Olympiad problems

1996 Romania National Olympiad, 1

Let $I \subset \mathbb{R}$ be a nondegenerate interval and $f:I \to \mathbb{R}$ a differentiable function. We denote $J= \left\{ \frac{f(b)-f(a)}{b-a} : a,b \in I, a<b \right\}.$ Prove that: $a)$ $J$ is an interval; $b)$ $J \subset f'(I),$ and the set $f'(I) \setminus J$ contains at most two elements; $c)$ Using parts $a)$ and $b),$ deduce that $f'$ has the intermediate value property.

2000 India Regional Mathematical Olympiad, 3

Tags: inequalities , integration , calculus , induction

Suppose $\{ x_n \}_{n\geq 1}$ is a sequence of positive real numbers such that $x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots$, and for all $n$ \[ \frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 . \] Show that for all $k$ \[ \frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3. \]

MIPT student olimpiad spring 2024, 3

Tags: calculus , function

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 Romania National Olympiad, 4

Tags: inequalities , calculus , integration

Let $ m,n$ be two natural numbers with $ m > 1$ and $ 2^{2m \plus{} 1} \minus{} n^2\geq 0$. Prove that: \[ 2^{2m \plus{} 1} \minus{} n^2\geq 7 .\]

2009 AIME Problems, 4

Tags: calculus , derivative

A group of children held a grape-eating contest. When the contest was over, the winner had eaten $ n$ grapes, and the child in $ k$th place had eaten $ n\plus{}2\minus{}2k$ grapes. The total number of grapes eaten in the contest was $ 2009$. Find the smallest possible value of $ n$.

2022 JHMT HS, 6

Tags: calculus

There is a unique choice of positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime and the infinite sums \[ \sum_{n=0}^{\infty} \left(\left(\frac{a - b\sqrt{c}}{10}\right)^{n-10}\cdot\prod_{k=0}^{9} (n - k)\right) \quad \text{and} \quad \sum_{n=0}^{\infty} \left((a - b\sqrt{c})^{n+1}\cdot\prod_{k=0}^{9} (n - k)\right) \] are equal (i.e., converging to the same finite value). Compute $a + b + c$.

2007 Today's Calculation Of Integral, 195

Tags: calculus , integration , function , real analysis , algebra , partial fractions , calculus computations

Find continuous functions $x(t),\ y(t)$ such that $\ \ \ \ \ \ \ \ \ x(t)=1+\int_{0}^{t}e^{-2(t-s)}x(s)ds$ $\ \ \ \ \ \ \ \ \ y(t)=\int_{0}^{t}e^{-2(t-s)}\{2x(s)+3y(s)\}ds$

Today's calculation of integrals, 879

Tags: calculus , integration , trigonometry , calculus computations , integral

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

2009 All-Russian Olympiad, 4

Tags: calculus , integration , analytic geometry , geometry , geometric transformation , reflection , combinatorics

Given a set $ M$ of points $ (x,y)$ with integral coordinates satisfying $ x^2 + y^2\leq 10^{10}$. Two players play a game. One of them marks a point on his first move. After this, on each move the moving player marks a point, which is not yet marked and joins it with the previous marked point. Players are not allowed to mark a point symmetrical to the one just chosen. So, they draw a broken line. The requirement is that lengths of edges of this broken line must strictly increase. The player, which can not make a move, loses. Who have a winning strategy?

1994 IMO Shortlist, 3

Tags: calculus , invariant , combinatorics , game , algorithm

Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. Prove that Peter can always transfer all his money into two accounts. Can Peter always transfer all his money into one account?

2005 Today's Calculation Of Integral, 57

Tags: calculus , integration , trigonometry , calculus computations

Find the value of $n\in{\mathbb{N}}$ satisfying the following inequality. \[\left|\int_0^{\pi} x^2\sin nx\ dx\right|<\frac{99\pi ^ 2}{100n}\]

PEN G Problems, 27

Tags: irrational number , limit , calculus , function , derivative , integration

Let $1<a_{1}<a_{2}<\cdots$ be a sequence of positive integers. Show that \[\frac{2^{a_{1}}}{{a_{1}}!}+\frac{2^{a_{2}}}{{a_{2}}!}+\frac{2^{a_{3}}}{{a_{3}}!}+\cdots\] is irrational.

