Olympiad problems

2013 Korea National Olympiad, 3

Tags: calculus , polynomial , quadratic , modular arithmetic , algebra , integration , induction

Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that (1) For all integer $m$, $f(m) \ne 0$. (2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.

2011 Today's Calculation Of Integral, 756

Tags: analytic geometry , integration , geometry , calculus , calculus computations

Let $a$ be real number. A circle $C$ touches the line $y=-x$ at the point $(a, -a)$ and passes through the point $(0,\ 1).$ Denote by $P$ the center of $C$. When $a$ moves, find the area of the figure enclosed by the locus of $P$ and the line $y=1$.

2011 Putnam, A5

Tags: derivative , calculus , function , vector

Let $F:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable functions with the following properties: • $F(u,u)=0$ for every $u\in\mathbb{R};$ • for every $x\in\mathbb{R},g(x)>0$ and $x^2g(x)\le 1;$ • for every $(u,v)\in\mathbb{R}^2,$ the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v)\rangle.$ Prove that there exists a constant $C$ such that for every $n\ge 2$ and any $x_1,\dots,x_{n+1}\in\mathbb{R},$ we have \[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]

2024-IMOC, A6

Tags: calculus , inequalities

Given positive real $a,b,c$ satisfying \[\frac{1}{\sqrt{a+1}}+\frac{3}{\sqrt{b+3}}+\frac{3}{\sqrt{c+3}}=\frac72\] Prove that $abc\leq 3$.\\ I was asked to propose a inequality for the condition of $abc<3$ inequality since <3 looks like a heart shape, then I construct a equality and with the help of wolfram, I gave the birth of this bad-looking inequality, I’m glad to see any method besides calculus.

2010 Today's Calculation Of Integral, 642

Tags: calculus computations , integration , calculus , trigonometry

Evaluate \[\int_0^{\frac{\pi}{6}} \frac{(\tan ^ 2 2x)\sqrt{\cos 2x}+2}{(\cos ^ 2 x)\sqrt{\cos 2x}}dx.\] Own

2012 Today's Calculation Of Integral, 792

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

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

2006 Putnam, A1

Tags: integration , trigonometry , analytic geometry , projective geometry , calculus , geometry

Find the volume of the region of points $(x,y,z)$ such that \[\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right). \]

2005 Today's Calculation Of Integral, 16

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

Calculate the following indefinite integrals. [1] $\int \sin (\ln x)dx$ [2] $\int \frac{x+\sin ^ 2 x}{x\sin ^ 2 x}dx$ [3] $\int \frac{x^3}{x^2+1}dx$ [4] $\int \frac{x^2}{\sqrt{2x-1}}dx$ [5] $\int \frac{x+\cos 2x +1}{x\cos ^ 2 x}dx$

1999 VJIMC, Problem 4

Tags: integration , calculus

Let $u_1,u_2,\ldots,u_n\in C([0,1]^n)$ be nonnegative and continuous functions, and let $u_j$ do not depend on the $j$-th variable for $j=1,\ldots,n$. Show that $$\left(\int_{[0,1]^n}\prod_{j=1}^nu_j\right)^{n-1}\le\prod_{j=1}^n\int_{[0,1]^n}u_j^{n-1}.$$

2012 Today's Calculation Of Integral, 790

Tags: parabola , geometry , analytic geometry , calculus , integration , conic , calculus computations

Define a parabola $C$ by $y=x^2+1$ on the coordinate plane. Let $s,\ t$ be real numbers with $t<0$. Denote by $l_1,\ l_2$ the tangent lines drawn from the point $(s,\ t)$ to the parabola $C$. (1) Find the equations of the tangents $l_1,\ l_2$. (2) Let $a$ be positive real number. Find the pairs of $(s,\ t)$ such that the area of the region enclosed by $C,\ l_1,\ l_2$ is $a$.

