HarvardMIT/md/en-122-2009-feb-alg-solutions.md

$12^{\text {th }}$ Annual Harvard-MIT Mathematics Tournament

Saturday 21 February 2009

Individual Round: Algebra Test

1. [3] If $a$ and $b$ are positive integers such that $a^{2}-b^{4}=2009$, find $a+b$.

Answer: 47 Solution: We can factor the equation as $\left(a-b^{2}\right)\left(a+b^{2}\right)=41 \cdot 49$, from which it is evident that $a=45$ and $b=2$ is a possible solution. By examining the factors of 2009 , one can see that there are no other solutions. 2. [3] Let $S$ be the sum of all the real coefficients of the expansion of $(1+i x)^{2009}$. What is $\log _{2}(S)$ ?

Answer: 1004 Solution: The sum of all the coefficients is $(1+i)^{2009}$, and the sum of the real coefficients is the real part of this, which is $\frac{1}{2}\left((1+i)^{2009}+(1-i)^{2009}\right)=2^{1004}$. Thus $\log _{2}(S)=1004$. 3. [4] If $\tan x+\tan y=4$ and $\cot x+\cot y=5$, compute $\tan (x+y)$.

Answer: 20 Solution: We have $\cot x+\cot y=\frac{\tan x+\tan y}{\tan x \tan y}$, so $\tan x \tan y=\frac{4}{5}$. Thus, by the tan sum formula, $\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}=20$. 4. [4] Suppose $a, b$ and $c$ are integers such that the greatest common divisor of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x+1$ (in the ring of polynomials in $x$ with integer coefficients), and the least common multiple of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x^{3}-4 x^{2}+x+6$. Find $a+b+c$. Answer: $\quad-6$ Solution: Since $x+1$ divides $x^{2}+a x+b$ and the constant term is $b$, we have $x^{2}+a x+b=(x+1)(x+b)$, and similarly $x^{2}+b x+c=(x+1)(x+c)$. Therefore, $a=b+1=c+2$. Furthermore, the least common multiple of the two polynomials is $(x+1)(x+b)(x+b-1)=x^{3}-4 x^{2}+x+6$, so $b=-2$. Thus $a=-1$ and $c=-3$, and $a+b+c=-6$. 5. [4] Let $a, b$, and $c$ be the 3 roots of $x^{3}-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$.

Answer: $\quad-2$ Solution: We can substitute $x=y-1$ to obtain a polynomial having roots $a+1, b+1, c+1$, namely, $(y-1)^{3}-(y-1)+1=y^{3}-3 y^{2}+2 y+1$. The sum of the reciprocals of the roots of this polynomial is, by Viete's formulas, $\frac{2}{-1}=-2$. 6. [5] Let $x$ and $y$ be positive real numbers and $\theta$ an angle such that $\theta \neq \frac{\pi}{2} n$ for any integer $n$. Suppose

$$\frac{\sin \theta }{x}=\frac{\cos \theta }{y}\backslash frac\{\backslash sin\; \backslash theta\}\{x\}=\backslash frac\{\backslash cos\; \backslash theta\}\{y\}$$

and

$$\frac{{\cos }^4\theta }{x^4}+\frac{{\sin }^4\theta }{y^4}=\frac{97\sin 2\theta }{x^{3}y+y^{3}x}\backslash frac\{\backslash cos\; ^\{4\}\; \backslash theta\}\{x^\{4\}\}+\backslash frac\{\backslash sin\; ^\{4\}\; \backslash theta\}\{y^\{4\}\}=\backslash frac\{97\; \backslash sin\; 2\; \backslash theta\}\{x^\{3\}\; y+y^\{3\}\; x\}$$

Compute $\frac{x}{y}+\frac{y}{x}$. Answer: 4 Solution: From the first relation, there exists a real number $k$ such that $x=k \sin \theta$ and $y=k \cos \theta$. Then we have

$$\frac{{\cos }^4\theta }{{\sin }^4\theta }+\frac{{\sin }^4\theta }{{\cos }^4\theta }=\frac{194\sin \theta \cos \theta }{\sin \theta \cos \theta ({\cos }^2\theta +{\sin }^2\theta )}=194\backslash frac\{\backslash cos\; ^\{4\}\; \backslash theta\}\{\backslash sin\; ^\{4\}\; \backslash theta\}+\backslash frac\{\backslash sin\; ^\{4\}\; \backslash theta\}\{\backslash cos\; ^\{4\}\; \backslash theta\}=\backslash frac\{194\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash theta\}\{\backslash sin\; \backslash theta\; \backslash cos\; \backslash theta\backslash left(\backslash cos\; ^\{2\}\; \backslash theta+\backslash sin\; ^\{2\}\; \backslash theta\backslash right)\}=194$$

