If $\log_2 x + \log_4 x + \log_8 x = 11$, what is $x$?

Convert to base $2$: $\log_4 x = \tfrac12 \log_2 x$ and $\log_8 x =...

For $x > 0$, what is the least possible value of $x +...

By AM-GM, $x + \frac{9}{x} \ge 2\sqrt{x \cdot \frac{9}{x}} =...

Real numbers satisfy $x + y = 7$ and $xy = 10$. Find $x^3 +...

Do not solve for $x$ and $y$ separately — the expression is...

For how many integers $k$ does $x^2 + kx + 36 = 0$ have two...

If the roots are integers $a$ and $b$ then $ab = 36$ and $k =...

Evaluate $\displaystyle\sum_{k=1}^{20} \frac{1}{k(k+1)}$.

Split the term: $\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$.The...

Find the value of $\sqrt{6 + \sqrt{6 + \sqrt{6 + \cdots}}}$.

Call the value $y$. The expression repeats inside itself, so $y =...

An arithmetic sequence starts $5, 8, 11, \ldots$. What is...

The $k$th term is $5 + (k-1)\cdot 3 = 3k + 2$.Solve $3k + 2 > 200$...

What is the smallest positive integer $n$ for which...

In polar form $1 + i = \sqrt{2}\,\mathrm{cis}\,45^\circ$, so $(1+i)^n...

How many questions are in the AMC 10 Algebra — Paper 2?

It has 30 questions and a time limit of 75 minutes.

Is the AMC 10 Algebra — Paper 2 free?

It costs Rs 899 for the whole series, which includes full worked solutions and a detailed score report.

Is there negative marking?

No. Wrong answers do not lose marks, so it is worth attempting every question.

Yes. After you submit, every question shows a step-by-step solution alongside your answer.

AMC 10 Mathematics 2026 Rs 899

AMC 10 Algebra — Paper 2

Premium AMC 10 algebra paper: 30 hard problems on Vieta, functional equations, logarithms, floor sums and polynomial remainders, with worked solutions.

30~~Questions~~

75~~Minutes~~

180~~Marks~~

None ~~Negative~~

9-10~~Grades~~

About this test

AMC 10 Algebra — Paper 2 is a full 30-question paper you attempt online in 75 minutes. It follows the American Mathematics Competition 10 pattern, so the timing, the question style and the marking match the real exam. There is no negative marking, so it is worth attempting every question. Leaving a question blank still earns 1.5 marks.

You get your score the moment you submit, question by question, with a step-by-step solution to every problem. 25 questions in 75 minutes. A high score qualifies for the AIME.

What it covers

Vieta relations Functional equations Logarithms Inequalities Symmetric systems Integer roots Telescoping series Nested radicals Arithmetic sequences Complex numbers Floor functions Polynomial remainders Absolute value Rates and work Recursions Linear systems Exponential equations Discriminants Floor sums Root transformations Constrained optimisation Change of base Binomial expansion Reciprocal equations Composed functions Geometric series Averages Lattice points Radical equations Symmetric substitution

Sections

Vieta relations — 1 question
Functional equations — 1 question
Logarithms — 1 question
Inequalities — 1 question
Symmetric systems — 1 question
Integer roots — 1 question
Telescoping series — 1 question
Nested radicals — 1 question
Arithmetic sequences — 1 question
Complex numbers — 1 question
Floor functions — 1 question
Polynomial remainders — 1 question
Absolute value — 1 question
Rates and work — 1 question
Recursions — 1 question
Linear systems — 1 question
Exponential equations — 1 question
Discriminants — 1 question
Floor sums — 1 question
Root transformations — 1 question
Constrained optimisation — 1 question
Change of base — 1 question
Binomial expansion — 1 question
Reciprocal equations — 1 question
Composed functions — 1 question
Geometric series — 1 question
Averages — 1 question
Lattice points — 1 question
Radical equations — 1 question
Symmetric substitution — 1 question

Difficulty spread

2 easy questions
11 medium questions
6 hard questions

Before you start

15 questions · 45 minutes.

Each correct answer scores 6 points.
Each unanswered question scores 1.5 points.
A wrong answer scores 0 — a blank is worth more than a guess unless you can eliminate at least three options.
No calculators. Rough work on paper.

This paper covers Algebra only, so a weak score points at one topic rather than leaving you guessing.

A look at the questions

Q1. The cubic $x^3 - 4x^2 + (-11)x - (-30) = 0$ has roots $r$, $s$ and $t$. Find $r^2 + s^2 + t^2$.

A) 38

B) 16

C) -6

D) -11

E) 11

Q2. A function satisfies $f(x) + 2f\!\left(\tfrac{1}{x}\right) = 3x$ for all $x \ne 0$. Find $f(3)$.

A) $\frac{7}{3}$

B) $-\frac{7}{3}$

C) 9

D) $-\frac{2}{3}$

E) $\frac{1}{3}$

28 more questions in the full paper, each with a worked solution.

Questions people ask

How does this differ from Algebra Paper 1?

This is the premium tier. Most questions need two ideas rather than one: Vieta combined with a symmetric identity, a substitution followed by a discriminant argument, or a floor function summed in blocks. That is the shape of AMC 10 questions 20 to 25.

Which topics appear?

Vieta relations, functional equations, logarithm change of base, AM-GM, telescoping and geometric series, nested radicals, complex numbers, floor sums, polynomial remainders, Simon's favourite factoring trick and radical equations with extraneous roots.

How is it scored?

Real AMC 10 rules: 6 points per correct answer, 1.5 per blank, 0 for a wrong one. At this difficulty the blank rule matters, because a random guess has a lower expected value than leaving the question alone.

Are full solutions included?

Yes. Every question has a worked solution that names the technique, so a wrong answer tells you which method to revise rather than only that you missed it.

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