Found problems: 85335
Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality:
$$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$
[i]Proposed by Mohammad Jafari[/i]
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integers $a$ and $b$, we have: $$f(a^2+ab)+f(b^2+ab)=(a+b)f(a+b).$$
Proposed by Heidar Shushtari
In the decimal expression of a $ 2009$-digit natural number there are only the digits $ 5$ and $ 8$. Prove that we can get a $ 2008$-digit number divisible by $ 11$ if we remove just one digit from the number.
Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base $10$)?
[i]Proposed byMuztaba Syed[/i]
Let $n > 1$ be a positive integer. Prove that every term of the sequence
$$
n - 1, n^n - 1, n^{n^2} - 1, n^{n^3} - 1, \dots
$$
has a prime divisor that does not divide any of the previous terms.
Compute the remainder when $$1002003004005006007008009$$ is divided by $13.$
The product of several distinct positive integers is divisible by ${2006}^{2}$. Determine the minimum value the sum of such numbers can take.
$ n$ sets $ S_1$, $ S_2$ $ \cdots$, $ S_n$ consists of non-negative numbers. $ x_i$ is the sum of all elements of $ S_i$, prove that there is a natural number $ k$, $ 1<k<n$, and:
\[ \sum_{i\equal{}1}^n x_i < \frac{1}{k\plus{}1} \left[ k \cdot \frac{n(n\plus{}1)(2n\plus{}1)}{6} \minus{} (k\plus{}1)^2 \cdot \frac{n(n\plus{}1)}{2} \right]\]
and there exists subscripts $ i$, $ j$, $ t$, and $ l$ (at least $ 3$ of them are distinct) such that $ x_i \plus{} x_j \equal{} x_t \plus{} x_l$.
A circle with radius $4$ and $251$ distinct points inside the circle are given. Show that it is possible to draw a circle with radius $1$ and containing at least $11$ of these points.
Let $n\ge 2$ be an integer. Solve in reals:
\[|a_1-a_2|=2|a_2-a_3|=3|a_3-a_4|=\cdots=n|a_n-a_1|.\]
A pirate ship spots, $10$ nautical miles to the east, an oblivious caravel sailing $60$ south of west at a steady $12 \text{ nm/hour}$. What is the minimum speed that the pirate ship must maintain at to be able to catch the caravel?
We say two sequence of natural numbers A=($a_1,...,a_n$) , B=($b_1,...,b_n$)are the exchange and we write $A\sim B$.
if $503\vert a_i - b_i$ for all $1\leq i\leq n$.
also for natural number $r$ : $A^r$ = ($a_1^r,a_2^r,...,a_n^r$).
Prove that there are natural number $k,m$ such that :
$i$)$250 \leq k $
$ii$)There are different permutations $\pi _1,...,\pi_k$ from {$1,2,3,...,502$} such that for $1\leq i \leq k-1$ we have $\pi _i^m\sim \pi _{i+1}$
(15 points)
Determine all natural integers $n$ for which there is no triplet $(a, b, c)$ of natural numbers such that:
$$n = \frac{a \cdot \,\,lcm(b, c) + b \cdot lcm \,\,(c, a) + c \cdot lcm \,\, (a, b)}{lcm \,\,(a, b, c)}$$
Let $m$ be a positive integer and $x_0, y_0$ integers such that $x_0, y_0$ are relatively prime, $y_0$ divides $x_0^2+m$, and $x_0$ divides $y_0^2+m$. Prove that there exist positive integers $x$ and $y$ such that $x$ and $y$ are relatively prime, $y$ divides $x^2 + m$, $x$ divides $y^2 + m$, and $x + y \leq m+ 1.$
Find all natural triples $(a,b,c)$, such that:
$a - )\,a \le b \le c$
$b - )\,(a,b,c) = 1$
$c - )\,\left. {{a^2}b} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{b^2}c} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{c^2}a} \right|{a^3} + {b^3} + {c^3}$.
Let $f: \mathbb{R}\to\mathbb{R}$ is a function such that $f( \cot x ) = \cos 2x+\sin 2x$ for all $0 < x < \pi$. Define $g(x) = f(x) f(1-x)$ for $-1 \leq x \leq 1$. Find the maximum and minimum values of $g$ on the closed interval $[-1, 1].$
Let $P_k$ be a point whose $x$-coordinate is $1+\frac{k}{n}\ (k=1,\ 2,\ \cdots,\ n)$ on the curve $y=\ln x$. For $A(1,\ 0)$, find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n} \overline{AP_k}^2.$
In each cell of an \(n \times n\) grid, one of the numbers \(0\), \(1,\) or \(2\) must be written. Determine all positive integers \(n\) for which there exists a way to fill the \(n \times n\) grid such that, when calculating the sum of the numbers in each row and each column, the numbers \(1, 2, \ldots, 2n\) are obtained in some order.
An integer-valued function $f$ satisfies $f(2) = 4$ and $f(mn) = f(m)f(n)$ for all integers $m$ and $n$. If $f$ is an increasing function, determine $f(2015)$.
You have an infinite stack of T-shaped tetrominoes (composed of four squares of side length 1), and an n × n board. You are allowed to place some tetrominoes on the board, possibly rotated, as long as no two tetrominoes overlap and no tetrominoes extend off the board. For which values of n can you cover the entire board?
There are$n$ balls numbered $1,2,\cdots,n$, respectively. They are painted with $4$ colours, red, yellow, blue, and green, according to the following rules:
First, randomly line them on a circle.
Then let any three clockwise consecutive balls numbered $i, j, k$, in order.
1) If $i>j>k$, then the ball $j$ is painted in red;
2) If $i<j<k$, then the ball $j$ is painted in yellow;
3) If $i<j, k<j$, then the ball $j$ is painted in blue;
4) If $i>j, k>j$, then the ball $j$ is painted in green.
And now each permutation of the balls determine a painting method.
We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods.
Find out the number of all distinct painting methods.
Let \( \mathbb R \) be the set of real numbers. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for all real numbers \( x \) and \( y \), the following equation holds:$$\big (x^2-y^2\big )f\big (xy\big )=xf\big (x^2y\big )-yf\big (xy^2\big ).$$
A triangle \( ABC \) is given with centers \( O \) and \( I \) of the circumscribed and inscribed circles, respectively. Point \( A_1 \) is the reflection of \( A \) with respect to \( I \). Point \( A_2 \) is such that lines \( BA_1 \) and \( BA_2 \) are symmetric with respect to \( BI \), and lines \( CA_1 \) and \( CA_2 \) are symmetric with respect to \( CI \). Prove that \( AO^2 = |A_2O^2 - A_2I^2| \).
Alice, Bob, and Carol are playing a game. Each turn, one of them says one of the $3$ players' names, chosen from {Alice, Bob, Carol} uniformly at random. Alice goes first, Bob goes second, Carol goes third, and they repeat in that order. Let $E$ be the expected number of names that are have been said when, for the first time, all $3$ names have been said twice. If $E = \tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. (Include the last name to be said twice in your count.)