Notice that if $t=\frac{x}{y}+\frac{y}{x}$ then $\left(t^{2}-2\right)^{2}-2=\frac{\cos ^{4} \theta}{\sin ^{4} \theta}+\frac{\sin ^{4} \theta}{\cos ^{4} \theta}=194$ and so $t=4$. 7. [5] Simplify the product

$$\prod _{m=1}^{100}\prod _{n=1}^{100}\frac{x^{n+m}+x^{n+m+2}+x^{2n+1}+x^{2m+1}}{x^{2n}+2x^{n+m}+x^{2m}}\backslash prod\_\{m=1\}^\{100\}\; \backslash prod\_\{n=1\}^\{100\}\; \backslash frac\{x^\{n+m\}+x^\{n+m+2\}+x^\{2\; n+1\}+x^\{2\; m+1\}\}\{x^\{2\; n\}+2\; x^\{n+m\}+x^\{2\; m\}\}$$

Express your answer in terms of $x$. Answer: $x^{9900}\left(\frac{1+x^{100}}{2}\right)^{2}\left(\right.$ OR $\left.\frac{1}{4} x^{9900}+\frac{1}{2} x^{10000}+\frac{1}{4} x^{10100}\right)$ Solution: We notice that the numerator and denominator of each term factors, so the product is equal to

$$\prod _{m=1}^{100}\prod _{n=1}^{100}\frac{(x^m+x^{n+1})(x^{m+1}+x^n)}{{(x^m+x^n)}^2}\backslash prod\_\{m=1\}^\{100\}\; \backslash prod\_\{n=1\}^\{100\}\; \backslash frac\{\backslash left(x^\{m\}+x^\{n+1\}\backslash right)\backslash left(x^\{m+1\}+x^\{n\}\backslash right)\}\{\backslash left(x^\{m\}+x^\{n\}\backslash right)^\{2\}\}$$

Each term of the numerator cancels with a term of the denominator except for those of the form $\left(x^{m}+x^{101}\right)$ and $\left(x^{101}+x^{n}\right)$ for $m, n=1, \ldots, 100$, and the terms in the denominator which remain are of the form $\left(x^{1}+x^{n}\right)$ and $\left(x^{1}+x^{m}\right)$ for $m, n=1, \ldots, 100$. Thus the product simplifies to

$${\left(\prod _{m=1}^{100}\frac{x^m+x^{101}}{x^1+x^m}\right)}^2\backslash left(\backslash prod\_\{m=1\}^\{100\}\; \backslash frac\{x^\{m\}+x^\{101\}\}\{x^\{1\}+x^\{m\}\}\backslash right)^\{2\}$$

Reversing the order of the factors of the numerator, we find this is equal to

as desired. 8. [7] If $a, b, x$, and $y$ are real numbers such that $a x+b y=3, a x^{2}+b y^{2}=7, a x^{3}+b y^{3}=16$, and $a x^{4}+b y^{4}=42$, find $a x^{5}+b y^{5}$. Answer: 20. Solution: We have $a x^{3}+b y^{3}=16$, so $\left(a x^{3}+b y^{3}\right)(x+y)=16(x+y)$ and thus

$$a{x}^4+b{y}^4+xy(a{x}^2+b{y}^2)=16(x+y)a\; x^\{4\}+b\; y^\{4\}+x\; y\backslash left(a\; x^\{2\}+b\; y^\{2\}\backslash right)=16(x+y)$$

