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AMC 12 trail

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Original AMC-style practice

New reasoning quests

20 new problems inspired by competition skills, with new settings, guided hints and step-by-step explanations.

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40 problems with answer checking, guided hints and worked explanations. Sign in as a student to save your attempts and share progress with your parent.

Foundation, Developing and Challenge are Numerive practice levels. These original exercises build competition skills; they are not official AMC questions or a timed mock exam.

Foundation

Original practice · Foundation

A logarithmic equation

For real x, log⁡2 (x)+log⁡2 (x−2)=3\displaystyle \operatorname{log}_{2}\, \left(x\right) + \operatorname{log}_{2}\, \left(x - 2\right) = 3. What is x?

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Original practice · Foundation

Cosine of twice an angle

An angle θ\displaystyle \theta satisfies sin⁡ θ=35\displaystyle \operatorname{sin}\, \theta = \frac{3}{5} and cos⁡ θ=45\displaystyle \operatorname{cos}\, \theta = \frac{4}{5}. What is cos⁡ (2 θ)\displaystyle \operatorname{cos}\, \left(2\, \theta\right)? Give a fraction.

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Original practice · Foundation

Cubes of cubic roots

The three roots of x3−6 x2+11 x−6=0\displaystyle x^{3} - 6\, x^{2} + 11\, x - 6 = 0 are a, b and c. What is a3+b3+c3\displaystyle a^{3} + b^{3} + c^{3}?

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Original practice · Foundation

A shrinking geometric sum

What is the sum of the infinite series 5+53+59+527\displaystyle 5 + \frac{5}{3} + \frac{5}{9} + \frac{5}{27} + ...? Give a fraction.

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Original practice · Foundation

Distance between foci

An ellipse has equation x225+y29=1\displaystyle \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1. What is the distance between its two foci?

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New reasoning quest · Foundation

A logarithmic balance

A real number x satisfies log⁡3 (x−1)=2−log⁡3 (x+1)\displaystyle \operatorname{log}_{3}\, \left(x - 1\right) = 2 - \operatorname{log}_{3}\, \left(x + 1\right). What is x2\displaystyle x^{2}?

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New reasoning quest · Foundation

A quotient in the complex plane

Let z=3+4 i1−2 i\displaystyle z = \frac{3 + 4\, i}{1 - 2\, i}, where i2=−1\displaystyle i^{2} = -1. What is ∣z∣2\displaystyle \left|z\right|^{2}?

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Developing

Original practice · Developing

The third side squared

Two sides of a triangle have lengths 5 and 7, and their included angle is 60 degrees. If the third side has length c, what is c2\displaystyle c^{2}?

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Original practice · Developing

A chain of logarithms

What is (log⁡2 3) (log⁡3 5) (log⁡5 16)\displaystyle \left(\operatorname{log}_{2}\, 3\right)\, \left(\operatorname{log}_{3}\, 5\right)\, \left(\operatorname{log}_{5}\, 16\right)?

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Original practice · Developing

At least one in each box

Six identical tokens are distributed among three distinct boxes, with at least one token in each box. How many distributions are possible?

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Original practice · Developing

No letter in its own envelope

Four distinct letters are put into four addressed envelopes, one letter per envelope. In how many arrangements is every letter in the wrong envelope?

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Original practice · Developing

Nearly full boxes

How many ordered triples of integers (a,b,c) satisfy a+b+c=10\displaystyle a + b + c = 10 and 0≤a\displaystyle 0 \le a,b,c≤4\displaystyle c \le 4?

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New reasoning quest · Developing

Adding two acute angles

Acute angles α\displaystyle \alpha and β\displaystyle \beta satisfy tan⁡ α=2\displaystyle \operatorname{tan}\, \alpha = 2 and tan⁡ β=13\displaystyle \operatorname{tan}\, \beta = \frac{1}{3}. What is tan⁡ (α+β)\displaystyle \operatorname{tan}\, \left(\alpha + \beta\right)?

