Algebra — The Complete Master Pack for

IOQM–RMO
Comprehensive Algebra Reference (Identities, Theorems, Summations, Recurrences,
Inequalities, Number-Theoretic Tools, Strategies, Worked Examples)
Revised Edition: August 31, 2025
Prepared for IOQM–RMO learners

How this revision helps you. We keep the original structure and topics and add many more
theorems, formulas, and friendly explanations with small-step examples. Nothing is removed;
only expanded and simplified.

Contents
1 Preface and how to use this pack 2

2 Core algebraic identities and factorizations 3

2.1 Elementary identities (must-know) . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Special classic identities (extended list) . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Cyclotomic and roots-of-unity identities . . . . . . . . . . . . . . . . . . . . . . . 3
2.4 Handy rearrangements (algebra cleanups) . . . . . . . . . . . . . . . . . . . . . . 3

3 Polynomials: machinery and root properties 4

3.1 Factor & remainder theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Vieta’s formulas (general) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Newton’s identities (power sums) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.4 Rational roots & bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.5 Discriminant, multiplicity, derivatives . . . . . . . . . . . . . . . . . . . . . . . . 4
3.6 Irreducibility tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.7 Resultant (elimination idea) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.8 Lagrange interpolation (construction) . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Symmetric polynomials and reduction 5

4.1 Elementary symmetric basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2 Power sums and reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3 Cyclic vs symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

5 Summation techniques and discrete calculus 5

5.1 Elementary closed forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5.2 Telescoping & partial fractions (templates) . . . . . . . . . . . . . . . . . . . . . 5
5.3 Abel summation (summation by parts) . . . . . . . . . . . . . . . . . . . . . . . . 5
5.4 Reordering double sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5.5 Finite differences (detect degree) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6 Generating functions and combinatorial algebra 6

6.1 Ordinary generating functions (OGF) . . . . . . . . . . . . . . . . . . . . . . . . 6
6.2 Standard OGFs to remember . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6.3 Exponential generating functions (EGF) . . . . . . . . . . . . . . . . . . . . . . . 6
6.4 Binomial identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6.5 Binomial inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1
7 Inequalities and algebraic optimization 6
7.1 Mean chain and weighted AM–GM . . . . . . . . . . . . . . . . . . . . . . . . . . 6
7.2 Cauchy–Schwarz & Engel (Titu) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
7.3 Rearrangement & Chebyshev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
7.4 Schur, Muirhead, classic olympiad ones . . . . . . . . . . . . . . . . . . . . . . . . 7

8 Number-theoretic algebraic tools 7

8.1 LTE (Lifting The Exponent) — practical forms . . . . . . . . . . . . . . . . . . . 7
8.2 CRT, orders, and Fermat–Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
8.3 Roots-of-unity filter (residue selector) . . . . . . . . . . . . . . . . . . . . . . . . 7
8.4 Gaussian integers, sums of two squares . . . . . . . . . . . . . . . . . . . . . . . . 7

9 Algebraic transforms, substitutions and manipulations 8

9.1 Homogenize & normalize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
9.2 Substitution patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
9.3 Symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

10 Recurrence relations: methods and detailed recipes 8

10.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
10.2 Characteristic polynomial method (go-to tool) . . . . . . . . . . . . . . . . . . . . 8
10.3 Generating functions for recurrences . . . . . . . . . . . . . . . . . . . . . . . . . 8
10.4 Undetermined coefficients (non-homogeneous) . . . . . . . . . . . . . . . . . . . . 8
10.5 Annihilator method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
10.6 Matrix (companion) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
10.7 Variable-coefficient and combinatorial recurrences . . . . . . . . . . . . . . . . . . 9
10.8 Asymptotics (dominant root) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
10.9 Worked mini-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

11 Advanced identities, factorials and summations 9

11.1 More binomial sums (handy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
11.2 Stirling numbers and falling factorials . . . . . . . . . . . . . . . . . . . . . . . . 9
11.3 Legendre & Kummer (valuation tools) . . . . . . . . . . . . . . . . . . . . . . . . 9
11.4 Bernoulli numbers (Faulhaber) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

12 Problem-solving strategies: a detailed checklist (50+) 10

13 Worked problems and illustrative examples 10

14 Common pitfalls and final checks 11

1 Preface and how to use this pack

This Master Pack is a do-it-fast, understand-deep reference for IOQM/RMO algebra. Each tool
has:
• What it says (short statement),
• Why it’s useful (contest intuition),
• How to use (steps/checklist),
• Example (a quick demo).
P P
Notation. νp (n) is the exponent of prime p in n; sym and cyc denote symmetric and cyclic
sums; a polynomial is monic if leading coefficient is 1.

