Olympiad problems

2014 Junior Balkan MO, 4

Tags: floor function , modular arithmetic , limit , logarithms , combinatorics solved , combinatorics

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

2019 PUMaC Team Round, 11

Tags: floor function , logarithms , combinatorics

The game Prongle is played with a special deck of cards: on each card is a nonempty set of distinct colors. No two cards in the deck contain the exact same set of colors. In this game, a “Prongle” is a set of at least $2$ cards such that each color is on an even number of cards in the set. Let k be the maximum possible number of prongles in a set of $2019$ cards. Find $\lfloor \log 2 (k) \rfloor$.

2010 Today's Calculation Of Integral, 571

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

Evaluate $ \int_0^{\pi} \frac{x\sin ^ 3 x}{\sin ^ 2 x\plus{}8}dx$.

2009 Today's Calculation Of Integral, 516

Tags: calculus , integration , trigonometry , logarithms , calculus computations

Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$. (1) Find the local minimum value of $ f(x)$. (2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$.

2009 Today's Calculation Of Integral, 451

Tags: calculus , integration , limit , logarithms , calculus computations

Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \ln \left(1\plus{}\frac{k^a}{n^{a\plus{}1}}\right).$

2012 Today's Calculation Of Integral, 835

Tags: calculus , integration , logarithms , trigonometry , calculus computations

Evaluate the following definite integrals. (a) $\int_1^2 \frac{x-1}{x^2-2x+2}\ dx$ (b) $\int_0^1 \frac{e^{4x}}{e^{2x}+2}\ dx$ (c) $\int_1^e x\ln \sqrt{x}\ dx$ (d) $\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx$

PEN G Problems, 20

Tags: floor function , logarithms , Irrational numbers

You are given three lists A, B, and C. List A contains the numbers of the form $10^{k}$ in base 10, with $k$ any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: \[\begin{array}{lll}A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}.\] Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists B or C that has exactly $n$ digits.

2003 Moldova National Olympiad, 10.8

Tags: logarithms , algebra unsolved , algebra

Find all integers n for which number $ \log_{2n\minus{}1}(n^2\plus{}2)$ is rational.

1998 AMC 12/AHSME, 22

Tags: logarithms , AMC , AIME , geometric series

What is the value of the expression \[ \frac {1}{\log_2 100!} \plus{} \frac {1}{\log_3 100!} \plus{} \frac {1}{\log_4 100!} \plus{} \cdots \plus{} \frac {1}{\log_{100} 100!}? \]$ \textbf{(A)}\ 0.01 \qquad \textbf{(B)}\ 0.1 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 10$

2009 Iran Team Selection Test, 12

Tags: logarithms , LaTeX , combinatorics proposed , combinatorics

$ T$ is a subset of $ {1,2,...,n}$ which has this property : for all distinct $ i,j \in T$ , $ 2j$ is not divisible by $ i$ . Prove that : $ |T| \leq \frac {4}{9}n + \log_2 n + 2$

2021 Purple Comet Problems, 17

Tags: Purple Comet , logarithms

For real numbers $x$ let $$f(x)=\frac{4^x}{25^{x+1}}+\frac{5^x}{2^{x+1}}.$$ Then $f\left(\frac{1}{1-\log_{10}4}\right)=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2006 Putnam, A2

Tags: Putnam , factorial , algebra , polynomial , logarithms , function , probability

Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)

1991 AIME Problems, 4

Tags: logarithms , trigonometry , function

How many real numbers $x$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$?

2004 China Team Selection Test, 2

Tags: floor function , logarithms , number theory

Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$

1996 Brazil National Olympiad, 3

Tags: logarithms , combinatorics proposed , combinatorics

Let $f(n)$ be the smallest number of 1s needed to represent the positive integer $n$ using only 1s, $+$ signs, $\times$ signs and brackets $(,)$. For example, you could represent 80 with 13 1s as follows: $(1+1+1+1+1)(1+1+1+1)(1+1+1+1)$. Show that $3 \log(n) \leq \log(3)f(n) \leq 5 \log(n)$ for $n > 1$.

1999 AIME Problems, 13

Tags: probability , logarithms , number theory , relatively prime

Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$

2010 Today's Calculation Of Integral, 659

Tags: calculus , integration , logarithms , geometric series , calculus computations

Evaluate $\int_0^1 \frac{\ln (x+2)}{x+1}dx.$

PEN E Problems, 24

Tags: inequalities , floor function , logarithms , limit , number theory , prime numbers

Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.

2000 Moldova National Olympiad, Problem 5

Tags: function , logarithms , induction , algebra unsolved , algebra

Let $ p$ be a positive integer. Define the function $ f: \mathbb{N}\to\mathbb{N}$ by $ f(n)\equal{}a_1^p\plus{}a_2^p\plus{}\cdots\plus{}a_m^p$, where $ a_1, a_2,\ldots, a_m$ are the decimal digits of $ n$ ($ n\equal{}\overline{a_1a_2\ldots a_m}$). Prove that every sequence $ (b_k)^\infty_{k\equal{}0}$ of positive integer that satisfy $ b_{k\plus{}1}\equal{}f(b_k)$ for all $ k\in\mathbb{N}$, has a finite number of distinct terms. $ \mathbb{N}\equal{}\{1,2,3\ldots\}$

2010 Today's Calculation Of Integral, 615

Tags: calculus , integration , logarithms , derivative , calculus computations

For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$ [i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i]

2007 Today's Calculation Of Integral, 198

Tags: calculus , integration , logarithms , trigonometry , calculus computations

Compare the values of the following definite integrals. \[\int_{0}^{\infty}\ln \left(x+\frac{1}{x}\right)\frac{dx}{1+x^{2}},\ \ \int_{0}^{\frac{\pi}{2}}\left(\frac{\theta}{\sin \theta}\right)^{2}d\theta\]

2010 Today's Calculation Of Integral, 637

Tags: calculus , integration , trigonometry , logarithms , calculus computations

For a non negative integer $n$, set t $I_n=\int_0^{\frac{\pi}{4}} \tan ^ n x\ dx$ to answer the following questions: (1) Calculate $I_{n+2}+I_n.$ (2) Evaluate the values of $I_1,\ I_2$ and $I_3.$ 1978 Niigata university entrance exam

IV Soros Olympiad 1997 - 98 (Russia), 11.7

Tags: algebra , inequalities , logarithms

Solve the inequality $$\log_{\frac12} x\ge 16^x$$

2014 ELMO Shortlist, 9

Tags: logarithms , number theory proposed , number theory

Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]

2004 South East Mathematical Olympiad, 3

Tags: logarithms , algebra , polynomial , inequalities , algebra unsolved

(1) Determine if there exists an infinite sequence $\{a_n\}$ with positive integer terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$. (2) Determine if there exists an infinite sequence $\{a_n\}$ with positive irrational terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$.