Super Calculator / Super Calculadora

Quick Start

Welcome to the olympiad calculator

1 Open the side menu (☰) and select Special Functions for the classic calculator keypad (~40 functions).
2 Enter a number and press any function button.
3 If the function needs more values, a pending operation indicator appears on screen.
4 For exact olympiad tools (fractions, geometry, polynomials, …) open the menu → Olympiad Tools.
5 The side panel (tablet) or bottom panel (mobile) shows automatic analysis of every entered number.

1-parameter functions execute immediately. 2+ parameter functions show an indicator and wait for more values.

Parameter System

1 parameter — immediate

Enter number → Press function → Result

E.g.: φ(12) → enter 12, press φ → shows 4

2-3-4 parameters — fixed

Enter value → Function → Value → = → (repeat if needed). Auto-executes when all parameters are filled.

E.g.: C(10,3) → enter 10 → C(n,k) → 3 → =

N parameters — variable

Enter value → Function → Value → = (add more). Press the same function again to execute.

E.g.: GCD(12,18,24) → 12 → GCD → 18 → = → 24 → GCD

Pending operation indicator "C(10, _)" means k is missing. "GCD(12, 18, _) [= add, GCD solve]" is a variable-length operation.

High-Precision Mode

Optional toggle in Settings → "High precision mode" (with a 5–100 digit slider).

When ON, the transcendental/irrational functions — sin, cos, tan, asin, acos,
atan, ln, log, exp, √ and ∛ — are computed with CONSTRUCTIVE (computable) real
numbers: the value is exact internally and only rounded when displayed.

  • No floating-point error.   √2 (30 digits) = 1.41421356237309504880168872421
  • sin(45°) (30 digits) = 0.707106781186547524400844362105
  • Singularities are detected by construction: tan(90°) → "undefined"
    (instead of a misleading 1.6×10¹⁶).
  • All computation runs in a background isolate with a loader — the UI never
    freezes. (Powers that would explode to billions of digits, e.g.
    1.0006^99999999, are rejected instantly as "too large".)

Powered by a vendored copy of computable_reals (a Dart port of Hans Boehm's
constructive reals). See Settings → Open source licenses.

Number Theory

φφ(n) — Euler's Totient Function1 param▼

Counts how many integers from 1 to n are coprime with n (GCD(k,n)=1).

φ(n) = n × ∏(1 − 1/p) for each prime p | n

Examples

φ(1)=1
φ(9)=6 → [1,2,4,5,7,8]
φ(12)=4 → [1,5,7,11]
φ(p)=p−1 for prime p

Tips

Multiplicative: φ(mn)=φ(m)φ(n) if GCD(m,n)=1
Euler's theorem: a^φ(n)≡1 (mod n) if GCD(a,n)=1
Σ φ(d) for d|n = n

λλ(n) — Carmichael's λ Function1 param▼

Smallest m>0 such that a^m≡1 (mod n) for ALL a coprime to n. Always divides φ(n).

λ(p^k)=φ(p^k) if p odd λ(2)=1, λ(4)=2, λ(2^k)=2^(k−2) if k≥3 λ(n) = lcm of parts

Examples

λ(8)=2
λ(15)=lcm(2,4)=4
λ(p)=p−1

Tips

λ(n)|φ(n) always
λ(n)=φ(n) iff n has a primitive root

μμ(n) — Möbius Function1 param▼

Detects whether n is square-free and counts prime factors.

μ(1)=1 μ(n)=(−1)^k if n=p₁·…·pₖ (distinct primes) μ(n)=0 if p²|n

Examples

μ(1)=1
μ(6)=1
μ(30)=−1
μ(12)=0 (has 2²)

Tips

Möbius inversion: if g(n)=Σf(d), then f(n)=Σμ(d)g(n/d)
Σμ(d) for d|n = [n=1]

λLλL(n) — Liouville Function1 param▼

Completely multiplicative: λL(n)=(−1)^Ω(n).

