Chapter
Theory
A mock olympiad tests method choice, not a single topic. Each variant contains an equation or factorization problem, a polynomial or sequence problem, an inequality, and a functional or mixed problem.
Before solving, classify the problem quickly: factors, substitution, Vieta, finite differences, AM-GM/Cauchy, substitution in a functional equation, or a modular check.
If the method is not visible after 2–3 minutes, write down the structure: repeated expressions, symmetric variables, integrality, and possible equality cases.
A system with \(x+y\), \(xy\) asks for Vieta. Several polynomial values ask for differences. Positive variables ask for inequalities. \(f(x+y)\) asks for \(0\) and \(1\).
Do not stay too long on one problem. In a mock variant, collect solvable ideas first, then return to the hard problems.
1. Is the method chosen? 2. Is the answer checked? 3. Are all integer cases included? 4. Is the equality case found? 5. Is the solution written without gaps?
Examples
Problem. Solve \(x^2-9x+20=0\).
\((x-4)(x-5)=0\), hence \(x=4\) or \(x=5\).
Problem. \(P(0)=1\), \(P(1)=4\), \(P(2)=9\), and \(\deg P\le2\). Find \(P(3)\).
The first differences are \(3,5\), so the next one is \(7\). Therefore \(P(3)=16\).
Problem. For \(x>0\), prove \(x+\frac{16}{x}\ge8\).
By AM-GM, \(x+\frac{16}{x}\ge2\sqrt{16}=8\).
Problem. \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)\), \(f(2)=10\). Find \(f\left(\frac{3}{5}\right)\).
\(f(1)=5\), hence \(f(q)=5q\). The answer is \(3\).
Problem. Prove that \(x^2+y^2=4k+3\) is impossible in integers.
Modulo \(4\), a square is only \(0\) or \(1\), so a sum of two squares cannot be \(3\).
Problem. \(u_0=0\), \(u_{n+1}-u_n=2n+1\). Find \(u_n\).
\(u_n=1+3+\cdots+(2n-1)=n^2\).
Problem. \(x+y=12\), \(xy=35\). Find \(x,y\).
The numbers are roots of \(t^2-12t+35=0\), so they are \(5\) and \(7\).
Problem. Why does \(x^2+y^2=5xy\) have no positive integer solutions?
In a minimal solution, the other root \(5y-x=\frac{y^2}{x}\) produces a smaller positive integer solution. This is a contradiction.
Problems
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 \(7\).
Ladders