It follows that

$$42+7xy=16(x+y)42+7\; x\; y=16(x+y)$$

From $a x^{2}+b y^{2}=7$, we have $\left(a x^{2}+b y^{2}\right)(x+y)=7(x+y)$ so $a x^{3}+b y^{3}+x y\left(a x^{2}+b y^{2}\right)=7(x+y)$. This simplifies to

$$16+3xy=7(x+y)16+3\; x\; y=7(x+y)$$

We can now solve for $x+y$ and $x y$ from (1) and (2) to find $x+y=-14$ and $x y=-38$. Thus we have $\left(a x^{4}+b y^{4}\right)(x+y)=42(x+y)$, and so $a x^{5}+b y^{5}+x y\left(a x^{3}+b y^{3}\right)=42(x+y)$. Finally, it follows that $a x^{5}+b y^{5}=42(x+y)-16 x y=20$ as desired. 9. [7] Let $f(x)=x^{4}+14 x^{3}+52 x^{2}+56 x+16$. Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the four roots of $f$. Find the smallest possible value of $\left|z_{a} z_{b}+z_{c} z_{d}\right|$ where ${a, b, c, d}={1,2,3,4}$. Answer: 8 Solution: Note that $\frac{1}{16} f(2 x)=x^{4}+7 x^{3}+13 x^{2}+7 x+1$. Because the coefficients of this polynomial are symmetric, if $r$ is a root of $f(x)$ then $\frac{4}{r}$ is as well. Further, $f(-1)=-1$ and $f(-2)=16$ so $f(x)$ has two distinct roots on $(-2,0)$ and two more roots on $(-\infty,-2)$. Now, if $\sigma$ is a permutation of ${1,2,3,4}$ : $\left|z_{\sigma(1)} z_{\sigma(2)}+z_{\sigma(3)} z_{\sigma(4)}\right| \leq \frac{1}{2}\left(z_{\sigma(1)} z_{\sigma(2)}+z_{\sigma(3)} z_{\sigma(4)}+z_{\sigma(4)} z_{\sigma(3)}+z_{\sigma(2)} z_{\sigma(1)}\right)$ Let the roots be ordered $z_{1} \leq z_{2} \leq z_{3} \leq z_{4}$, then by rearrangement the last expression is at least: $\frac{1}{2}\left(z_{1} z_{4}+z_{2} z_{3}+z_{3} z_{2}+z_{4} z_{1}\right)$ Since the roots come in pairs $z_{1} z_{4}=z_{2} z_{3}=4$, our expression is minimized when $\sigma(1)=1, \sigma(2)=$ $4, \sigma(3)=3, \sigma(4)=2$ and its minimum value is 8 . 10. [8] Let $f(x)=2 x^{3}-2 x$. For what positive values of $a$ do there exist distinct $b, c, d$ such that $(a, f(a))$, $(b, f(b)),(c, f(c)),(d, f(d))$ is a rectangle? Answer: $\left[\frac{\sqrt{3}}{3}, 1\right]$ Solution: Say we have four points $(a, f(a)),(b, f(b)),(c, f(c)),(d, f(d))$ on the curve which form a rectangle. If we interpolate a cubic through these points, that cubic will be symmetric around the center of the rectangle. But the unique cubic through the four points is $f(x)$, and $f(x)$ has only one point of symmetry, the point $(0,0)$. So every rectangle with all four points on $f(x)$ is of the form $(a, f(a)),(b, f(b)),(-a, f(-a)),(-b, f(-b))$, and without loss of generality we let $a, b>0$. Then for any choice of $a$ and $b$ these points form a parallelogram, which is a rectangle if and only if the distance from $(a, f(a))$ to $(0,0)$ is equal to the distance from $(b, f(b))$ to $(0,0)$. Let $g(x)=x^{2}+(f(x))^{2}=4 x^{6}-8 x^{4}+5 x^{2}$, and consider $g(x)$ restricted to $x \geq 0$. We are looking for all the values of $a$ such that $g(x)=g(a)$ has solutions other than $a$. Note that $g(x)=h\left(x^{2}\right)$ where $h(x)=4 x^{3}-8 x^{2}+5 x$. This polynomial $h(x)$ has a relative maximum of 1 at $x=\frac{1}{2}$ and a relative minimum of $25 / 27$ at $x=\frac{5}{6}$. Thus the polynomial $h(x)-h(1 / 2)$ has the double root $1 / 2$ and factors as $\left(4 x^{2}-4 x+1\right)(x-1)$, the largest possible value of $a^{2}$ for which $h\left(x^{2}\right)=h\left(a^{2}\right)$ is $a^{2}=1$, or $a=1$. The smallest such value is that which evaluates to $25 / 27$ other than $5 / 6$, which is similarly found to be $a^{2}=1 / 3$, or $a=\frac{\sqrt{3}}{3}$. Thus, for $a$ in the range $\frac{\sqrt{3}}{3} \leq a \leq 1$ the equation $g(x)=g(a)$ has nontrivial solutions and hence an inscribed rectangle exists.