λL(n) = (−1)^Ω(n)

Examples

λL(12)=(−1)³=−1 (Ω=3)
λL(36)=(−1)⁴=1 (Ω=4)

Tips

ΣλL(d) for d|n = 1 if n is a perfect square, 0 otherwise

ωω(n) — Distinct Prime Factors1 param▼

Counts the number of distinct primes dividing n.

ω(n) = k if n = p₁^a₁ × … × pₖ^aₖ

Examples

ω(12)=2 → [2,3]
ω(30)=3 → [2,3,5]
ω(p^k)=1

ΩΩ(n) — Prime Factors with Multiplicity1 param▼

Total number of prime factors counting repetitions.

Ω(n) = a₁ + a₂ + … + aₖ

Examples

Ω(12)=3 → 2×2×3
Ω(72)=5 → 2³×3² → 3+2
Ω(p)=1, Ω(p²)=2

σ₀σ₀(n) — Divisor Count1 param▼

Total number of positive divisors of n.

σ₀(n) = (a₁+1)(a₂+1)···(aₖ+1) if n=p₁^a₁×…

Examples

σ₀(12)=6 → [1,2,3,4,6,12]
σ₀(p)=2
σ₀(p²)=3

σσ(n) — Sum of Divisors1 param▼

Sum of all positive divisors of n.

σ(n) = Σ d for d | n

Examples

σ(6)=12 (perfect)
σ(12)=28
σ(p)=p+1

Tips

n is perfect ⟺ σ(n)=2n
n is abundant ⟺ σ(n)>2n

sopfrsopfr(n) — Sum of Primes with Repetition1 param▼

Sums the prime factors counting multiplicity.

sopfr(n) = a₁p₁ + a₂p₂ + … + aₖpₖ

Examples

sopfr(12)=2+2+3=7
sopfr(60)=12

sopfsopf(n) — Sum of Distinct Primes1 param▼

Sum of distinct primes dividing n.

sopf(n) = p₁ + p₂ + … + pₖ

Examples

sopf(12)=2+3=5
sopf(60)=10

radrad(n) — Radical1 param▼

Product of distinct primes dividing n (ABC Conjecture function).

rad(n) = ∏ p for p prime, p | n

Examples

rad(72)=6
rad(480)=30
rad(p)=p

n#n# — Primorial1 param▼

Product of all primes ≤ n.

n# = ∏ p for p prime, p ≤ n

Examples

5#=30
7#=210
11#=2310

π(n)π(n) — Prime Counting Function1 param▼

Counts primes ≤ n. Exact for n ≤ 1,000,000; Li(x) approximation for larger values.

π(n) ~ n/ln(n) (Prime Number Theorem)

Examples

π(10)=4
π(100)=25
π(1,000,000)=78,498

drdr(n) — Digital Root1 param▼

Iterative digit sum until a single digit is obtained.

dr(n) = 1 + (n−1) mod 9 (for n > 0)

Examples

dr(493)=7
dr(999)=9
dr(n)≡n (mod 9)

⌊⌉⌊x⌋ / ⌈x⌉ — Floor and Ceiling1 param▼

Floor: greatest integer ≤ x. Ceiling: smallest integer ≥ x. Toggles between both.

⌊x⌋ ≤ x < ⌊x⌋+1 ⌈x⌉−1 < x ≤ ⌈x⌉

Examples

⌊3.7⌋=3, ⌈3.7⌉=4
⌊−2.3⌋=−3, ⌈−2.3⌉=−2
⌊5⌋=⌈5⌉=5

VₚVₚ(n) — p-adic Valuation2 params: n → Vₚ → p → =▼

Maximum power of prime p dividing n.

Vₚ(n) = max[k : p^k | n]

Examples

V₂(24)=3
V₃(81)=4
V₅(100)=2

Tips

Legendre's formula: Vₚ(n!)=Σ⌊n/pⁱ⌋
Vₚ(ab)=Vₚ(a)+Vₚ(b)

Modular Arithmetic

moda mod b — Division Remainder2 params: a → mod → b → =▼

Remainder of dividing a by b.

a mod b = a − b × ⌊a/b⌋

Examples

17 mod 5=2
23 mod 7=2
−8 mod 3=1

a%na^b mod n — Modular Exponentiation3 params: a → a%n → b → = → n → =▼

Computes a^b mod n efficiently using repeated squaring O(log b).