2
2 Core algebraic identities and factorizations
2.1 Elementary identities (must-know)
• Squares/cubes. a2 − b2 = (a − b)(a + b); a3 − b3 = (a − b)(a2 + ab + b2 ); a3 + b3 =
(a + b)(a2 − ab + b2 ).
• Difference of powers. For integer n ≥ 1,
n−1
X
n n
a − b = (a − b) an−1−k bk .
k=0

• Sum of even powers. If n is odd, an + bn = (a + b) n−1 k n−1−k bk .

P
k=0 (−1) a
1 − xn
• Geometric series. For x ̸= 1, 1 + x + · · · + xn−1 = .
1−x
• Algebraic expansions. (x + y + z)2 = x2 + y 2 + z 2 + 2(xy + yz + zx) and
X
(x + y + z)3 = x3 + y 3 + z 3 + 3 x2 y + 6xyz.
sym

Why useful? Quickly factor, telescope sums, or check divisibility.

Example 2.1 (Fast factor). Factor x4 − 1. Solution: (x2 − 1)(x2 + 1) = (x − 1)(x + 1)(x2 + 1).

2.2 Special classic identities (extended list)

Theorem 2.2 (Sophie Germain). a4 + 4b4 = (a2 − 2ab + 2b2 )(a2 + 2ab + 2b2 ).

Theorem 2.3 (Brahmagupta–Fibonacci (product of sums of two squares)). (a2 + b2 )(c2 + d2 ) =

(ac ∓ bd)2 + (ad ± bc)2 .

Theorem 2.4 (Aurifeuillean-type idea (template)). For certain n, x2n ±1 can factor nontrivially
over Z[x] beyond cyclotomic pieces. In contests, first try cyclotomic (xn ± 1) then check special
n (like powers of 2) for extra patterns.

2.3 Cyclotomic and roots-of-unity identities

Definition 2.5. Φn (x) is the monic polynomial whose roots are primitive n-th roots of unity.
Then Y
xn − 1 = Φd (x), deg Φn (x) = φ(n).
d|n

Useful facts.
Pn−1 km
•
P
ζ n =1 ζ = 0 for n > 1. More generally, k=0 ζ = 0 unless n | m, in which case it equals
n.
• For odd n, xn + 1 = d|2n, d∤n Φd (x).
Q

2.4 Handy rearrangements (algebra cleanups)

b2
• Completing square: x2 + bx = (x + 2b )2 − 4.
• Symmetric 2-variable: With s = x + y, p = xy, one has x2 + y 2 = s2 − 2p.
• Three variables: p = x + y + z, q = xy + yz + zx, r = xyz reduce many expressions.

3
3 Polynomials: machinery and root properties
3.1 Factor & remainder theorems
Proposition 3.1. P (c) = 0 ⇐⇒ (x − c) | P (x). The remainder on division by (x − c) equals
P (c).

3.2 Vieta’s formulas (general)

For monic xn + an−1 xn−1 + · · · + a0 with roots r1 , . . . , rn ,
X X Y
ri = −an−1 , ri rj = an−2 , . . . , ri = (−1)n a0 .
i<j

3.3 Newton’s identities (power sums)

k
P
Let Sk = i ri . Then

S1 + an−1 = 0, S2 + an−1 S1 + 2an−2 = 0, S3 + an−1 S2 + an−2 S1 + 3an−3 = 0, . . .

Use: Convert between power sums and coefficients.

3.4 Rational roots & bounds

p
Lemma 3.2 (Rational Root Test). If P ∈ Z[x] has root q in lowest terms, then p | a0 and
q | an .

