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Problem-Solving Strategies / LEVEL 3 · DIFFICULTY 3/5
Change the Representation
Use residues, bijections and shared-altitude geometry.
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STAGE PS 3.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS
Rule Out a Possibility
Useful preparation: Patterns and Induction
Goal: Understand and apply rule out a possibility.
Before you begin: Patterns and Induction
Understand the idea
A remainder classification can disprove a claim without finding the alleged object. Every integer is congruent to zero, one or two modulo three, so its square has only two possible residues.
Choose and carry out a method
Square each residue class. Compare the proposed total with the resulting list; a residue of two cannot be a square.
Check the reasoning
Passing this test does not prove a number is a square. It is a necessary condition used to rule out impossible cases.
An integer N equals 5. What is its remainder modulo 3? Explain to yourself why this rules out N being a perfect square; enter the remainder only.
- Classify all integer inputs by their remainder modulo 3.
- , and modulo 3; .
- The remainder is 2. A square cannot have this residue.
2
An integer N equals 8. What is its remainder modulo 3? Explain to yourself why this rules out N being a perfect square; enter the remainder only.
- Classify all integer inputs by their remainder modulo 3.
- , and modulo 3; .
- The remainder is 2. A square cannot have this residue.
2
Common pitfalls
Possible mix-up: Remainder one guarantees a square.
It passes one necessary test but does not establish squareness.
Possible mix-up: A correct numerical answer alone explains the method.
State the governing relationship and check the conditions described above.
Explain it to yourself
Find a nonsquare that still has remainder one modulo three.
Preview the eight practice prompts
An integer N equals 14. What is its remainder modulo 3? Explain to yourself why this rules out N being a perfect square; enter the remainder only.
An integer N equals 17. What is its remainder modulo 3? Explain to yourself why this rules out N being a perfect square; enter the remainder only.
An integer N equals 20. What is its remainder modulo 3? Explain to yourself why this rules out N being a perfect square; enter the remainder only.
An integer N equals 23. What is its remainder modulo 3? Explain to yourself why this rules out N being a perfect square; enter the remainder only.
An integer N equals 26. What is its remainder modulo 3? Explain to yourself why this rules out N being a perfect square; enter the remainder only.
An integer N equals 29. What is its remainder modulo 3? Explain to yourself why this rules out N being a perfect square; enter the remainder only.
A square array is claimed to contain exactly 32 dots. What is the total’s remainder modulo 3, which shows that an integer-sided square array is impossible?
A square array is claimed to contain exactly 35 dots. What is the total’s remainder modulo 3, which shows that an integer-sided square array is impossible?
STAGE PS 3.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS
Build a Bijection
Useful preparation: Rule Out a Possibility
Goal: Understand and apply build a bijection.
Before you begin: Rule Out a Possibility
Understand the idea
A bijection matches objects in two collections in both directions without duplication. Integer pairs with match two-element subsets because sorting gives a unique ordered representation.
Choose and carry out a method
Describe how to turn one counted object into the other and how to reverse the process. Then use the simpler collection’s count.
Check the reasoning
Both directions matter: every target must arise and no two source objects may map to the same target.
How many integer pairs (a,b) satisfy ?
- Match each allowed pair to a two-element subset.
- Choose two distinct labels without order: .
- There are 15 pairs. Strict inequality excludes equal labels and prevents reversed duplicates.
15
How many integer pairs (a,b) satisfy ?
- Match each allowed pair to a two-element subset.
- Choose two distinct labels without order: .
- There are 21 pairs. Strict inequality excludes equal labels and prevents reversed duplicates.
21
Common pitfalls
Possible mix-up: Count both (a,b) and (b,a).
The strict inequality selects exactly one order per pair.
Possible mix-up: A correct numerical answer alone explains the method.
State the governing relationship and check the conditions described above.
Explain it to yourself
Give the forward and reverse correspondence for a round-robin match.
Preview the eight practice prompts
How many integer pairs (a,b) satisfy ?
How many integer pairs (a,b) satisfy ?
How many integer pairs (a,b) satisfy ?
How many integer pairs (a,b) satisfy ?
How many integer pairs (a,b) satisfy ?
How many integer pairs (a,b) satisfy ?
15 players each meet every other player exactly once. How many matches take place?
16 players each meet every other player exactly once. How many matches take place?
STAGE PS 3.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS
Share an Altitude
Useful preparation: Build a Bijection
Goal: Understand and apply share an altitude.
Before you begin: Build a Bijection
NOTICE → MARK → REASON
One big triangle, two useful pieces
Mark BD and DC, then the shared height. Label the information from your problem. This is a separate learning diagram; it does not supply the values for your practice question.
Highlight parts and add your information
Choose a part here, or select it in the diagram. Mark what you know, what you need, or a reason. These are your scratch notes; the website does not grade or save them.
One small move: Select a part above. Find its given measurement in the question, then label that part below.
Use the problem’s stated values. A drawing can look shorter after rotation even when the actual length stays the same. Changing the model or reloading clears these notes.
Understand the idea
Two triangles with bases on the same line and a common opposite vertex have equal perpendicular heights. Their areas therefore inherit the ratio of their bases.
Choose and carry out a method
Express the desired base as a fraction of the whole base, then multiply the whole triangle’s area by that fraction.
Check the reasoning
The pieces must reconstruct the original triangle. The area ratio uses matching base segments, not unrelated slanted side lengths.
Point D lies on BC with . Triangle ABC has area 27. Find the area of triangle ABD.
- Triangles sharing an altitude have areas proportional to their bases.
- [ABD]/[ABC]=; multiply by 27.
- The area is 9. The other piece has area 18, and the two sum correctly.
9
Point D lies on BC with . Triangle ABC has area 33. Find the area of triangle ABD.
- Triangles sharing an altitude have areas proportional to their bases.
- [ABD]/[ABC]=; multiply by 33.
- The area is 12. The other piece has area 21, and the two sum correctly.
12
Common pitfalls
Possible mix-up: The segment ratio means the first piece is of the whole.
It is of the whole.
Possible mix-up: A correct numerical answer alone explains the method.
State the governing relationship and check the conditions described above.
Explain it to yourself
Why is no side-length or height calculation needed?
Preview the eight practice prompts
Point D lies on BC with . Triangle ABC has area 45. Find the area of triangle ABD.
Point D lies on BC with . Triangle ABC has area 51. Find the area of triangle ABD.
Point D lies on BC with . Triangle ABC has area 57. Find the area of triangle ABD.
Point D lies on BC with . Triangle ABC has area 63. Find the area of triangle ABD.
Point D lies on BC with . Triangle ABC has area 69. Find the area of triangle ABD.
Point D lies on BC with . Triangle ABC has area 75. Find the area of triangle ABD.
A triangular field of area 81 is divided from its top vertex to a point splitting the opposite edge in ratio . Find the area beside the first edge segment.
A triangular field of area 87 is divided from its top vertex to a point splitting the opposite edge in ratio . Find the area beside the first edge segment.