Decompose b in binary and square successively

Examples

2¹⁰⁰ mod 7=2
3¹³ mod 11=5
Fundamental in RSA

Tips

Flow: enter a → a%n → enter b → = → enter n → =

a⁻¹a⁻¹ mod n — Modular Inverse2 params: a → a⁻¹ → n → =▼

Finds b such that a×b≡1 (mod n). Only exists if GCD(a,n)=1.

Extended Euclidean algorithm

Examples

3⁻¹ mod 7=5 → 3×5=15≡1
5⁻¹ mod 11=9
Does not exist if GCD(a,n)≠1

ordord_n(a) — Multiplicative Order2 params: a → ord → n → =▼

Smallest k>0 with a^k≡1 (mod n). Requires GCD(a,n)=1.

ord_n(a) = min[k > 0 : a^k ≡ 1 (mod n)]

Examples

ord₇(2)=3
ord₁₀(3)=4

Tips

ord_n(a) always divides φ(n)
a is a primitive root ⟺ ord_n(a)=φ(n)

(a/p)(a/p) — Legendre Symbol2 params: a → (a/p) → p → =▼

1 if a is a quadratic residue mod p, −1 if not, 0 if p|a. Requires p odd prime.

(a/p) ≡ a^((p−1)/2) (mod p) — Euler's Criterion

Examples

(2/7)=1
(3/7)=−1
(5/5)=0

(a/n)ⱼ(a/n)ⱼ — Jacobi Symbol2 params: a → (a/n)ⱼ → n → =▼

Generalization of Legendre for odd composite n. Uses quadratic reciprocity.

(a/n) = ∏(a/pᵢ)^eᵢ where n = ∏pᵢ^eᵢ

Examples

(2/15)=(2/3)(2/5)=(−1)(−1)=1
(a/n)=−1 ⟹ a is NOT a QR

gg — Primitive Root1 param▼

Smallest primitive root mod n (if exists). g is primitive if ord_n(g)=φ(n).

[g, g², …, g^φ(n)] = (ℤ/nℤ)*

Examples

g(7)=3
g(11)=2
Exists only for n=1,2,4,p^k,2p^k

GCDGCD — Greatest Common DivisorN params (variable, min. 2)▼

Largest integer dividing all values. Accepts 2 or more numbers.

GCD(a,b) via Euclidean algorithm

Examples

GCD(12,18)=6
GCD(12,18,24)=6
GCD(a,b)×LCM(a,b)=a×b

Tips

Flow: 12→GCD→18→GCD (executes)
For 3+: 12→GCD→18→=→24→GCD

LCMLCM — Least Common MultipleN params (variable, min. 2)▼

Smallest positive integer divisible by all values.

LCM(a,b) = a×b / GCD(a,b)

Examples

LCM(4,6)=12
LCM(3,5,7)=105

DiofDioph — Linear Diophantine Equation3 params: a → Dioph → b → = → c → =▼

Solves ax+by=c. Gives particular and general solution.

ax+by=c has solution ⟺ GCD(a,b)|c x=x₀+(b/g)t, y=y₀−(a/g)t

Examples

3x+5y=1 → x=2+5t, y=−1−3t
4x+6y=3 → No solution

TCRCRT — Chinese Remainder TheoremVariable (pairs a,m)▼

Solves a system of congruences x≡aᵢ (mod mᵢ).

x≡a₁(mod m₁), x≡a₂(mod m₂) → x≡r(mod lcm(m₁,m₂))

Examples

x≡2(mod 3), x≡3(mod 5) → x≡8(mod 15)
x≡1(mod 4), x≡2(mod 3) → x≡5(mod 12)

Tips

Flow: a₁→CRT→m₁→=→a₂→=→m₂→CRT
Moduli must be compatible

Combinatorics

n!n! — Factorial1 param▼

Product from 1 to n. Arbitrary precision.

n! = 1×2×…×n, 0!=1

Examples

5!=120
10!=3,628,800
20!=2,432,902,008,176,640,000

n!!n!! — Double Factorial1 param▼

Product of integers with the same parity as n.

n!! = n×(n−2)×(n−4)×…

Examples

7!!=105
8!!=384
0!!=1!!=1

C(n,k)C(n,k) — Combinations2 params: n → C(n,k) → k → =▼

Ways to choose k from n regardless of order.

