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AMC 10 trail

Algebra, geometry, number theory, counting and probability.

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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

Festival tickets

Two adult tickets and three student tickets cost $29. Four adult tickets and one student ticket cost $33. What is the price of one adult ticket in dollars?

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Original practice · Foundation

Crossing lines

The lines y=2 x+3\displaystyle y = 2\, x + 3 and y=−x+12\displaystyle y = -x + 12 intersect. What is the y-coordinate of their intersection?

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New reasoning quest · Foundation

When the pass pays off

At a climbing center, Plan A costs $12 plus $3 per climb. Plan B costs $30 plus $1.50 per climb. What is the smallest whole number of climbs for which Plan B costs strictly less than Plan A?

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New reasoning quest · Foundation

A rule applied twice

A linear function f satisfies f (2)=7\displaystyle f\, \left(2\right) = 7 and f (5)=−2\displaystyle f\, \left(5\right) = -2. What is f(f(1))?

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Developing

Original practice · Developing

Two distances on a line

The equation ∣x−2∣+∣x+1∣=9\displaystyle \left|x - 2\right| + \left|x + 1\right| = 9 has two real solutions. What is their product?

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Original practice · Developing

Cubes without finding the numbers

Real numbers x and y satisfy x+y=10\displaystyle x + y = 10 and xy=21\displaystyle xy = 21. What is x3+y3\displaystyle x^{3} + y^{3}?

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Original practice · Developing

Reciprocals reveal a product

Positive real numbers x and y satisfy x+y=10\displaystyle x + y = 10 and 1x+1y=56\displaystyle \frac{1}{x} + \frac{1}{y} = \frac{5}{6}. What is xy?

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New reasoning quest · Developing

Integer roots, smallest sum

The polynomial x2\displaystyle x^{2} − kx + 24 has two distinct positive integer roots. What is the smallest possible value of k?

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New reasoning quest · Developing

Equal percentages, unequal result

A game token’s price rises by p percent and then falls by p percent of its new price, where 0<p<100\displaystyle 0 < p < 100. Its final price is 91%\displaystyle 91\% of its original price. What is p?

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New reasoning quest · Developing

Cutting off a corner

Triangle ABC has vertices A=(0,0)\displaystyle A = \left(0, 0\right), B=(12,0)\displaystyle B = \left(12, 0\right), and C=(0,9)\displaystyle C = \left(0, 9\right). Point P is the midpoint of AB, and point Q lies on AC with AQ=AC3\displaystyle AQ = \frac{AC}{3}. What is the area of quadrilateral PBCQ?

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New reasoning quest · Developing

Card sums modulo five

Two distinct cards are chosen uniformly at random from cards numbered 1 through 12. What is the probability that the sum of their numbers is divisible by 5? Give a reduced fraction.

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New reasoning quest · Developing

The chord between two tangents

A circle has center O and radius 5. A point P is 13 units from O. The two tangents from P touch the circle at A and B. What is the length of chord AB? Give a reduced fraction.

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New reasoning quest · Developing

Two package sizes

A shop sells tokens only in packs of 4 and packs of 7. How many different pairs of nonnegative pack counts let a customer buy exactly 85 tokens?

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New reasoning quest · Developing

Alternating square totals

What is the value of 12−22+32−42\displaystyle 1^{2} - 2^{2} + 3^{2} - 4^{2} + ... +392−402\displaystyle +39^{2} - 40^{2}?

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New reasoning quest · Developing

Two circles of one triangle

A right triangle has legs 5 and 12. Let R be its circumradius and r its inradius. What is Rr\displaystyle \frac{R}{r}? Give a reduced fraction.

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New reasoning quest · Developing

Clear denominators carefully

What is the product of all real solutions of 1x−1+1x+1=34\displaystyle \frac{1}{x - 1} + \frac{1}{x + 1} = \frac{3}{4}?

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Challenge

Original practice · Challenge

Repeated roots

There are two real values of k for which x2−2\displaystyle x^{2} - 2kx +k+6=0\displaystyle +k + 6 = 0 has a repeated real root. What is the sum of those two values of k?

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Original practice · Challenge

Neighbors at a round table

Six distinct people sit around a round table. Arrangements differing only by rotation are considered the same, but reflections are different. How many arrangements place two specified people next to each other?

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Original practice · Challenge

A telescoping fraction sum

What is the value of 11×2+12×3\displaystyle \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + ... +119×20\displaystyle \frac{+1}{19 \times 20}? Give a fraction.

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New reasoning quest · Challenge

A diameter chosen from lattice points

Two distinct points are selected uniformly from all points with integer coordinates on x2+y2=25\displaystyle x^{2} + y^{2} = 25. What is the probability that the segment joining them is a diameter? Give a reduced fraction.

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New reasoning quest · Challenge

One return to the diagonal

A path from (0,0) to (4,4) uses four east steps and four north steps. It never has y>x\displaystyle y > x and visits exactly one point with y=x\displaystyle y = x other than its two endpoints. How many such paths are there?

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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

124 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 21 topics
Geometry & measurement 21 topics
Counting & probability 26 topics
Data & statistics 3 topics
Problem-solving strategies 14 topics

Past-exam questions

32 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

2023 AMC 10B · Q13

Absolute value and symmetry

Related skills

Also appeared as 2023 AMC 12B · Q9.

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

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