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New reasoning quest · Developing

Two sums reveal a series

An infinite geometric sequence has first term a>0\displaystyle a > 0 and common ratio r with 0<r<1\displaystyle 0 < r < 1. Its terms sum to 12, and the squares of its terms sum to 48. What is a?

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New reasoning quest · Developing

Real parts of four roots

The complex equation z4=−16\displaystyle z^{4} = -16 has four roots. What is the sum of the squares of the real parts of those four roots?

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New reasoning quest · Developing

A line touches an ellipse

Real numbers x and y satisfy x29+y216=1\displaystyle \frac{x^{2}}{9} + \frac{y^{2}}{16} = 1. If M is the greatest possible value of x+y\displaystyle x + y, what is M2\displaystyle M^{2}?

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New reasoning quest · Developing

A condition changes the sample space

Two fair six-sided dice are rolled. Given that their sum is greater than 8, what is the probability that at least one die shows an even number? Give a reduced fraction.

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New reasoning quest · Developing

An uneven choice of bags

A fair die chooses a bag: rolls 1 or 2 select bag A, and rolls 3 through 6 select bag B. Bag A has 3 red and 1 blue bead; bag B has 1 red and 2 blue beads. One bead is chosen uniformly from the selected bag. Given that it is red, what is the probability that bag A was selected? Give a reduced fraction.

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New reasoning quest · Developing

How many colors appear?

A spinner has four equally likely colors. It is spun three independent times. What is the expected number of distinct colors that appear? Give a reduced fraction.

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New reasoning quest · Developing

A repeated-root recurrence

A sequence satisfies a0=0\displaystyle a_{0} = 0, a1=1\displaystyle a_{1} = 1, and an+2=4 an+1−4 an\displaystyle a_{n+2} = 4\, a_{n+1} - 4\, a_{n} for every integer n≥0\displaystyle n \ge 0. What is a6\displaystyle a_{6}?

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New reasoning quest · Developing

A canceled factor still matters

Define f (x)=x2−1x−1\displaystyle f\, \left(x\right) = \frac{x^{2} - 1}{x - 1} for x≠1\displaystyle x \ne 1. There are exactly two real values of x for which f(f(x)) is undefined. What is their sum?

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New reasoning quest · Developing

Three factors under one budget

Positive real numbers a and b satisfy a+2 b=12\displaystyle a + 2\, b = 12. What is the greatest possible value of ab2\displaystyle ab^{2}?

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New reasoning quest · Developing

Binary strings with breathing room

An 8-character binary string is chosen uniformly from all strings of zeros and ones. What is the probability that it has no two adjacent ones? Give a reduced fraction.

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New reasoning quest · Developing

Every team gets a task

Five different tasks are assigned to three labeled teams. Each task goes to exactly one team, and every team must receive at least one task. How many assignments are possible?

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New reasoning quest · Developing

Large powers, small remainder

When x2026+x1013\displaystyle x^{2026} + x^{1013} is divided by x2+x+1\displaystyle x^{2} + x + 1, the remainder has degree less than 2. What is the constant term of the remainder?

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Challenge

Original practice · Challenge

Reciprocals of cubic roots

The roots of x3−4 x2+2 x+1=0\displaystyle x^{3} - 4\, x^{2} + 2\, x + 1 = 0 are r, s and t, counted with multiplicity. What is 1r+1s+1t\displaystyle \frac{1}{r} + \frac{1}{s} + \frac{1}{t}?

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Original practice · Challenge

A sine-cosine sum

A real angle θ\displaystyle \theta satisfies sin⁡ θ+cos⁡ θ=65\displaystyle \operatorname{sin}\, \theta + \operatorname{cos}\, \theta = \frac{6}{5}. What is sin⁡ (2 θ)\displaystyle \operatorname{sin}\, \left(2\, \theta\right)? Give a fraction.

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Original practice · Challenge

Which box supplied the red counter?

Box A has 2 red and 1 blue counter. Box B has 1 red and 3 blue counters. A box is chosen with equal probability, then one counter is drawn uniformly from it. Given that the counter is red, what is the probability that box A was chosen?