C(n,k) = n! / (k!(n−k)!)

Examples

C(5,2)=10
C(10,3)=120
C(n,0)=C(n,n)=1

Tips

Pascal's identity: C(n,k)=C(n−1,k−1)+C(n−1,k)
C(n,k)=C(n,n−k)

V(n,k)V(n,k) — Variations2 params: n → V(n,k) → k → =▼

Ways to choose k from n WITH order.

V(n,k) = n! / (n−k)!

Examples

V(5,2)=20
V(10,3)=720

CatCat(n) — Catalan Numbers1 param▼

Counts binary trees, triangulations, Dyck paths, balanced parentheses.

Cₙ = C(2n,n)/(n+1)

Examples

C₀=1, C₁=1, C₂=2, C₃=5, C₄=14, C₅=42

D(n)D(n) — Derangements1 param▼

Permutations where no element remains in its original position.

D(n) = (n−1)(D(n−1)+D(n−2))

Examples

D(3)=2
D(4)=9
D(n)/n! → 1/e ≈ 0.3679

BellB(n) — Bell Numbers1 param▼

Total number of partitions of a set of n elements.

B(n) = Σ S₂(n,k) for k=0..n

Examples

B(3)=5
B(4)=15
B(5)=52

p(n)p(n) — Integer Partitions1 param▼

Ways to write n as a sum of positive integers (order does not matter).

Dynamic programming

Examples

p(4)=5 → [4,3+1,2+2,2+1+1,1+1+1+1]
p(10)=42
p(100)=190,569,292

S₂S₂(n,k) — Stirling 2nd Kind2 params: n → S₂ → k → =▼

Ways to partition n elements into exactly k non-empty subsets.

S₂(n,k) = k·S₂(n−1,k) + S₂(n−1,k−1)

Examples

S₂(4,2)=7
S₂(5,3)=25

s₁s₁(n,k) — Stirling 1st Kind (unsigned)2 params: n → s₁ → k → =▼

Permutations of n elements with exactly k cycles.

|s₁(n,k)| = (n−1)·|s₁(n−1,k)| + |s₁(n−1,k−1)|

Examples

s₁(4,2)=11
s₁(4,1)=6

F(n)F(n) — nth Fibonacci1 param▼

Computes F(n) with fast doubling O(log n). Supports very large n.

F(0)=0, F(1)=1, F(n)=F(n−1)+F(n−2)

Examples

F(10)=55
F(50)=12,586,269,025
F(100)=354,224,848,179,261,915,075

Tips

F(n) mod m is periodic (Pisano period)
GCD(F(m),F(n))=F(GCD(m,n))

ΣdigBΣdigB — Digit Sum in Base b2 params: n → ΣdigB → b → =▼

Sums the digits of n written in base b.

If n=Σdᵢ×bⁱ, ΣdigB=Σdᵢ

Examples

ΣdigB(255,2)=8 → 11111111₂
ΣdigB(100,10)=1
ΣdigB(100,16)=10 → 64₁₆

Statistics

AMArithmetic MeanN params▼

Classic average of N numbers.

AM = (x₁+x₂+…+xₙ)/n

Examples

AM(3,7)=5
AM(2,4,6)=4

Tips

Flow: 3→Med A→7→Med A (executes)

GMGeometric MeanN params▼

nth root of the product. Positive values only.

GM = (x₁×x₂×…×xₙ)^(1/n)

Examples

GM(2,8)=4
GM(1,4,9)≈3.30

HMHarmonic MeanN params▼

Reciprocal of the arithmetic mean of the reciprocals. Positive values only.

