Olympiad problems

2009 Harvard-MIT Mathematics Tournament, 5

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

Compute \[\lim_{h\to 0}\dfrac{\sin(\frac{\pi}{3}+4h)-4\sin(\frac{\pi}{3}+3h)+6\sin(\frac{\pi}{3}+2h)-4\sin(\frac{\pi}{3}+h)+\sin(\frac{\pi}{3})}{h^4}.\]

2008 IMS, 3

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

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$.

2022 JHMT HS, 1

Tags: derivative , integration , calculus

Compute the value of \[ \frac{d}{dx}\int_{1}^{10} x^3\,dx. \]

2009 Today's Calculation Of Integral, 510

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

(1) Evaluate $ \int_0^{\frac{\pi}{2}} (x\cos x\plus{}\sin ^ 2 x)\sin x\ dx$. (2) For $ f(x)\equal{}\int_0^x e^t\sin (x\minus{}t)\ dt$, find $ f''(x)\plus{}f(x)$.

2012 Online Math Open Problems, 40

Tags: calculus , algebra , derivative , trigonometry , circumcircle , geometry

Suppose $x,y,z$, and $w$ are positive reals such that \[ x^2 + y^2 - \frac{xy}{2} = w^2 + z^2 + \frac{wz}{2} = 36 \] \[ xz + yw = 30. \] Find the largest possible value of $(xy + wz)^2$. [i]Author: Alex Zhu[/i]

2005 Today's Calculation Of Integral, 35

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

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

1998 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , graphing lines , slope , derivative , analytic geometry

Find the slopes of all lines passing through the origin and tangent to the curve $y^2=x^3+39x-35$.

2010 Putnam, A3

Tags: calculus , probability , derivative , function , complex analysis , vector

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

2013 ELMO Problems, 2

Tags: function , inequalities , calculus , derivative , 3-variable inequality , logarithm

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

2011 Today's Calculation Of Integral, 739

Tags: functional equation , calculus , algebra , derivative , integration , trigonometry , function

Find the function $f(x)$ such that : \[f(x)=\cos x+\int_0^{2\pi} f(y)\sin (x-y)\ dy\]

2022 VTRMC, 6

Tags: calculus , derivative , function

Let $f : \mathbb{R} \to \mathbb{R}$ be a function whose second derivative is continuous. Suppose that $f$ and $f''$ are bounded. Show that $f'$ is also bounded.

2018 Korea USCM, 6

Tags: inequalities , derivative , real analysis , function

Suppose a continuous function $f:[0,1]\to\mathbb{R}$ is differentiable on $(0,1)$ and $f(0)=1$, $f(1)=0$. Then, there exists $0<x_0<1$ such that $$|f'(x_0)| \geq 2018 f(x_0)^{2018}$$

2009 Stanford Mathematics Tournament, 7

Tags: rectangle , calculus , trigonometry , derivative , trapezoid , geometry , function

An isosceles trapezoid has legs and shorter base of length $1$. Find the maximum possible value of its area

2005 Swedish Mathematical Competition, 4

Tags: arithmetic sequence , derivative , algebra

The zeroes of a fourth degree polynomial $f(x)$ form an arithmetic progression. Prove that the three zeroes of the polynomial $f'(x)$ also form an arithmetic progression.

2002 Putnam, 1

Tags: function , derivative , limit , calculus

Let $k$ be a fixed positive integer. The $n$th derivative of $\tfrac{1}{x^k-1}$ has the form $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$, where $P_n(x)$ is a polynomial. Find $P_n(1)$.

1999 Putnam, 4

Tags: calculus , derivative , limit , function

Let $f$ be a real function with a continuous third derivative such that $f(x)$, $f^\prime(x)$, $f^{\prime\prime}(x)$, $f^{\prime\prime\prime}(x)$ are positive for all $x$. Suppose that $f^{\prime\prime\prime}(x)\leq f(x)$ for all $x$. Show that $f^\prime(x)<2f(x)$ for all $x$.

2009 Princeton University Math Competition, 5

Tags: trigonometry , calculus , inequalities , derivative

Find the maximal positive integer $n$, so that for any real number $x$ we have $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$.

2011 Morocco National Olympiad, 1

Tags: derivative , inequalities , calculus

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 $x,y$.

2009 ISI B.Stat Entrance Exam, 6

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

Let $f(x)$ be a function satisfying \[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\] Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.

2001 Romania National Olympiad, 1

Tags: derivative , real analysis , function , calculus

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, derivable on $R\backslash\{x_0\}$, having finite side derivatives in $x_0$. Show that there exists a derivable function $g:\mathbb{R}\rightarrow\mathbb{R}$, a linear function $h:\mathbb{R}\rightarrow\mathbb{R}$ and $\alpha\in\{-1,0,1\}$ such that: \[ f(x)=g(x)+\alpha |h(x)|,\ \forall x\in\mathbb{R} \]

2007 Moldova Team Selection Test, 2

Tags: algebra , derivative , calculus , polynomial , induction

If $b_{1}, b_{2}, \ldots, b_{n}$ are non-negative reals not all zero, then prove that the polynomial \[x^{n}-b_{1}x^{n-1}-b_{2}x^{n-2}-\ldots-b_{n}=0\] has only one positive root $p$, which is simple. Moreover prove that any root of the polynomial does not exceed $p$ in absolute value.

2004 Poland - First Round, 3

Tags: inequalities , geometry , vector , derivative , calculus , trigonometry

3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and: $\frac{EF}{FD}=\frac{AD}{DB}$ Prove that $CF \bot AE$

1989 IMO Longlists, 61

Tags: function , inequalities , calculus , derivative

Prove for $ 0 < k \leq 1$ and $ a_i \in \mathbb{R}^\plus{},$ $ i \equal{} 1,2 \ldots, n$ the following inequality holds: \[ \left( \frac{a_1}{a_2 \plus{} \ldots \plus{} a_n} \right)^k \plus{} \ldots \plus{} \left( \frac{a_n}{a_1 \plus{} \ldots \plus{} a_{n\minus{}1}} \right)^k \geq \frac{n}{(n\minus{}1)^k}.\]

2011 Romania National Olympiad, 2

Tags: real analysis , calculus , function , derivative , abstract algebra , inequalities

[color=darkred]Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .[/color]

1994 IMC, 3

Tags: real analysis , derivative , calculus , function

Let $f$ be a real-valued function with $n+1$ derivatives at each point of $\mathbb R$. Show that for each pair of real numbers $a$, $b$, $a<b$, such that $$\ln\left( \frac{f(b)+f'(b)+\cdots + f^{(n)} (b)}{f(a)+f'(a)+\cdots + f^{(n)}(a)}\right)=b-a$$ there is a number $c$ in the open interval $(a,b)$ for which $$f^{(n+1)}(c)=f(c)$$