1951 AMC 12/AHSME, 15

Tags: calculus , integration

The largest number by which the expression $ n^3 \minus{} n$ is divisible for all possible integral values of $ n$, is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$

2025 India STEMS Category C, 4

Tags: function , calculus

Does there exist a function $f:[0,1]\rightarrow (0,\infty)$ such that [list] [*]$f$ is differentiable on $[0,1]$ [*] It's derivative $f'$ is continuous on $[0,1]$. [*] $(f'(x))^3-x^{\frac{1}{3}}>6(1-f(x)^{\frac{1}{5}})$ for all $x\in [0,1]$. [*] $f(1)=1$ [/list] [i]Proposed by Medhansh Tripathi[/i]

2003 Romania Team Selection Test, 5

Tags: algebra , polynomial , calculus , integration , function , complex number , absolute value

Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial over the ring of integer polynomials, such that $|f(0)|$ is not a perfect square. Prove that if the leading coefficient of $f$ is 1 (the coefficient of the term having the highest degree in $f$) then $f(X^2)$ is also irreducible in the ring of integer polynomials. [i]Mihai Piticari[/i]

2005 Today's Calculation Of Integral, 25

Tags: calculus , integration , trigonometry , calculus computations

Let $|a|<\frac{\pi}{2}$. Evaluate \[\int_0^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^2}\]

2011 China National Olympiad, 2

Tags: function , calculus , derivative , rearrangement inequality , inequalities

Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$. Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$

2010 Today's Calculation Of Integral, 669

Tags: calculus , integration , function , calculus computations

Find the differentiable function defined in $x>0$ such that ${\int_1^{f(x)} f^{-1}(t)dt=\frac 13(x^{\frac {3}{2}}-8}).$

2009 Today's Calculation Of Integral, 465

Tags: calculus , integration , calculus computations

Compute $ \int_0^1 x^{2n\plus{}1}e^{\minus{}x^2}dx\ (n\equal{}1,\ 2,\ \cdots)$ , then use this result, prove that $ \sum_{n\equal{}0}^{\infty} \frac{1}{n!}\equal{}e$.

2012 Today's Calculation Of Integral, 781

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

Let $l,\ m$ be the tangent lines passing through the point $A(a,\ a-1)$ on the line $y=x-1$ and touch the parabola $y=x^2$. Note that the slope of $l$ is greater than that of $m$. (1) Exress the slope of $l$ in terms of $a$. (2) Denote $P,\ Q$ be the points of tangency of the lines $l,\ m$ and the parabola $y=x^2$. Find the minimum area of the part bounded by the line segment $PQ$ and the parabola $y=x^2$. (3) Find the minimum distance between the parabola $y=x^2$ and the line $y=x-1$.

2008 AMC 12/AHSME, 21

Tags: probability , analytic geometry , geometry , integration , calculus

Two circles of radius 1 are to be constructed as follows. The center of circle $ A$ is chosen uniformly and at random from the line segment joining $ (0,0)$ and $ (2,0)$. The center of circle $ B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $ (0,1)$ to $ (2,1)$. What is the probability that circles $ A$ and $ B$ intersect? $ \textbf{(A)} \; \frac{2\plus{}\sqrt{2}}{4} \qquad \textbf{(B)} \; \frac{3\sqrt{3}\plus{}2}{8} \qquad \textbf{(C)} \; \frac{2 \sqrt{2} \minus{} 1}{2} \qquad \textbf{(D)} \; \frac{2\plus{}\sqrt{3}}{4} \qquad \textbf{(E)} \; \frac{4 \sqrt{3} \minus{} 3}{4}$

2012 Today's Calculation Of Integral, 811

Tags: calculus , integration , trigonometry , calculus computations

Let $a$ be real number. Evaluate $\int_a^{a+\pi} |x|\cos x\ dx.$

2021 CMIMC Integration Bee, 12

Tags: calculus , integration

$$\int_1^\infty \frac{1 + 2x \ln 2}{x\sqrt{x 4^x - 1}}\,dx$$ [i]Proposed by Vlad Oleksenko[/i]

2012 Grigore Moisil Intercounty, 2

Tags: calculus , real analysis , definite integral , integration

$ \int_0^{\pi^2/4} \frac{dx}{1+\sin\sqrt x +\cos\sqrt x} $

2003 National Olympiad First Round, 16

Tags: calculus , derivative , 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} $