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Original practice · Challenge

Shifting a recurrence

A sequence satisfies a1=2\displaystyle a_{1} = 2 and an+1=3 an+2\displaystyle a_{n+1} = 3\, a_{n} + 2 for n≥1\displaystyle n \ge 1. What is a5\displaystyle a_{5}?

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Original practice · Challenge

A nonreal cube root of unity

A complex number z satisfies z3=1\displaystyle z^{3} = 1 and z≠1\displaystyle z \ne 1. What real number equals (1−z) (1−z2)\displaystyle \left(1 - z\right)\, \left(1 - z^{2}\right)?

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New reasoning quest · Challenge

Fourth powers without solving the cubic

Let r, s, and t be the three complex roots, counted with multiplicity, of x3−2 x2−3 x+1=0\displaystyle x^{3} - 2\, x^{2} - 3\, x + 1 = 0. What is r4+s4+t4\displaystyle r^{4} + s^{4} + t^{4}?

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New reasoning quest · Challenge

Two angles, one product

The equation sin⁡ x+sin⁡ (3 x)=cos⁡ x\displaystyle \operatorname{sin}\, x + \operatorname{sin}\, \left(3\, x\right) = \operatorname{cos}\, x has two solutions in 0<x<π2\displaystyle 0 < x < \frac{\pi}{2}. What is the product of the tangents of those two solutions?

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New reasoning quest · Challenge

Four roots in a window

For how many integer values of k does x4\displaystyle x^{4} − kx² +16=0\displaystyle +16 = 0 have four distinct real roots, all in the closed interval [−4,4]?

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EXPLORE REAL COMPETITION QUESTIONS

Past papers

Open the original paper, try it on scratch paper, then check the linked solutions. Each question also connects to a Numerive lesson.

Original questions and diagrams open on the source website. You can search related topics here in either language. External work does not add points or attempts to your Numerive report.

Choose a topic quest

139 related preparation topics. Open a lesson to learn, or sign in as a student to practise original Numerive problems.

Numbers & number theory 39 topics
Algebra & functions 35 topics
Geometry & measurement 21 topics
Counting & probability 26 topics
Data & statistics 3 topics
Problem-solving strategies 15 topics

Past-exam questions

45 verified question references. Follow the related skills below to build foundations with Numerive lessons and original practice.

Use “View original” to read the question and diagrams at the source. The listed page is the PDF viewer page, including any cover. Entries without a web link are topic references from your supplied collection. About the indexed collection.

Past AMC reference

2024 AMC 12A · Q10

Transforming trigonometric products

Related skills

Source: 2025 AMC 12 preparation collection (supplied PDF) · PDF page 3 · collection item 7

Past AMC reference

2024 AMC 12A · Q16

Distributing objects among distinct recipients

Related skills

Source: 2025 AMC 12 preparation collection (supplied PDF) · PDF page 8 · collection item 18

Past AMC reference

2024 AMC 12A · Q19

Triangle constraints and the cosine rule

Related skills

Source: 2025 AMC 12 preparation collection (supplied PDF) · PDF page 6 · collection item 13

Past AMC reference

2024 AMC 12B · Q15

Coordinate area in a polygon

Related skills

Source: 2025 AMC 12 preparation collection (supplied PDF) · PDF page 7 · collection item 16

Past AMC reference

2024 AMC 12B · Q19

Angles and areas in a hexagon

Related skills

Source: 2025 AMC 12 preparation collection (supplied PDF) · PDF page 6 · collection item 14

Past AMC reference

2023 AMC 12A · Q19

Changing bases in a logarithmic equation

Related skills

Source: 2025 AMC 12 preparation collection (supplied PDF) · PDF page 2 · collection item 4

Past AMC reference

2023 AMC 12B · Q9

Absolute value and symmetry

Related skills

Also appeared as 2023 AMC 10B · Q13.

Source: 2025 AMC 12 preparation collection (supplied PDF) · PDF page 3 · collection item 6

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