HM = n/(1/x₁+1/x₂+…+1/xₙ)

Examples

HM(2,8)=3.2
HM(1,4,9)≈2.08

QMQuadratic Mean (RMS)N params▼

Root of the mean of squares (Root Mean Square).

QM = √((x₁²+x₂²+…+xₙ²)/n)

Examples

QM(3,4)≈3.54
QM(1,2,3)≈2.16

min/maxmin / max — Minimum and MaximumN params▼

Finds the smallest/largest value in a set of N numbers.

min(a₁,…,aₙ) and max(a₁,…,aₙ)

Examples

min(3,7,1)=1
max(3,7,1)=7

Mean Inequality (AM-GM-HM)

For positive numbers it always holds that:

HM ≤ GM ≤ AM ≤ QM

Equality holds only when all values are equal.
This inequality is fundamental in olympiads.

Olympiad Tools

A dedicated training section — open the side menu → Olympiad Tools. Each tool is a small form: fill the inputs, press Compute, and read the exact answer. Results use exact arithmetic (fractions, surds) wherever possible.

Fractions — Exact Rational Arithmetic

In olympiads, answers are exact: 22/7, not 3.14… The Fractions tool uses arbitrary-precision rationals p/q, always reduced to lowest terms with canonical sign.

Fraction arithmeticTwo fractions and an operation (+−×÷). Result as fraction, mixed number and decimal.

Simplify / convertReduces any fraction or decimal to lowest terms.

1/2 + 1/3 = 5/6 | 18/12 → 3/2 = "1 1/2" = 1.5 | 7/3 → "2 1/3"

Radicals — Exact Simplification

Simplifies radicals to the form coefficient·√radicand with no perfect-square factors inside the root.

√nSimplify a square root.

ⁿ√nSimplify the k-th root by extracting d^k factors.

a/√bRationalize the denominator: a/√b → (a/b)·√b.

a/(c+√d)Rationalize with conjugate: multiply by (c−√d)/(c²−d).

√72 → 6√2 | ³√54 → 3·³√2 | 1/√2 → √2/2 | 1/(1+√2) = −1+√2

Geometry

Geometry covers ~25% of IMO problems. Results are exact where possible: Heron area as a surd, circumradius R and inradius r as surds, shoelace areas as exact fractions.

Triangle from sidesType (scalene/isosceles/equilateral + right/acute/obtuse), perimeter, exact Heron area, circumradius R, inradius r.

(3,4,5) → scalene, right, area=6, R=5/2, r=1 (13,14,15) → scalene, acute, area=84, R=65/8, r=4

Pythagorean triplesLists primitive triples (a,b,c) with a²+b²=c², gcd=1, using Euclid's formula m>n, gcd(m,n)=1, different parity.

Up to 50: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (20,21,29)

ShoelaceExact polygon area from rational coordinates, with a drawing of the polygon. Enter vertices as "x,y" separated by ";".

"0,0; 4,0; 0,3" → area = 6

Pick's theoremFor integer-vertex polygons: A = I + B/2 − 1. Computes exact area, boundary lattice points B and interior points I, and draws them on the polygon.

"0,0; 5,0; 5,4; 0,4" → A=20, B=18, I=12

Triangle centersCentroid G, circumcenter O and orthocenter H as exact fractions; incenter approximated. Draws the triangle, the four centers and the Euler line (H = 3G − 2O).

A=(0,0), B=(4,0), C=(0,3) → O=(2, 3/2), H=(0,0), G=(4/3, 1)

Polynomials — Exact Analysis

Polynomials with rational coefficients. Accepts natural notation: x^2-5x+6, 3/2x-1, -x^3+x.

Analyze polynomialDegree, rational roots (rational root theorem), Vieta's relations, discriminant (degree 2 and 3), derivative.

x²−5x+6 → rational roots: 2,3 · sum=5 · product=6 · disc=1 · derivative: 2x−5

Solve quadraticDiscriminant, nature of roots, exact rational roots when possible, approximate real roots.

