Practice
Solve \(x^2-7x+12=0\).
Factor.
\((x-3)(x-4)=0\). Answer: \(3,4\).
\(u_0=1\), \(u_{n+1}=u_n+2n\). Find \(u_n\).
Sum the differences.
\(u_n=1+\sum_{k=0}^{n-1}2k=1+n(n-1)\).
For \(x>0\), prove \(x+\frac{1}{x}\ge2\).
AM-GM.
\(x+\frac{1}{x}\ge2\sqrt{1}=2\).
Find all linear \(f(x)=ax+b\) such that \(f(x+1)=f(x)+3\).
Compare coefficients.
We get \(a=3\), \(b\) arbitrary. Answer: \(f(x)=3x+b\).
Find integer roots of \(x^3-4x^2-x+4\).
Group terms.
\(x^2(x-4)-1(x-4)=(x-4)(x-1)(x+1)\). Roots: \(4,1,-1\).
\(P\) has degree at most \(2\), \(P(0)=1\), \(P(1)=4\), \(P(2)=9\). Find \(P(3)\).
Second differences are constant.
Differences \(3,5\), next \(7\). Answer \(16\).
If \(a,b>0\), \(a+b=4\), prove \(ab\le4\).
AM-GM.
\(\sqrt{ab}\le\frac{a+b}{2}=2\), hence \(ab\le4\).
\(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)\), \(f(2)=6\). Find \(f\left(\frac{5}{3}\right)\).
Find \(f(1)\).
\(f(1)=3\), so \(f(q)=3q\). Answer: \(5\).
Solve \(x+y+xy=11\), \(x^2+y^2=13\).
Let \(s=x+y\), \(p=xy\).
\(s+p=11\), \(s^2-2p=13\). We get \(s=5\), \(p=6\). Answer: \((2,3),(3,2)\).
Find integer roots of \(x^3-x^2-10x+10\).
Grouping.
\((x-1)(x^2-10)\), the only integer root is \(1\).
If \(a,b,c>0\), \(a+b+c=6\), prove \(abc\le8\).
AM-GM.
\(\sqrt[3]{abc}\le2\), hence \(abc\le8\).
\(f:\mathbb Z\to\mathbb Z\), \(f(m+n)=f(m)+f(n)+mn\), \(f(1)=1\). Find \(f(n)\).
Compare with \(\frac{n(n-1)}{2}\).
\(f(n)-\frac{n(n-1)}{2}\) is additive and equals \(1\) at \(n=1\). Answer: \(f(n)=\frac{n(n+1)}{2}\).
Find positive integer solutions of \(x^2-y^2=45\).
\((x-y)(x+y)=45\).
Pairs \(1,45\), \(3,15\), \(5,9\) give \((23,22),(9,6),(7,2)\).
\(u_0=3\), \(u_1=7\), \(u_{n+2}=2u_{n+1}-u_n\). Find \(u_n\).
The difference is constant.
\(u_{n+1}-u_n=4\), hence \(u_n=4n+3\).
Prove \(\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}\ge\frac{x+y+z}{2}\) for \(x,y,z>0\).
Cauchy.
The left side is at least \(\frac{(x+y+z)^2}{2(x+y+z)}\).
Find linear \(f\) such that \(f(f(x))=x+2\), \(f(0)>0\).
Let \(f(x)=ax+b\).
\(a^2=1\), \(b(a+1)=2\). Only \(a=1,b=1\) works. Answer: \(f(x)=x+1\).
Prove that \(x^2+y^2=4k+3\) has no integer solutions.
Modulo \(4\).
Squares give \(0,1\); the sum cannot give \(3\).
For which integers \(a\) does \(x^2+ax+20\) have integer roots?
The roots have product \(20\).
Sums of factor pairs: \(\pm21,\pm12,\pm9\). Thus \(a\in\{\pm21,\pm12,\pm9\}\).
If \(x,y,z>0\), \(xyz=1\), prove \((1+x)(1+y)(1+z)\ge8\).
\(1+x\ge2\sqrt{x}\).
Multiplying the three estimates gives \(8\sqrt{xyz}=8\).
\(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)+2xy\), \(f(1)=1\). Find \(f\).
Subtract \(x^2\).
\(g(x)=f(x)-x^2\) is additive, \(g(1)=0\), hence \(g=0\). Answer: \(f(x)=x^2\).
Find all integers \(n\) for which \(n^2+3n+5\) is divisible by \(n+1\).
Use \(n\equiv-1\pmod{n+1}\).
The remainder is \(1-3+5=3\). Need \(n+1\mid3\). Answer: \(n=0,-2,2,-4\).
\(P\) has degree at most \(3\), values \(1,2,5,10\) at \(0,1,2,3\). Find \(P(4)\).
Third differences are constant.
Differences: \(1,3,5\); second differences \(2,2\); next first difference \(7\). Answer \(17\).
Find positive integer solutions of \(\frac1x+\frac1y=\frac17\).
\((x-7)(y-7)=49\).
We get \((8,56),(14,14),(56,8)\).
Find linear \(f\) if \(f(x+y)=f(x)+f(y)+4\).
Let \(f=ax+b\).
\(b=2b+4\), so \(b=-4\). Answer: \(f(x)=ax-4\).
Prove that \(x^2+y^2=8xy\) has no positive integer solutions.
Vieta descent.
For a minimal solution \(x>y\). The other root \(x'=8y-x=\frac{y^2}{x}\) is positive, integral, and smaller than \(y\). Contradiction.
Find \(a,b,c\) if \(a+b+c=0\), \(a^2+b^2+c^2=18\), \(a^3+b^3+c^3=0\).
\(3abc=0\).
\(abc=0\), one variable is \(0\), the other two are opposite and their squares give \(18\). Answer: permutations of \((3,-3,0)\).
Prove for \(a,b,c\ge0\): \(a^3+b^3+c^3+3abc\ge\sum_{\mathrm{sym}}a^2b\).
This is Schur's inequality.
Let \(a\ge b\ge c\). The difference equals \(\sum a(a-b)(a-c)=(a-b)^2(a+b-c)+c(a-c)(b-c)\ge0\).
Increasing \(f:\mathbb R\to\mathbb R\), \(f(x+f(y))=f(x)+y\). Find \(f\).
First \(y=0\), then \(x=0\).
\(f(0)=0\), \(f(f(y))=y\). Putting \(y=f(t)\), we get additivity. An increasing additive function is \(cx\); from \(c^2=1\) and increasing, \(c=1\). Answer \(f(x)=x\).
Find all integers \(n\) for which \(n^4+4\) is prime.
Sophie Germain factorization.
\(n^4+4=(n^2-2n+2)(n^2+2n+2)\). Prime only for \(n=\pm1\).
A polynomial of degree \(\le3\) has values \(0,1,8,27\) at \(0,1,2,3\). Find \(P(4)\).
This is a cubic sequence.
Differences \(1,7,19\), second \(6,12\), third \(6\). Next: second \(18\), first \(37\), value \(64\).
Let \(x,y,z>0\). Prove \(\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\ge\frac{x+y+z}{2}\).
Cauchy.
By Cauchy, the left side is at least \(\frac{(x+y+z)^2}{2(x+y+z)}\).
Prove that \(x^2+y^2+z^2=8k+7\) has no integer solutions.
Squares modulo \(8\).
Squares give \(0,1,4\); the sum of three such residues is not \(7\).