Olympiad problems

2010 Princeton University Math Competition, 4

Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$. (Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)

2007 Today's Calculation Of Integral, 252

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

Compare $ \displaystyle f(\theta) \equal{} \int_0^1 (x \plus{} \sin \theta)^2\ dx$ and $ \ g(\theta) \equal{} \int_0^1 (x \plus{} \cos \theta)^2\ dx$ for $ 0\leqq \theta \leqq 2\pi .$

2011 Math Prize For Girls Problems, 18

Tags: algebra , polynomial , quadratic , calculus , integration , number theory , relatively prime

The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$?

2012 Bogdan Stan, 2

Tags: function , find all functions , calculus , integration

Find the continuous functions $ f:\left[ 0,\frac{1}{3} \right] \longrightarrow (0,\infty ) $ that satisfy the functional relation $$ 54\int_0^{1/3} f(x)dx +32\int_0^{1/3} \frac{dx}{\sqrt{x+f(x)}} =21. $$ [i]Cristinel Mortici[/i]

1989 IMO Longlists, 20

Tags: geometry , rectangle , calculus , integration , number theory

Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions: [b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$ [b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint. [b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length. Prove that $ R$ has at least one side of integral length. [i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.

2009 Today's Calculation Of Integral, 500

Tags: calculus , integration , calculus computations

Let $ a,\ b,\ c$ be positive real numbers. Prove the following inequality. \[ \int_1^e \frac {x^{a \plus{} b \plus{} c \minus{} 1}[2(a \plus{} b \plus{} c) \plus{} (c \plus{} 2a)x^{a \minus{} b} \plus{} (a \plus{} 2b)x^{b \minus{} c} \plus{} (b \plus{} 2c)x^{c \minus{} a} \plus{}(2a \plus{} b)x^{a \minus{} c} \plus{} (2b \plus{} c)x^{b \minus{} a} \plus{} (2c \plus{} a)x^{c \minus{} b}]}{(x^a \plus{} x^b)(x^b \plus{} x^c)(x^c \plus{} x^a)}\geq a \plus{} b \plus{} c.\] I have just posted 500 th post. [color=blue]Thank you for your cooperations, mathLinkers and AOPS users.[/color] I will keep posting afterwards. Japanese Communities Modeartor kunny

2005 Balkan MO, 2

Tags: quadratic , calculus , integration , number theory

Find all primes $p$ such that $p^2-p+1$ is a perfect cube.

2006 Princeton University Math Competition, 5

Tags: calculus , integration , function , inequalities , floor function

Find the greatest integer less than the number $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$

2014 Taiwan TST Round 1, 1

Tags: function , integration , logarithm

Find all increasing functions $f$ from the nonnegative integers to the integers satisfying $f(2)=7$ and \[ f(mn) = f(m) + f(n) + f(m)f(n) \] for all nonnegative integers $m$ and $n$.

2010 Today's Calculation Of Integral, 568

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

Throw $ n$ balls in to $ 2n$ boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes. Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$

1994 India Regional Mathematical Olympiad, 1

Tags: integration , inequalities

A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is $15000$. What are the page numbers on the torn leaf?

2013 Bogdan Stan, 3

Tags: integration , indefinite integral , antiderivative , calculus

$ \int \frac{1+2x^3}{1+x^2-2x^3+x^6} dx $ [i]Ion Nedelcu[/i] and [i]Lucian Tutescu[/i]

2014 VJIMC, Problem 4

Tags: inequalities , integration , calculus

Let $0<a<b$ and let $f:[a,b]\to\mathbb R$ be a continuous function with $\int^b_af(t)dt=0$. Show that $$\int^b_a\int^b_af(x)f(y)\ln(x+y)dxdy\le0.$$

2011 Today's Calculation Of Integral, 739

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

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

2009 Today's Calculation Of Integral, 409

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \sqrt{\frac{x\plus{}\sqrt{x^2\plus{}1}}{x^2\plus{}1}}\ dx$.

2005 Vietnam Team Selection Test, 3

Tags: function , geometry , geometric transformation , 3d geometry , induction , calculus , integration

Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$

2021 JHMT HS, 9

Tags: calculus , integration , function

Define a sequence $\{ a_n \}_{n=0}^{\infty}$ by $a_0 = 1,$ $a_1 = 8,$ and $a_n = 2a_{n-1} + a_{n-2}$ for $n \geq 2.$ The infinite sum \[ \sum_{n=1}^{\infty} \int_{0}^{2021\pi/14} \sin(a_{n-1}x)\sin(a_nx)\,dx \] can be expressed as a common fraction $\tfrac{p}{q}.$ Compute $p + q.$

1976 Miklós Schweitzer, 8

Tags: algebra , polynomial , integration , real analysis

Prove that the set of all linearly combinations (with real coefficients) of the system of polynomials $ \{ x^n\plus{}x^{n^2} \}_{n\equal{}0}^{\infty}$ is dense in $ C[0,1]$. [i]J. Szabados[/i]

2006 ISI B.Math Entrance Exam, 5

Tags: geometry , rectangle , calculus , integration

A domino is a $2$ by $1$ rectangle . For what integers $m$ and $n$ can we cover an $m*n$ rectangle with non-overlapping dominoes???

PEN G Problems, 16

Tags: irrational number , algebra , polynomial , integration , inequalities

For each integer $n \ge 1$, prove that there is a polynomial $P_{n}(x)$ with rational coefficients such that $x^{4n}(1-x)^{4n}=(1+x)^{2}P_{n}(x)+(-1)^{n}4^{n}$. Define the rational number $a_{n}$ by \[a_{n}= \frac{(-1)^{n-1}}{4^{n-1}}\int_{0}^{1}P_{n}(x) \; dx,\; n=1,2, \cdots.\] Prove that $a_{n}$ satisfies the inequality \[\left\vert \pi-a_{n}\right\vert < \frac{1}{4^{5n-1}}, \; n=1,2, \cdots.\]