Real roots of cubicDepressed cubic / Cardano / trigonometric method (3 real roots case).

x³−6x²+11x−6 → 1, 2, 3

RuffiniSynthetic division of p(x) by (x − c) with every step shown; verifies the remainder theorem p(c) = remainder.

x³−6x²+11x−6 ÷ (x−1) → x²−5x+6, remainder 0

Linear system 2×2 / 3×3Cramer's rule with exact fractions. Rows "a,b,…,k" (coefficients + constant) separated by ";". Shows the determinant.

"2,1,5; 1,-1,1" → x=2, y=1 (det=−3)

Analyze polynomial also plots y = p(x) marking rational roots and extrema. It works on a single variable x, already expanded — for several variables or parentheses use Algebra.

Algebra — Polynomials in Several Variables

Symbolic algebra with exact rational coefficients: expanding a product, collecting like terms and comparing two expressions are all the same operation here, so an identity is checked by expanding both sides.

Notation: variables are a letter with an optional subscript (a, x, x1); multiplication may be implicit (2ab, (a+b)(a−b), 3(x+1)); exponents accept ^ or superscripts ((a+b+c)²). Terms are printed in graded lexicographic order, the usual convention.

Expand and simplifyExpands the expression and collects like terms. Also reports the degree, the number of terms, the variables, and whether the result is homogeneous or symmetric.

(a+b+c)² → a² + 2ab + 2ac + b² + 2bc + c² (degree 2, 6 terms, homogeneous, symmetric) (a+b)⁵ → a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

Check identityExpands both sides and compares them exactly. If they differ, it shows the difference, which is where the error is.

(a+b)³ vs a³+3a²b+3ab²+b³ → ✓ (a+b)² vs a²+b² → difference = 2ab

Common factorPulls out the greatest common monomial together with the rational content, so the factor inside the parentheses has integer coefficients.

2a²b + 4ab² → 2ab(a + 2b) x/2 + y/3 → (1/6)(3x + 2y)

Substitute / evaluateValues may be numbers or expressions, and substitution is simultaneous. Any variable left unassigned stays in the result.

(a+b+c)² with a=1, b=2, c=3 → 36 (a+b+c)² with a=1, b=2 → c² + 6c + 9

Coefficient of a monomialThe coefficient of one monomial in the expansion, without reading the whole result.

a²b in (a+b+c)³ → 3 x⁵ in (1+x)¹⁰ → 252

Partial derivative∂/∂x of the expansion, exact (no numerical approximation).

∂/∂a (a+b)³ → 3a² + 6ab + 3b²

Notable productsThe classic identities for your own A and B, which may themselves be expressions: (A±B)², (A+B)(A−B), (A±B)³ and the sum/difference of cubes.

A=x, B=2y → (A+B)² = x² + 4xy + 4y² | (A+B)(A²−AB+B²) = x³ + 8y³

(a+b)ⁿSymbolic binomial expansion term by term, with its list of coefficients.

n=5 → a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵ (1, 5, 10, 10, 5, 1)

Division is allowed by a number or by a monomial that divides every term: (a²b+ab²)/(ab) → a + b. An expansion that would be too large to read (or to compute quickly) is refused with a message instead of freezing the app.

Advanced Number Theory

Tools for the harder number-theory problems that appear in national and international olympiads.

√a mod pModular square root (Tonelli-Shanks). Returns ±r such that r²≡a (mod p) for prime p.

√2 mod 7 → ±3 (3²=9≡2)

ax≡b (mod n)Linear congruence — all solutions in [0,n). Has solutions iff gcd(a,n)|b. Shows the solutions highlighted on a "modular clock" (n ≤ 120).

3x≡6 (mod 9) → x∈{2,5,8}

Sieve of EratosthenesNumbered grid up to n (≤ 400) with primes highlighted; lists the primes and π(n).

n=100 → π(100)=25: 2, 3, 5, 7, 11, …, 97

x²−Dy²=1Pell equation — fundamental solution via continued fraction of √D.

D=61 → x=1766319049, y=226153980

CF(√n)Periodic continued fraction [a₀;(a₁,…,aₖ)] of √n.

