Olympiad problems

2007 Today's Calculation Of Integral, 252

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 .$

2003 Mediterranean Mathematics Olympiad, 2

Tags: derivative , trigonometry , geometry , hyperbola , conic , calculus

In a triangle $ABC$ with $BC = CA + \frac 12 AB$, point $P$ is given on side $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2 \angle CPA.$

1981 Canada National Olympiad, 2

Tags: trigonometry , derivative , function , calculus , geometric transformation , reflection , geometry

Given a circle of radius $r$ and a tangent line $\ell$ to the circle through a given point $P$ on the circle. From a variable point $R$ on the circle, a perpendicular $RQ$ is drawn to $\ell$ with $Q$ on $\ell$. Determine the maximum of the area of triangle $PQR$.

2006 Moldova National Olympiad, 10.5

Tags: derivative , function , algebra , inequalities , calculus , logarithm

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]

1999 Putnam, 2

Tags: polynomial , quadratic , derivative , algebra , calculus

Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P^{\prime\prime}(x)$, where $Q(x)$ is a quadratic polynomial and $P^{\prime\prime}(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.

2002 Vietnam Team Selection Test, 2

Tags: diophantine equation , parameterization , derivative , calculus , polynomial , algebra

Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.

2005 Today's Calculation Of Integral, 6

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

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

2014 USA TSTST, 3

Tags: induction , derivative , polynomial , trigonometry , calculus , inequalities , algebra

Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.

2001 Federal Math Competition of S&M, Problem 1

Tags: derivative , calculus , number theory

Solve in positive integers \[ x^y + y = y^x + x \]

2006 ISI B.Math Entrance Exam, 4

Tags: calculus computations , derivative , induction , function , calculus

Let $f:\mathbb{R} \to \mathbb{R}$ be a function that is a function that is differentiable $n+1$ times for some positive integer $n$ . The $i^{th}$ derivative of $f$ is denoted by $f^{(i)}$ . Suppose- $f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0$. Prove that $f^{(n+1)}(x)=0$ for some $x \in (0,1)$

2024 Brazil Undergrad MO, 2

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

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 $.

2024 VJIMC, 1

Tags: calculus , derivative , function , integration , inequalities

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuously differentiable function. Prove that \[\left\vert f(1)-\int_0^1 f(x) dx\right\vert \le \frac{1}{2} \max_{x \in [0,1]} \vert f'(x)\vert.\]

2003 National Olympiad First Round, 16

Tags: derivative , calculus , function

For which of the following values of real number $t$, the equation $x^4-tx+\dfrac 1t = 0$ has no root on the interval $[1,2]$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ \text{None of the preceding} $

2009 Stanford Mathematics Tournament, 5

Tags: derivative , calculus

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

2005 National Olympiad First Round, 19

Tags: derivative , function , calculus

What is the greatest real root of the equation $x^3-x^2-x-\frac 13 = 0$? $ \textbf{(A)}\ \dfrac{\sqrt {3} - \sqrt{2}}{2} \qquad\textbf{(B)}\ \dfrac{\sqrt [3]{3} - \sqrt[3]{2}}{2} \qquad\textbf{(C)}\ \dfrac 1{\sqrt[3] {3} - 1} \qquad\textbf{(D)}\ \dfrac 1{\sqrt[3] {4} - 1} \qquad\textbf{(E)}\ \text{None of above} $

2012 Today's Calculation Of Integral, 833

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

Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$ For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$

2009 Princeton University Math Competition, 7

Tags: calculus , trigonometry , derivative , inequalities

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

2010 Today's Calculation Of Integral, 611

Tags: derivative , search , logarithm , inequalities , integration , function , calculus

Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$. Evaluate $\int_0^2 g(t)dt$. [i]2010 Kumamoto University entrance exam/Science[/i]

1994 AIME Problems, 4

Tags: derivative , logarithm , geometric series , floor function , calculus

Find the positive integer $n$ for which \[ \lfloor \log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994. \] (For real $x$, $\lfloor x\rfloor$ is the greatest integer $\le x.$)

2012 Today's Calculation Of Integral, 804

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

For $a>0$, find the minimum value of $I(a)=\int_1^e |\ln ax|\ dx.$

1969 Miklós Schweitzer, 7

Tags: advanced fields , derivative , calculus

Prove that if a sequence of Mikusinski operators of the form $ \mu e^{\minus{}\lambda s}$ ( $ \lambda$ and $ \mu$ nonnegative real numbers, $ s$ the differentiation operator) is convergent in the sense of Mikusinski, then its limit is also of this form. [i]E. Geaztelyi[/i]

1987 IMO Longlists, 73

Tags: algebra , calculus , derivative , function

Let $f(x)$ be a periodic function of period $T > 0$ defined over $\mathbb R$. Its first derivative is continuous on $\mathbb R$. Prove that there exist $x, y \in [0, T )$ such that $x \neq y$ and \[f(x)f'(y)=f'(x)f(y).\]

2014 All-Russian Olympiad, 3

Tags: algebra , polynomial , derivative , calculus , function

If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x)$, $f(x)g(x)$, $f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3-3x^2+5$ and $x^2-4x$ are written on the blackboard. Can we write a nonzero polynomial of form $x^n-1$ after a finite number of steps?

2010 Today's Calculation Of Integral, 547

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

Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.

2007 Today's Calculation Of Integral, 240

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

2 curves $ y \equal{} x^3 \minus{} x$ and $ y \equal{} x^2 \minus{} a$ pass through the point $ P$ and have a common tangent line at $ P$. Find the area of the region bounded by these curves.