Proposition 3.3 (Cauchy bound). For monic xn + cn−1 xn−1 + · · · + c0 , every root r satisfies
|r| ≤ 1 + max{|c0 |, . . . , |cn−1 |}.

Proposition 3.4 (Descartes’ rule of signs). #(positive real roots, counting multiplicity) ≤
#(sign changes of coefficients). Similarly for negative roots via P (−x).

3.5 Discriminant, multiplicity, derivatives

• Quadratic: ∆ = b2 − 4ac. ∆ = 0 indicates a repeated root.
• In general: r is a repeated root ⇐⇒ gcd(P, P ′ ) has positive degree.

3.6 Irreducibility tests

Theorem 3.5 (Eisenstein). If a prime p divides a0 , . . . , an−1 , does not divide an , and p2 ∤ a0 ,
then P is irreducible over Q.

Proposition 3.6 (Gauss). A primitive polynomial in Z[x] factors over Q[x] iff it factors over
Z[x].

Tip 3.7. After a shift x 7→ x + 1 (or x 7→ x + k), Eisenstein often becomes applicable.

3.7 Resultant (elimination idea)

The resultant Res(P, Q) is 0 iff P, Q share a root in C. For contest use, know that eliminating a
variable by resultants or by clever substitution can show finiteness or force constraints.

4
3.8 Lagrange interpolation (construction)
Given distinct xi and values yi , there is a unique polynomial P of degree < n with P (xi ) = yi :
n
X Y x − xj
P (x) = yi .
xi − xj
i=1 j̸=i

Trick. Degree-counting: if two polynomials of degree ≤ d agree at d + 1 points, they are equal.

4 Symmetric polynomials and reduction

4.1 Elementary symmetric basis
For x, y, z, set p = x + y + z, q = xy + yz + zx, r = xyz. Any symmetric polynomial in x, y, z
can be expressed in p, q, r.

4.2 Power sums and reductions

If x, y, z are roots of t3 − pt2 + qt − r = 0, then Sk = xk + y k + z k satisfies the recurrence induced
by the characteristic equation:

Sk = pSk−1 − qSk−2 + rSk−3 (k ≥ 3),

with S0 = 3, S1 = p, S2 = p2 − 2q.

4.3 Cyclic vs symmetric

Cyclic symmetry is weaker than full symmetry; reductions to p, q, r may fail for purely cyclic
problems. Treat carefully.

5 Summation techniques and discrete calculus

5.1 Elementary closed forms
n n n  2
X n(n + 1) X
2 n(n + 1)(2n + 1) X
3 n(n + 1)
k= , k = , k = .
2 6 2
k=1 k=1 k=1

5.2 Telescoping & partial fractions (templates)

 
1 1 1 1 1 1 1
= − , = − .
k(k + m) m k k+m k(k + 1) k k+1
Use: Write ak = bk − bk+1 so partial sums collapse.

5.3 Abel summation (summation by parts)

P
Let A(n) = k≤n ak . Then for sequences (ak ), (bk ),

n
X n−1
X
ak bk = A(n)bn − A(m − 1)bm−1 − A(k)(bk+1 − bk ).
k=m k=m

Use: When A(k) is simple/bounded and bk has monotone differences.

5
5.4 Reordering double sums
n X
X i n X
X n
f (i, j) = f (i, j).
i=1 j=1 j=1 i=j

Tip. Draw the i–j lattice to see the region.

5.5 Finite differences (detect degree)

∆f (k) = f (k + 1) − f (k). If ∆d+1 f ≡ 0, then f is a polynomial of degree ≤ d. This proves
Faulhaber-type formulas and helps guess closed forms.

6 Generating functions and combinatorial algebra

6.1 Ordinary generating functions (OGF)
n.
P
For (an ), A(x) = n≥0 an x Convolution in n becomes multiplication of OGFs.

6.2 Standard OGFs to remember

1 X 1 X n + m − 1 x
= xn , = xn , = Fibonacci OGF.
1−x (1 − x)m m−1 1 − x − x2
n≥0 n≥0

6.3 Exponential generating functions (EGF)

P an n
E(x) = n≥0 n! x (useful when labels matter).

