Olympiad problems

1982 Dutch Mathematical Olympiad, 4

Determine $ \gcd (n^2\plus{}2,n^3\plus{}1)$ for $ n\equal{}9^{753}$.

1994 Vietnam Team Selection Test, 2

Tags: calculus , integration , number theory

Consider the equation \[x^2 + y^2 + z^2 + t^2 - N \cdot x \cdot y \cdot z \cdot t - N = 0\] where $N$ is a given positive integer. a) Prove that for an infinite number of values of $N$, this equation has positive integral solutions (each such solution consists of four positive integers $x, y, z, t$), b) Let $N = 4 \cdot k \cdot (8 \cdot m + 7)$ where $k,m$ are no-negative integers. Prove that the considered equation has no positive integral solutions.

2010 Harvard-MIT Mathematics Tournament, 10

Tags: calculus

Let $f(n)=\displaystyle\sum_{k=1}^n \dfrac{1}{k}$. Then there exists constants $\gamma$, $c$, and $d$ such that \[f(n)=\ln(x)+\gamma+\dfrac{c}{n}+\dfrac{d}{n^2}+O\left(\dfrac{1}{n^3}\right),\] where the $O\left(\dfrac{1}{n^3}\right)$ means terms of order $\dfrac{1}{n^3}$ or lower. Compute the ordered pair $(c,d)$.

2016 Ukraine Team Selection Test, 10

Tags: algebra , polynomial , calculus

Let $a_1,\ldots, a_n$ be real numbers. Define polynomials $f,g$ by $$f(x)=\sum_{k=1}^n a_kx^k,\ g(x)=\sum_{k=1}^n \frac{a_k}{2^k-1}x^k.$$ Assume that $g(2016)=0$. Prove that $f(x)$ has a root in $(0;2016)$.

2013 Today's Calculation Of Integral, 865

Tags: calculus , integration , geometry , geometric transformation , rotation , analytic geometry , rectangle

Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.

2000 Brazil Team Selection Test, Problem 1

Tags: circumcircle , trigonometry , calculus , trig identities , law of sines , geometry

Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$. [color=red][Moderator edited: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=5088 .][/color]

2008 Estonia Team Selection Test, 3

Tags: inequalities , function , algebra , calculus

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

2005 Today's Calculation Of Integral, 31

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

Evaluate \[\lim_{n\to\infty} \int_0^{\pi} x^2 |\sin nx| dx\]

2006 Moldova National Olympiad, 10.5

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

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: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]

1991 Arnold's Trivium, 14

Tags: integration , function , geometry , calculus

Calculate with at most $10\%$ relative error \[\int_{-\infty}^{\infty}(x^4+4x+4)^{-100}dx\]

2005 Today's Calculation Of Integral, 28

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^{\frac{\pi}{4}} \frac{x\cos 5x}{\cos x}dx\]

2009 Today's Calculation Of Integral, 432

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

Define the function $ f(t)\equal{}\int_0^1 (|e^x\minus{}t|\plus{}|e^{2x}\minus{}t|)dx$. Find the minimum value of $ f(t)$ for $ 1\leq t\leq e$.

2005 IMO Shortlist, 3

Tags: inequalities , algebra , calculus

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

1986 IMO Longlists, 25

Tags: algebra , polynomial , system of equations , calculus

Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations: \[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\] Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$

2013 Today's Calculation Of Integral, 880

Tags: calculus , integration , trigonometry , limit , geometry , geometric transformation , rotation

For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows. (1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$. (2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$ (3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.

2008 IMO Shortlist, 2

Tags: inequalities , algebra , vieta , calculus

[b](a)[/b] Prove that \[\frac {x^{2}}{\left(x \minus{} 1\right)^{2}} \plus{} \frac {y^{2}}{\left(y \minus{} 1\right)^{2}} \plus{} \frac {z^{2}}{\left(z \minus{} 1\right)^{2}} \geq 1\] for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$. [b](b)[/b] Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$. [i]Author: Walther Janous, Austria[/i]

2012 Today's Calculation Of Integral, 800

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

For a positive constant $a$, find the minimum value of $f(x)=\int_0^{\frac{\pi}{2}} |\sin t-ax\cos t|dt.$

2009 Today's Calculation Of Integral, 398

Tags: calculus , integration , inequalities , geometry , search , calculus computations

In $ xyz$ space, find the volume of the solid expressed by the sytem of inequality: $ 0\leqq x\leqq 1,\ 0\leqq y\leqq 1,\ 0\leqq z\leqq 1$ $ x^2 \plus{} y^2 \plus{} z^2 \minus{} 2xy \minus{} 1\geqq 0$

2009 Canada National Olympiad, 3

Tags: calculus , inequalities

Define $f(x,y,z)=\frac{(xy+yz+zx)(x+y+z)}{(x+y)(y+z)(z+x)}$. Determine the set of real numbers $r$ for which there exists a triplet of positive real numbers satisfying $f(x,y,z)=r$.

2007 Today's Calculation Of Integral, 217

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

Evaluate $ \int_{0}^{1}e^{\sqrt{e^{x}}}\ dx\plus{}2\int_{e}^{e^{\sqrt{e}}}\ln (\ln x)\ dx$.

1956 AMC 12/AHSME, 41

Tags: calculus , integration

The equation $ 3y^2 \plus{} y \plus{} 4 \equal{} 2(6x^2 \plus{} y \plus{} 2)$ where $ y \equal{} 2x$ is satisfied by: $ \textbf{(A)}\ \text{no value of }x \qquad\textbf{(B)}\ \text{all values of }x \qquad\textbf{(C)}\ x \equal{} 0\text{ only}$ $ \textbf{(D)}\ \text{all integral values of }x\text{ only} \qquad\textbf{(E)}\ \text{all rational values of }x\text{ only}$

2007 AIME Problems, 8

Tags: geometry , rectangle , calculus , integration , derivative , quadratic , modular arithmetic

A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called [i]basic [/i]if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle. Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000.