2009 Today's Calculation Of Integral, 494

Tags: integration , hyperbola , analytic geometry , conic , calculus computations , calculus

Suppose the curve $ C: y \equal{} ax^3 \plus{} 4x\ (a\neq 0)$ has a common tangent line at the point $ P$ with the hyperbola $ xy \equal{} 1$ in the first quadrant. (1) Find the value of $ a$ and the coordinate of the point $ P$. (2) Find the volume formed by the revolution of the solid of the figure bounded by the line segment $ OP$ and the curve $ C$ about the line $ OP$. [color=green][Edited.][/color]

2009 Today's Calculation Of Integral, 440

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

For $ a>1$, find $ \lim_{n\to\infty} \int_0^a \frac{e^x}{1\plus{}x^n}dx.$

1985 AIME Problems, 10

Tags: function , floor function , integration , calculus

How many of the first 1000 positive integers can be expressed in the form \[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \] where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?

1998 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , trigonometry , logarithm , limit

Evaluate $\displaystyle\lim_{x\to 1}x^{\dfrac{x}{\sin(1-x)}}$.

1999 AMC 12/AHSME, 26

Tags: integration , perimeter , geometry , calculus

Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $ 1$. The polygons meet at a point $ A$ in such a way that the sum of the three interior angles at $ A$ is $ 360^\circ$. Thus the three polygons form a new polygon with $ A$ as an interior point. What is the largest possible perimeter that this polygon can have? $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 21\qquad \textbf{(E)}\ 24$

2014 India Regional Mathematical Olympiad, 2

Tags: algebra , arithmetic sequence , integration , calculus

The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.

2013 Today's Calculation Of Integral, 860

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

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

2021 Alibaba Global Math Competition, 6

Tags: measure theory , inequality , function , integration , calculus , inequalities

Let $M(t)$ be measurable and locally bounded function, that is, \[M(t) \le C_{a,b}, \quad \forall 0 \le a \le t \le b<\infty\] with some constant $C_{a,b}$, from $[0,\infty)$ to $[0,\infty)$ such that \[M(t) \le 1+\int_0^t M(t-s)(1+t)^{-1}s^{-1/2} ds, \quad \forall t \ge 0.\] Show that \[M(t) \le 10+2\sqrt{5}, \quad \forall t \ge 0.\]

1999 Harvard-MIT Mathematics Tournament, 7

Tags: geometry , derivative , calculus

If a right triangle is drawn in a semicircle of radius $1/2$ with one leg (not the hypotenuse) along the diameter, what is the triangle's maximum possible area?

2013 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra , integration , floor function , trigonometry , calculus

Find the minimum positive noninteger root of $ \sin x=\sin \lfloor x \rfloor $. F. Petrov

2007 Today's Calculation Of Integral, 177

Tags: parabola , geometry , vector , conic , calculus , integration

On $xy$plane the parabola $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$ intersects with the line $y=x$ at the point $P$ that is different from the origin. Assumed that the circle $C$ is touched to $K$ at $P$ and $y$ axis at the point $Q.$ Let $S_{1}$ be the area of the region surrounded by the line passing through two points $P,\ Q$ and $K,$ or $S_{2}$ be the area of the region surrounded by the line which is passing through $P$ and parallel to $x$ axis and $K.$ Find the value of $\frac{S_{1}}{S_{2}}.$

1983 Putnam, A6

Tags: limit , calculus , real analysis , integration

Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

2003 Putnam, 4

Tags: derivative , quadratic , absolute value , calculus , function , inequalities

Suppose that $a, b, c, A, B, C$ are real numbers, $a \not= 0$ and $A \not= 0$, such that \[|ax^2+ bx + c| \le |Ax^2+ Bx + C|\] for all real numbers $x$. Show that \[|b^2- 4ac| \le |B^2- 4AC|\]

Today's calculation of integrals, 876

Tags: function , definite integral , integration , calculus computations , calculus

Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition : 1) $f(-1)\geq f(1).$ 2) $x+f(x)$ is non decreasing function. 3) $\int_{-1}^ 1 f(x)\ dx=0.$ Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

2013 Today's Calculation Of Integral, 866

Tags: calculus computations , integration , trigonometry , calculus , geometry

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$