6.4 Binomial identities

n   n  
X n n
X n k
=2 , (−1) = 0 (n ≥ 1),
k k
k=0 k=0
n   n     
X n X r s r+s
k = n2n−1 , = (Vandermonde),
k k n−k n
k=0 k=0
n    
X k n+1
= (hockey-stick).
m m+1
k=m

6.5 Binomial inversion

Pn n

Pn n−k n


If bn = k=0 k ak , then an = k=0 (−1) k bk .

7 Inequalities and algebraic optimization

7.1 Mean chain and weighted AM–GM
For nonnegative xi ,
RMS ≥ AM ≥ GM ≥ HM.
P
Weighted AM–GM: for wi ≥ 0, wi = 1,
Y w X
xi i ≤ wi xi (equality when xi equal where wi > 0).

6
7.2 Cauchy–Schwarz & Engel (Titu)
n
a2 ( ai )2
X X  X 2 X P
x2i yi2 ≥ x i yi , i
≥ P (bi > 0).
bi bi
i=1

7.3 Rearrangement & Chebyshev

P
If a1 ≤ · · · ≤ an and b1 ≤ · · · ≤ bn , then
ai bi is maximal when similarly
 ordered and
 minimal

1X 1X 1X
when oppositely ordered. Chebyshev (for similarly sorted): ai bi ≥ ai bi .
n n n

7.4 Schur, Muirhead, classic olympiad ones

For a, b, c ≥ 0, Schur (r = 1):
X
a3 + b3 + c3 + 3abc ≥ a2 b.
sym

Muirhead: if α majorizes β, then the symmetric sum at α is ≥ that at β (for nonnegative

a b c 3
variables). Nesbitt: For a, b, c > 0, + + ≥ .
b+c c+a a+b 2

8 Number-theoretic algebraic tools

8.1 LTE (Lifting The Exponent) — practical forms
Theorem 8.1 (Basic LTE (odd p)). Let p odd prime, p | a − b, p ∤ ab. Then for n ≥ 1,
νp (an − bn ) = νp (a − b) + νp (n) .
Theorem 8.2 (Sum form). If p | a + b, p ∤ ab, and n odd, then
νp (an + bn ) = νp (a + b) + νp (n) .
Remark 8.3 (The case p = 2). Handle separately. For example, if a ≡ b ≡ 1 (mod 2) and n even,
special adjustments are needed. Always check hypotheses!

8.2 CRT, orders, and Fermat–Euler

If gcd(a, m) = 1, then aφ(m) ≡ 1 (mod m). The multiplicative order ordm (a) divides φ(m).
Chinese Remainder Theorem: simultaneous congruences modulo coprime moduli have a unique
solution modulo the product.

8.3 Roots-of-unity filter (residue selector)

If ω is a primitive m-th root of unity, then
m−1
1 X −jr X
ω · f (ω j ) = [xn ] F (x),
m
j=0 n≡r (mod m)

which picks coefficients (or terms) with n ≡ r (mod m). Great for separating arithmetic
progression terms in sums.

8.4 Gaussian integers, sums of two squares

In Z[i], N (a + bi) = a2 + b2 . A prime p ≡ 1 (mod 4) splits in Z[i] and is a sum of two squares.
Useful for Diophantine equations with x2 + y 2 .

7
9 Algebraic transforms, substitutions and manipulations
9.1 Homogenize & normalize
If an inequality is homogeneous, you may set x + y + z = 1 (or xyz = 1, etc.) to simplify. If not
homogeneous, multiply by a positive symmetric expression to homogenize.

9.2 Substitution patterns

• Two variables: s = x + y, p = xy.
• Three variables: p = x + y + z, q = xy + yz + zx, r = xyz.
• Reciprocal: x 7→ 1/x for rational symmetric forms.
• “Equal-case” hint: If you suspect x = y at equality, set x = y to reduce.

9.3 Symmetry breaking

Assume order x ≥ y ≥ z to cut cases, or analyze boundary cases to guess extremals.

