Practice (EndingDigits,TheDivideByNineMethod,MODBasic)
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Let $x_1$ and $x_2$ be the two roots of $x^2 - 3mx +2(m-1)=0$. If $\frac{1}{x_1}+\frac{1}{x_2}=\frac{3}{4}$, what is the value of $m$?
Let $x_1$ and $x_2$ be the two roots of $2x^2 -7x -4=0$, compute the values of the following expressions using as many different ways as possible.
(1) $x_1^2 + x_2^2$
(2) $(x_1+1)(x_2+1)$
(3) $\mid x_1 - x_2 \mid$
If one root of $x^2 + \sqrt{2}x + a = 0$ is $1-\sqrt{2}$, find the other root as well as the value of $a$.
Consider the equation $x^2 +(m-2)x + \frac{1}{2}m-3=0$.
(1) Show that this equation always have two distinct real roots
(2) Let $x_1$ and $x_2$ be its roots. If $x_1+x_2=m+1$, what is the value of $m$?
If $n>0$ and $x^2 -(m-2n)x + \frac{1}{4}mn=0$ has two equal positive real roots, what is the value of $\frac{m}{n}$?
If real number $m$ and $n$ satisfy $mn\ne 1$ and $19m^2+99m+1=0$ and $19+99n+n^2=0$, what is the value of $\frac{mn+4m+1}{n}$?
Let $x_1$ and $x_2$ be two real roots of $m^2x^2 +2(3-m)x+1=0$. If $m=\frac{1}{x_1}+\frac{1}{x_2}$, what is the value of $m$?
Let $x_1$ and $x_2$ be the two real roots of the equation $x^2 - 2(k+1)x+k^2 + 2 = 0$. If $(x_1+1)(x_2+1) =8$, find the value of $k$
Let $x_1$ and $x_2$ be the two real roots of the equation $x^2 - 2mx + (m^2+2m+3)=0$. Find the minimal value of $x_1^2 + x_2^2$.
How many integer pairs $(a,b)$ with $1 < a, b\le 2015$ are there such that $log_a b$ is an integer?
Define the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n =\frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. What is the least $k$ such that $a_k < a_{k-1}$?
A word is an ordered, non-empty sequence of letters, such as word or wrod. How many distinct words can be made from a subset of the letters $\textit{c, o, m, b, o}$, where each letter in the list is used no more than the number of times it appears?
What is the $22^{nd}$ positive integer $n$ such that $22^n$ ends in a $2$?
Find the sum of all positive integers $n$ such that the least common multiple of $2n$ and $n^2$ equals $(14n - 24)$?
What is the largest positive integer $n$ less than $10,000$ such that in base 4, $n$ and $3n$ have the same number of digits; in base 8, $n$ and $7n$ have the same number of digits; and in base 16, $n$ and $15n$ have the same number of digits? Express your answer in base 10.
What is the smallest positive integer $n$ such that $20\equiv n^{15} \pmod{29}$?
Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3\le p<100$, and $1\le a < p$ such that the sum $a+a^2+a^3+ \cdots + a^{(p-2)!}$ is not divisible by $p$?
Alice places down $n$ bishops on a $2015\times 2015$ chessboard such that no two bishops are attacking each other. (Bishops attack each other if they are on a diagonal.)
- Find, with proof, the maximum possible value of $n$.
- For this maximal $n$, find, with proof, the number of ways she could place her bishops on the chessboard.
For every integer $n$, let $m$ denote the integer made up of the last four digit of $n^{2015}$. Consider all positive integer $n < 10000$, let $A$ be the number of cases when $n > m$, and $B$ be the number of cases when $n < m$. Compute $A-B$.
In the diagram, AB is the diameter of a semicircle, D is the midpoint of the semicircle, angle $BAC$ is a right angle, $AC=AB$, and $E$ is the intersection of $AB$ and $CD$. Find the ratio between the areas of the two shaded regions.
Let $\frac{p}{q}=1+ \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{100000}$ where $p$ and $p$ are both positive integers and do not have common divisor greater than 1. How many ending zeros does $q$ have?
Suppose that $(u_n)$ is a sequence of real numbers satisfying $u_{n+2}=2u_{n+1}+u_n$, and that $u_3=9$ and $u_6=128$. What is $u_{2015}$?
If for any integer $k\ne 27$ and $\big(a-k^{2015}\big)$ is divisible by $(27-k)$, what is the last two digits of $a$?
Let $P(x)$ be a polynomial with integer coefficients. Show that $P(7)=5$ and $P(15)=9$ cannot hold simultaneously.
Let $m$ be a positive odd integer, $m\ge 2$. Find the smallest positive integer $n$ such that $2^{2015}$ divides $m^n-1$.