Practice (TheColoringMethod)

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Solve the equation $$\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6$$


Find all pairs of integers $(x,y)$ such that $$x^3+y^3=(x+y)^2$$.

Find all real numbers $m$ such that $$x^2+my^2-4my+6y-6x+2m+8 \ge 0$$ for every pair of real numbers $x$ and $y$.


Does there exist a polynomial $P(x)$ such that $P(1)=2015$ and $P(2015)=2016$?

Show that $a=1+\sqrt{2}$ is irrational using the following steps: (a) Find a polynomial with integer coefficients that has $a$ as a root. (b) Use the Rational Root Theorem to show $a$ is irrational. Show that $\sqrt{2}+\sqrt{3}$ is irrational using the same steps.

Find $a$, $b$, so that $(x-1)^2$ divides $ax^4+bx^3+1$.

Find all polynomials with integer coefficients $p(x)$ that satisfy the following identity $$2p(2x)=p(3x)+p(x)$$.

Show that for each integer $n$ the polynomial $(\cos\theta+x\sin\theta)^n-\cos n\theta-x\sin n\theta$ is divisible by $x^2+1$

Show that $$\sum_{i=0}^n \binom{n}{i}^2 = \binom{2n}{n}$$.

Let $p(x) = x^3-3x+1$. Show that if a complex number $a$ is a root of $p(x)$, then $a^2-2$ is also a root.

Given $$P(x)=(1+x+x^2)^{100}=a_0+a_1x+\cdots+a_{200}x^{200}$$

Compute the following sums:

  • $S_1=a_0+a_1+a_2+a_3 +\cdots+a_{200}$
  • $S_2=a_0+a_2+a_4+a_6 +\cdots+a_{200}$.

Find the minimal value of $\sqrt{x^2 - 4x + 5} + \sqrt{x^2 +4x +8}$.

What is the value of \[\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?\]

Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?

A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?

Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?

Suppose that real number $x$ satisfies \[\sqrt{49-x^2}-\sqrt{25-x^2}=3\]. What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$?

When $7$ fair standard $6$-sided die are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as \[\frac{n}{6^{7}}\], where $n$ is a positive integer. What is $n$?

How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \[x+3y=3\]\[||x|-|y||=1\]

A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? [asy] draw((0,0)--(4,0)--(4,3)--(0,0)); label("$A$", (0,0), SW); label("$B$", (4,3), NE); label("$C$", (4,0), SE); label("$4$", (2,0), S); label("$3$", (4,1.5), E); label("$5$", (2,1.5), NW); fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray); [/asy]

What is the greatest integer less than or equal to \[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\]

Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [asy] draw(circle((0,0),13)); draw(circle((5,-6.2),5)); draw(circle((-5,-6.2),5)); label("$B$", (9.5,-9.5), S); label("$A$", (-9.5,-9.5), S); [/asy]

Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\overline{AB}$ and $\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$?

Let $S$ be a set of 6 integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a