Olympiad problems

2025 SEEMOUS, P2

Calculate $$\lim_{n\rightarrow\infty}n\int_0^{\infty} e^{-x}\sqrt[n]{e^x - 1 -\frac{x}{1!} - \frac{x^2}{2!} - \dots -\frac{x^n}{n!}}\,dx.$$

2005 Today's Calculation Of Integral, 23

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

Evaluate \[\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)\]

2022 CIIM, 1

Tags: function , quadratic , integral , calculus

Given the function $f(x) = x^2$, the sector of $f$ from $a$ to $b$ is defined as the limited region between the graph of $y = f(x)$ and the straight line segment that joins the points $(a, f(a))$ and $(b, f(b))$. Define the increasing sequence $x_0$, $x_1, \cdots$ with $x_0 = 0$ and $x_1 = 1$, such that the area of the sector of $f$ from $x_n$ to $x_{n+1}$ is constant for $n \geq 0$. Determine the value of $x_n$ in function of $n$.

1972 Poland - Second Round, 6

Tags: algebra , calculus

Prove that there exists a function $ f $ defined and differentiable in the set of all real numbers, satisfying the conditions $|f'(x) - f'(y)| \leq 4|x-y|$.

2005 Today's Calculation Of Integral, 50

Tags: logarithm , limit , derivative , integration , polynomial , algebra , calculus

Let $a,b$ be real numbers such that $a<b$. Evaluate \[\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}\].

1986 AMC 12/AHSME, 29

Tags: inequalities , calculus , triangle inequality , integration

Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{none of these} $

1999 Romania Team Selection Test, 7

Tags: arithmetic sequence , integration , algebra , calculus , geometric sequence , geometric series , binomial theorem

Prove that for any integer $n$, $n\geq 3$, there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that \[ b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . \] Give an example of two such progressions having at least five terms. [i]Mihai Baluna[/i]

1998 IMC, 6

Tags: function , calculus , inequalities , trigonometry , real analysis , integration

Let $f: [0,1]\rightarrow\mathbb{R}$ be a continuous function satisfying $xf(y)+yf(x)\le 1$ for every $x,y\in[0,1]$. (a) Show that $\int^1_0 f(x)dx \le \frac{\pi}4$. (b) Find such a funtion for which equality occurs.

1991 Arnold's Trivium, 34

Tags: calculus , derivative

Investigate the singular points on the curve $y=x^3$ in the projective plane.

Today's calculation of integrals, 768

Tags: limit , symmetry , geometry , integration , 3d geometry , calculus , sphere

Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying \[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\] in $xyz$-space. (1) Find $V(r)$. (2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$ (3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$

2011 Today's Calculation Of Integral, 758

Tags: conic , geometry , analytic geometry , graphing lines , slope , calculus , integration

Find the slope of a line passing through the point $(0,\ 1)$ with which the area of the part bounded by the line and the parabola $y=x^2$ is $\frac{5\sqrt{5}}{6}.$

1964 Miklós Schweitzer, 8

Tags: real analysis , calculus , function , derivative , topology

Let $ F$ be a closed set in the $ n$-dimensional Euclidean space. Construct a function that is $ 0$ on $ F$, positive outside $ F$ , and whose partial derivatives all exist.

MIPT student olimpiad spring 2024, 3

Tags: function , calculus

Is it true that if a function $f: R \to R$ is continuous and takes rational values at rational points, then at least at one point it is differentiable?

2007 Tournament Of Towns, 2

Tags: integration , calculus

Initially, the number $1$ and a non-integral number $x$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write $x^2$ on the blackboard in a finite number of moves?

2012 Centers of Excellency of Suceava, 4

Tags: real analysis , function , calculus , derivative , lagrange , convexity

Let be two real numbers $ a<b $ and a differentiable function $ f:[a,b]\longrightarrow\mathbb{R} $ that has a bounded derivative. Show that if $ \frac{f(b)-f(a)}{b-a} $ is equal to the global supremum or infimum of $ f', $ then $ f $ is polynomial with degree $ 1. $ [i]Cătălin Țigăeru[/i]

2005 Romania National Olympiad, 3

Tags: number theory , function , greatest common divisor , linear algebra , matrix , least common multiple , calculus

Let $X_1,X_2,\ldots,X_m$ a numbering of the $m=2^n-1$ non-empty subsets of the set $\{1,2,\ldots,n\}$, $n\geq 2$. We consider the matrix $(a_{ij})_{1\leq i,j\leq m}$, where $a_{ij}=0$, if $X_i \cap X_j = \emptyset$, and $a_{ij}=1$ otherwise. Prove that the determinant $d$ of this matrix does not depend on the way the numbering was done and compute $d$.

2005 Moldova Team Selection Test, 3

Tags: calculus , integration , function , number theory

\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\] where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$.

2005 Today's Calculation Of Integral, 84

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

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

2001 Vietnam Team Selection Test, 1

Tags: algebra , calculus , floor function , integration

Let a sequence of integers $\{a_n\}$, $n \in \mathbb{N}$ be given, defined by \[a_0 = 1, a_n= a_{n-1} + a_{[n/3]}\] for all $n \in \mathbb{N}^{*}$. Show that for all primes $p \leq 13$, there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$. (Here $[x]$ denotes the integral part of real number $x$).

2008 Moldova Team Selection Test, 4

Tags: calculus , polynomial , factorial , algebra , integration

A non-zero polynomial $ S\in\mathbb{R}[X,Y]$ is called homogeneous of degree $ d$ if there is a positive integer $ d$ so that $ S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y)$ for any $ \lambda\in\mathbb{R}$. Let $ P,Q\in\mathbb{R}[X,Y]$ so that $ Q$ is homogeneous and $ P$ divides $ Q$ (that is, $ P|Q$). Prove that $ P$ is homogeneous too.

2012 Today's Calculation Of Integral, 857

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

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

2009 Today's Calculation Of Integral, 432

Tags: function , calculus , integration , 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$.

2014 Macedonia National Olympiad, 4

Tags: inequalities , three variable inequality , calculus

Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that: \[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]

2010 Abels Math Contest (Norwegian MO) Final, 4b

Tags: calculus , number theory , integration

Let $n > 2$ be an integer. Show that it is possible to choose $n$ points in the plane, not all of them lying on the same line, such that the distance between any pair of points is an integer (that is, $\sqrt{(x_1 -x_2)^2 +(y_1 -y_2)^2}$ is an integer for all pairs $(x_1, y_1)$ and $(x_2, y_2)$ of points).

1968 IMO Shortlist, 1

Tags: function , calculus , minimum value , minimization , algebra

Two ships sail on the sea with constant speeds and fixed directions. It is known that at $9:00$ the distance between them was $20$ miles; at $9:35$, $15$ miles; and at $9:55$, $13$ miles. At what moment were the ships the smallest distance from each other, and what was that distance ?