f'(x)+F(x)=2f(x)+x^2/2, $$ for any real numbers $ x; $ where $ F $ is a primitive of $ f. $ [i]Carmen Botea[/i]

2013 Today's Calculation Of Integral, 887

Tags: calculus , integration , function , trigonometry , derivative , calculus computations , logarithm

For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx. (1) Find $f(\sqrt{3})$ (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.

2007 Today's Calculation Of Integral, 255

Tags: calculus , integration , geometry , derivative , calculus computations , logarithm

Find the value of $ a$ for which the area of the figure surrounded by $ y \equal{} e^{ \minus{} x}$ and $ y \equal{} ax \plus{} 3\ (a < 0)$ is minimized.

PEN I Problems, 8

Tags: floor function , calculus , derivative , inequalities

Prove that $\lfloor \sqrt[3]{n}+\sqrt[3]{n+1}+\sqrt[3]{n+2}\rfloor =\lfloor \sqrt[3]{27n+26}\rfloor$ for all positive integers $n$.

2009 Serbia Team Selection Test, 1

Tags: inequalities , trigonometry , calculus , derivative , angle bisector , geometry

Let $ \alpha$ and $ \beta$ be the angles of a non-isosceles triangle $ ABC$ at points $ A$ and $ B$, respectively. Let the bisectors of these angles intersect opposing sides of the triangle in $ D$ and $ E$, respectively. Prove that the acute angle between the lines $ DE$ and $ AB$ isn't greater than $ \frac{|\alpha\minus{}\beta|}3$.

1989 APMO, 5

Tags: function , induction , calculus , derivative , limit , algebra , recursion

Determine all functions $f$ from the reals to the reals for which (1) $f(x)$ is strictly increasing and (2) $f(x) + g(x) = 2x$ for all real $x$, where $g(x)$ is the composition inverse function to $f(x)$. (Note: $f$ and $g$ are said to be composition inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all real $x$.)

2014 AIME Problems, 7

Tags: trigonometry , calculus , derivative , vector , ratio , analytic geometry , graphing lines

Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.

2000 Putnam, 3

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

2013 Today's Calculation Of Integral, 893

Tags: calculus , integration , trigonometry , function , derivative , calculus computations , logarithm

Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$