Practice (TheColoringMethod)
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At Euler Middle School, $198$ students voted on two issues in a school referendum with the following results: $149$ voted in favor of the first issue and $119$ voted in favor of the second issue. If there were exactly $29$ students who voted against both issues, how many students voted in favor of both issues?
In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If $\tfrac{1}{3}$ of all the ninth graders are paired with $\tfrac{2}{5}$ of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?
Jeremy's father drives him to school in rush hour traffic in $20$ minutes. One day there is no traffic, so his father can drive him $18$ miles per hour faster and gets him to school in $12$ minutes. How far in miles is it to school?
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. What is the value of $X$?
A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?
Ralph went to the store and bought 12 pairs of socks for a total of $24. Some of the socks he bought cost $1 a pair, some of the socks he bought cost $3 a pair, and some of the socks he bought cost $4 a pair. If he bought at least one pair of each type, how many pairs of $1 socks did Ralph buy?
In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?
On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?
Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are $2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4,$ and $4.5$. If a slip with 2 goes into cup $E$ and a slip with 3 goes into cup $B$, then the slip with 3.5 must go into what cup?
A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a 76 game schedule. How many games does a team play within its own division?
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space?
To build roads between $16$ cities so that one can travel from any city to any other city by passing through at most one other city. What is the minimum number of roads that the city with the most road has?
Let $m$ and $n$ be two positive integers between $2$ and $99$, inclusive. Mr. $S$ knows their sum, and Mr. $P$ knows their product. Following are their conversations:
- Mr. $S$: I am certain that you don't know these two numbers individually. But I don't know them either.
- Mr. $P$: Yes, I didn't know. But I know them now.
- Mr. $S$: If this is the case, I know them now too.
What are the two numbers?
Factorize: $f(a)=4a^4-3a^3-2a^2+3a-2$
Factorize: $(ab+bc+ca)(a+b+c)-abc+(a+b)(b+c)(c+a)$
Factorize: $f(x,y,z)=(x+y+z)^5-x^5-y^5-z^5$
Factorize $f(x,y,z) = x^3+y^3 +z^3 - 3xyz$.
Simplify $$\frac{(y-z)^3 +(z-x)^3+(x-y)^3}{(y-z)(z-x)(x-y)}$$
If equation $x^2 - (1-2a)x+a^2-3 = 0$ has two distinct real roots, and equation $x^2 -2x+2a-1=0$ is not solvable in real numbers, find the values of $a$ such that the roots of the first equation are integers.
If one root of the equation $x^2 -6x+m^2-2m+5=0$ is $2$. Find the value of the other root and $m$.
If the equation $x^2+2(m-2)x + m^2 + 4 = 0 $ has two real roots, and the sum of their square is 21 more than their product, find the value of $m$.
Let $\alpha$ and $\beta$ be the two roots of $x^2 + 2x -5=0$. Evaluate $\alpha^2 + \alpha\beta + 2\alpha$.
If at least one real root of equation $x^2 - mx +5+m=0$ equals one root of $x^2 - (7m+1)x+13m+7=0$, compute the product of the four roots of these two equations.
If the difference of the two roots of the equation $x^2 + 6x + k=0$ is 2, what is the value of $k$?
If the two roots of $(a^2 -1)x^2 -(a+1)x+1=0$ are reciprocal, find the value of $a$.