10 Recurrence relations: methods and detailed recipes

10.1 Classification
• Linear homogeneous, constant coefficients: an = c1 an−1 + · · · + ck an−k .
• Linear non-homogeneous: plus a known f (n).
• Variable-coefficient: coefficients depend on n.
• Nonlinear: need invariants/creative tricks.

10.2 Characteristic polynomial method (go-to tool)

Form P (r) = rk − c1 rk−1 − · · · − ck = 0. If roots ri are distinct,
X
an = αi rin .
i

If r has multiplicity s, include nj rn for j = 0, . . . , s − 1.

Example 10.1 (Repeated root). an = 2an−1 − an−2 with a0 = 0, a1 = 1. Here (r − 1)2 = 0, so

an = αn · 1n + β · 1n = αn + β. Fit a0 = 0 ⇒ β = 0, a1 = 1 ⇒ α = 1. Thus an = n.

10.3 Generating functions for recurrences

an xn , convert the recurrence into an equation for A(x), solve rationally, then
P
Define A(x) =
extract coefficients (partial fractions).

10.4 Undetermined coefficients (non-homogeneous)

Solve homogeneous part, then guess particular form similar to f (n):
• If f (n) is polynomial of degree d, try a polynomial of degree d.
• If f (n) = λn , try Cλn (multiply by ns if λ is a root of characteristic polynomial with
multiplicity s).

8
10.5 Annihilator method
Find Q(E) (in the shift operator Ean = an+1 ) that kills f (n); apply to both sides to get a
higher-order homogeneous equation.

10.6 Matrix (companion) method

Write vn+1 = M vn ; then vn = M n v0 . If diagonalizable, use M = P DP −1 ; otherwise use Jordan
form (polynomial factors nj appear).

10.7 Variable-coefficient and combinatorial recurrences

EGFs often help when factorial-like coefficients appear. For combinatorial recurrences, interpret
an as counting objects and derive closed forms (e.g., Catalan, Bell, Stirling numbers).

10.8 Asymptotics (dominant root)

n
For linear constant-coefficient recurrences, the root with largest |r| controls growth: an ∼ Crmax
(times a polynomial if repeated).

10.9 Worked mini-examples

Example 10.2 (Linear non-homogeneous). an = 2an−1 + n, a0 = 0. Homogeneous solution
(p)
C · 2n . Guess an = An + B. Substitute: An + B = 2(A(n − 1) + B) + n. Compare coefficients
⇒ A = −1, B = −2. Then an = C2n − n − 2, and a0 = 0 ⇒ C = 2. So an = 2n+1 − n − 2.

11 Advanced identities, factorials and summations

11.1 More binomial sums (handy)

n  
X n 2
k = n(n + 1)2n−2 ,
k
k=0
n  
k n 1 1
X
(−1) = ,
k k+1 n+1
k=0
n    
X r+k r+n+1
= .
k n
k=0

11.2 Stirling numbers and falling factorials

Falling factorial: xn = x(x − 1) · · · (x − n + 1).
n
X n
X
xn = S(n, k) xk , xn = s(n, k) xk ,
k=0 k=0

where S(n, k) are Stirling numbers of the second kind, s(n, k) the (signed) Stirling numbers of
the first kind.

11.3 Legendre & Kummer (valuation tools)

X n   
m
νp (n!) = , Kummer: νp = # of carries when adding n and m−n in base p.
pj n
j≥1

9
11.4 Bernoulli numbers (Faulhaber)
Sums of powers nk=1 k m are polynomials in n with Bernoulli numbers. Know up to m = 4 by
P
heart (given in Section 5 and standard tables).