√7=[2;(1,1,1,4)] | √2=[1;(2)]

Sums of squaresDecompose n as a²+b² (when possible) and always as a²+b²+c²+d² (Lagrange's theorem).

25=3²+4² | 7=0²+1²+1²+√5... → 0²+1²+1²+√5

Frobenius numberLargest integer that CANNOT be represented as a non-negative combination of the given denominations (gcd must be 1).

{6,9,20} → 43 | {3,5} → 7 (Chicken McNugget)

Quadratic residues mod nThe set {x² mod n} and its size. For odd prime p there are (p+1)/2 residues.

mod 11 → {0, 1, 3, 4, 5, 9} (6 residues)

φ, τ, σ, μTable of multiplicative functions (totient, divisor count, divisor sum, Möbius) for n in a range (max 30 rows).

n=12 → φ=4, τ=6, σ=28, μ=0

Step by Step — Worked Procedures

Shows the algorithm step by step, not just the answer. Essential for understanding and teaching the methods.

Euclidean algorithmEach division step plus the Bézout identity gcd(a,b)=x·a+y·b.

gcd(240,46): 240=5·46+10 / 46=4·10+6 / 10=1·6+4 / 6=1·4+2 / 4=2·2+0 → gcd=2

Prime factorizationEach successive division shown: n÷d=quotient.

360: 360÷2=180 / 180÷2=90 / 90÷2=45 / 45÷3=15 / 15÷3=5 → 2³×3²×5

CRTCombines congruences pairwise showing each intermediate step.

x≡2(3), x≡3(5), x≡2(7) → x≡23(mod 105)

RuffiniAlso available under Polynomials: synthetic division by (x − c) with each column shown.

Complex Numbers & Sequences

Roots of unityAll n-th roots e^(2πik/n), drawn on the unit circle as a regular polygon. Their sum is always 0 for n>1.

De Moivre powerComputes (re+im·i)^n. Example: (1+i)^8=16, (1+i)^4=−4.

n-th roots of complexAll n roots, evenly spaced on a circle of radius |z|^(1/n).

Linear recurrenceGenerates terms of aₙ=c₁aₙ₋₁+…+cₖaₙ₋ₖ with given seeds. Works with rational coefficients.

Fibonacci: coeffs=[1,1], seeds=[0,1] → 0,1,1,2,3,5,8,13,21,34 Lucas: seeds=[2,1] → 2,1,3,4,7,11,18 Pell: coeffs=[2,1], seeds=[0,1] → 0,1,2,5,12,29

Pascal row n[C(n,0),…,C(n,n)]. Row sum = 2^n. Also: multinomial coefficients.

Row 6 = 1, 6, 15, 20, 15, 6, 1 (sum=64)

(a+b)ⁿFull binomial expansion with superscript formatting (n ≤ 30).

(a+b)³ = a³ + 3a²b + 3ab² + b³

Pascal mod mPascal's triangle modulo m drawn as a colored grid (up to 128 rows); counts coefficients divisible by m. With m=2 the Sierpiński fractal appears (Kummer/Lucas theorems).

32 rows, m=2 → Sierpiński

Statistics — Exact Descriptive

Descriptive statisticsCount, min/max/range, mean, median and variance as exact fractions; mode(s); standard deviation approximated.

2, 4, 4, 5, 7 → mean=22/5, median=4, mode=4, σ²=66/25

QM ≥ AM ≥ GM ≥ HMThe four classical means of a positive list — AM and HM exact, QM and GM approximate — verifying the chain of inequalities.

1, 2, 4 → AM=7/3, GM=2, HM=12/7

Matrices — Exact Linear Algebra

Fraction-exact (no floating-point error). Rows separated by ";", entries by ","; two matrices separated by "|".

DeterminantDeterminant of a square matrix by Gaussian elimination.

"6,1,1; 4,-2,5; 2,8,7" → det = −306

InverseExact inverse (fractions); reports singular matrices.

"4,7; 2,6" → [[3/5, −7/10], [−1/5, 2/5]]

A × BMatrix product of two matrices.

