Practice (TheColoringMethod)

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Let $A_n$ be the average of all the integers between 1 and 101 which are the multiples of $n$ . Which is the largest among $A_2, A_3, A_4, A_5$ and $A_6$?

$\textbf{Passing the Bridge}$

It is a dark and stormy night. Four people must evacuate from an island to the mainland. The only link is a narrow bridge which allows passage of two people at a time. Moreover, the bridge must be illuminated, and the four people have only one lantern among them. After each passage to the mainland, if there are still people on the island, someone must bring the lantern back. When they cross the bridge individually, the four people take $2$, $4$, $8$ and $16$ minutes, respectively. Crossing the bridge in pairs, the slower speed is used. What is the minimum time for the entire evacuation?


In triangle $ABC$, $E$ is a point on $AC$ and $F$ is a point on $AB$. $BE$ and $CF$ intersect at $D$. If the areas of triangles $BDF$, $BCD$ and $CDE$ are 3, 7 and 7 respectively, what is the area of the quadrilateral $AEDF$?

A regiment had 48 soldiers but only half of them had uniforms. During inspection, they form a 6 × 8 rectangle, and it was just enough to conceal in its interior everyone without a uniform. Later, some new soldiers joined the regiment, but again only half of them had uniforms. During the next inspection, they used a different rectangular formation, again just enough to conceal in its interior everyone without a uniform. How many new soldiers joined the regiment?

The sum of 2008 consecutive positive integers is a perfect square. What is the minimum value of the largest of these integers?

Find the largest positive integer $n$ such that $(3^{1024} - 1)$ is divisible by $2^n$.


A farmer has four straight fences, with respective lengths 1, 4, 7 and 8 metres. What is the maximum area of the quadrilateral the farmer can enclose?

In the diagram , $PA = QB = PC = QC = PD = QD = 1, CE = CF = EF$ and $EA = BF = 2AB$. Determine $BD$.


Each of the numbers 2, 3, 4, 5, 6, 7, 8 and 9 is used once to fill in one of the boxes in the equation below to make it correct. Of the three fractions being added, what is the value of the largest one?


Let $x$ be a positive number. Denote by $[x]$ the integer part of $x$ and by $\{x\}$ the decimal part of $x$. Find the sum of all positive numbers satisfying $5\{x\} + 0.2[x] = 25$.

A positive integer $n$ is said to be good if there exists a perfect square whose sum of digits in base $10$ is equal to $n$. For instance, $13$ is good because $7^2 = 49$ and $4 + 9 = 13$. How many good numbers are among $1, 2, 3, \cdots , 2007$?

A prime number is called an absolute prime if every permutation of its digits in base 10 is also a prime number. For example: 2, 3, 5, 7, 11, 13 (31), 17 (71), 37 (73) 79 (97), 113 (131, 311), 199 (919, 991) and 337 (373, 733) are absolute primes. Prove that no absolute prime contains all of the digits 1, 3, 7 and 9 in base 10.

Use each of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 exactly once to fill in the nine small circles in the Olympic symbol below, so that the numbers inside each large circle is 14.


There are 14 points of intersection in the seven-pointed star in the diagram. Label these points with the numbers 1, 2, 3, $\dots$, 14 such that the sum of the labels of the four points on each line is the same. Give one of solution.


Mary found a $3$-digit number that, when multiplied by itself, produced a number which ended in her original $3$-digit number. What is the sum of all the numbers which have this property?

Determine all positive integers $m$ and $n$ such that $m^2+1$ is a prime number and $10(m^2 + 1) = n^2 + 1$.

Find $8$ prime numbers, not necessarily distinct such that the sum of the squares of these numbers is $992$ less than $4$ times of the product of these numbers.

A base-10 three digit number $n$ is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of $n$ are both three-digit numerals?

Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?


In rectangle $ABCD$, we have $AB=8$, $BC=9$, $H$ is on $BC$ with $BH=6$, $E$ is on $AD$ with $DE=4$, line $EC$ intersects line $AH$ at $G$, and $F$ is on line $AD$ with $GF \perp AF$. Find the length of $GF$.


A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles?


Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?

Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$?


Find all right triangles whose sides' lengths are all integers, and areas equal circumstance numerically.

A triangle's perimeter is 2016, and the ratio of its three altitudes is 3:5:7. Find the area of this triangle.