12 Problem-solving strategies: a detailed checklist (50+)

1. Try 0, ±1 quickly; check parity and small mods (2, 3, 4, 5, 7).
2. If symmetric, try x = y = z or reduce to p, q, r.
3. Treat as quadratic in one variable; use discriminant (integer/perfect-square checks).
4. Use Vieta jumping when you see symmetric quadratic Diophantine patterns.
5. For sums, seek telescoping; attempt partial fractions.
6. Reorder double sums; visualize lattice region.
7. Use Abel summation when partial sums are simple.
8. Detect polynomial sequences via finite differences.
9. Use generating functions for recurrences/convolutions.
10. Bound with AM–GM/Cauchy; homogenize if needed; check equality cases to guess structure.
11. Apply Schur/Muirhead for symmetric inequalities.
12. For an ± bn valuations, try LTE (verify hypotheses carefully).
13. For irreducibility, try shift then Eisenstein.
14. Use orders/CRT to constrain exponents in congruences.
15. Consider matrix method for fast an or pattern-finding.
16. Try substitution x 7→ 1/x or s = x + y, p = xy to simplify.
17. Use interpolation/degree-counting to confirm polynomial identities.
18. Combine bounding with modular contradictions.
19. Break symmetry by ordering x ≥ y ≥ z or checking boundary cases.
20. Always re-check for extraneous roots after squaring/dividing.

13 Worked problems and illustrative examples

Example 13.1 (Discriminant finite search). Solve in integers x2 + xy + y 2 = 4. Solution. View
as quadratic in x: x2 + yx + (y 2 − 4) = 0. Discriminant ∆ = y 2 − 4(y 2 − 4) = −3y 2 + 16 must
be a perfect square ≥ 0. So 3y 2 + t2 = 16. Try small y (0, ±1, ±2) to find solutions.
Pn 1 1
Example 13.2 (Telescoping with partial fractions). Sn = k=1 k(k+2) . Since k(k+2) =
 
1 1 1
2 k − k+2 , sums collapse to
 
1 1 1 1
Sn = 1+ − − .
2 2 n+1 n+2
Example 13.3 (Vieta jumping skeleton). x2 − kxy + y 2 = 1. Fix y; the two roots in x satisfy
x + x′ = ky and xx′ = y 2 − 1. Jump from (x, y) to (x′ , y) and use minimality for contradictions
or infinite descent.
Example 13.4 (Undetermined coefficients, repeated root). an −2an−1 +an−2 = n. Homogeneous:
(h)
(r − 1)2 = 0 ⇒ an = αn + β. Try particular An + B: substitute to solve for A, B; fit a0 , a1 to
get full an .

10
 X a2 (a + b + c)2
Example 13.5 (Inequality via Engel). For a, b, c > 0, show ≥ . Apply
ab + bc ab + bc + ca
Engel with ai = a, bi = ab + bc cyclically.
3n − 1 . Since 3 | 23 − 1 and 3 ∤ 2, LTE gives


Example
13.6 (LTE quick count). Find ν3 2
ν3 23n − 1 = ν3 23 − 1 + ν3 (n) = 1 + ν3 (n).

14 Common pitfalls and final checks

• Never divide by an expression that might be 0 without splitting cases.
• After squaring, plug back to remove extraneous roots.
• For integer roots, ensure discriminant is a perfect square and parity fits.
• With LTE, confirm exact hypotheses (p | a − b or p | a + b; special p = 2 cases).
• In telescoping, check end terms to avoid off-by-one mistakes.
• For inequalities, state domain (nonnegativity) and check equality conditions.

Appendix A: Extra formula bank (quick lookup)

Algebra/Factorization
√ √
a4 + b4 = (a2 + 2ab + b2 )(a2 − 2ab + b2 ) (over R).
x4 + 4x2 + 1 = (x2 + 2x + 1) + (x2 + 1) = (x2 + 1)2 + 2x.

More binomial/finite sums

n   n  
k n k n
X X
(−1) k = 0, (−1) k 2 = 0 (n ≥ 2),
k k
k=0 k=0
n  2   n     
X n 2n X r+k s+n−k r+s+n+1
= , = .
k n k n−k n
k=0 k=0

Catalan numbers
  √ n
1 2n X
n 1− 1 − 4x X
Cn = , Cn x = , Cn+1 = Ci Cn−i .
n+1 n 2x
n≥0 i=0

Orders and primitive roots If p is prime and a is a primitive root mod p, then every nonzero
residue is some power of a. Useful to solve ax ≡ b (mod p).

End of the Revised Algebra Master Pack for IOQM–RMO.

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