"1,2; 3,4 | 5,6; 7,8" → [[19,22],[43,50]]

Solve A·x = b (n×n)Exact solution of a square system, or "no unique solution" if singular.

"2,1,1; 1,2,1; 1,1,2 | 1,1,1" → x=y=z=1/4

Rank & transposeMatrix rank and its transpose.

Calculus — Numerical

Numerical analysis of any function f(x) (e.g. x^2+sin(x), 1/x, exp(x)). Trigonometry in radians.

Derivative f'(x₀)Central differences with Richardson extrapolation.

d/dx (x²+sin x) at 1 ≈ 2.540302306

Definite integral ∫Composite Simpson's rule over [a,b].

∫₀¹ x² dx ≈ 0.3333333333

LimitTwo-sided numerical limit; reports "does not exist" when the sides differ.

lim_x→0 sin(x)/x ≈ 1

Practice Mode

Access via Olympiad Tools → Practice. The app presents a problem; type
your answer — it tells you immediately if you are right and tracks your
score for the session.

Topics: φ(n) Euler's totient, GCD, n! factorial, C(n,k) combinations,
σ₀(n) divisor count, digit sum, a mod b, F(n) Fibonacci.

Tip: The checker tolerates leading zeros and extra spaces.

Numeric Analysis Panel

When any number is entered the right panel (tablet) or bottom panel
(mobile) automatically shows:

  • Properties: digits, parity, sign
  • Representations: binary, octal, hexadecimal
  • Primality: Miller-Rabin (deterministic for n < 3.2×10¹⁸), factorization
  • Neighboring primes: previous and next
  • Divisors: complete list, sum, count
  • Classifications: perfect square/cube, perfect power, Fibonacci,
    triangular, palindrome

For numbers with ≤ 15 digits, it also shows:

  • Arithmetic functions: φ, λ, μ, ω, Ω, sopfr, sopf, rad, dr
  • Classifications: square-free, powerful, Harshad, semiprime,
    abundant / deficient / perfect

Note: pressing CE (Clear Entry) clears the analysis panel immediately.

Key Olympiad Formulas

Fundamental Identities

• Euler's theorem:    a^φ(n) ≡ 1 (mod n)  if GCD(a,n)=1
• Fermat's little:   a^(p−1) ≡ 1 (mod p)  if p prime
• Wilson:             (p−1)! ≡ −1 (mod p) ⟺ p is prime
• Legendre's formula: Vₚ(n!) = Σᵢ ⌊n/pⁱ⌋
• Lucas:              C(n,k) mod p = ∏ C(nᵢ,kᵢ) mod p
• Σ φ(d) for d|n = n
• Σ μ(d) for d|n = [n=1]
• φ(mn) = φ(m)φ(n)·GCD(m,n)/φ(GCD(m,n))
• GCD(F(m),F(n)) = F(GCD(m,n))
• AM ≥ GM ≥ HM  (mean inequality)

Quick Reference Table

n     φ(n)  λ(n)  μ(n)  σ(n)   ω   Ω
1     1     1     1     1      0   0
6     2     2     1     12     2   2
12    4     2     0     28     2   3
30    8     4     −1    72     3   3
60    16    4     0     168    3   4
100   40    20    0     217    2   4

Super Calculator / Super Calculadora

📲 Now on Google Play — free

Quick Start

Welcome to the olympiad calculator

Parameter System

High-Precision Mode

Number Theory

Modular Arithmetic

Combinatorics

Statistics

Mean Inequality (AM-GM-HM)

Olympiad Tools

Fractions — Exact Rational Arithmetic

Radicals — Exact Simplification

Geometry

Polynomials — Exact Analysis

Algebra — Polynomials in Several Variables

Advanced Number Theory

Step by Step — Worked Procedures

Complex Numbers & Sequences

Statistics — Exact Descriptive

Matrices — Exact Linear Algebra

Calculus — Numerical

Practice Mode

Numeric Analysis Panel

Key Olympiad Formulas

Fundamental Identities

Quick Reference Table