Try these four original exercises without hints. This is a study checkpoint, not an official competition score or a prediction of qualification. Allow roughly 40–60 minutes and write explanations. Use the solutions only after recording an attempt.
1. Algebra
For positive real a,b with a + b = 8, prove 1/a + 1/b ≥ 1/2. When does equality hold?
Hint
Write the sum as (a+b)/(ab).
Full solution
(a − b)² ≥ 0 implies 4ab ≤ (a + b)² = 64, so 0 < ab ≤ 16. Thus 1/a + 1/b = 8/(ab) ≥ 8/16 = 1/2. Equality requires a = b; the sum condition gives a = b = 4.
Check your reasoning: If you missed why dividing reverses the comparison of denominators, revisit positive reciprocals and the algebra lesson.
2. Number theory
Show that 5 cannot divide n² + 2 for any integer n.
Hint
List the possible square remainders modulo 5.
Full solution
An integer is congruent to 0,1,2,3 or 4 modulo 5. Squaring gives remainders 0,1,4,4,1. Adding 2 gives 2,3 or 1 modulo 5, never 0. Therefore 5 does not divide n² + 2.
Check your reasoning: List all remainder classes. Checking only a few ordinary integers does not cover all integers.
3. Geometry
In triangle ABC, D lies on AB, E lies on AC, DE ∥ BC and AD:AB = 1:3. What fraction of the area of ABC lies in quadrilateral DBCE?
Hint
Subtract the small similar triangle from the whole triangle.
Full solution
Triangles ADE and ABC are similar by corresponding angles. Their linear ratio is 1/3, so area(ADE)/area(ABC) = 1/9. The remaining quadrilateral has area fraction 1 − 1/9 = 8/9. The diagram and theorem in the geometry practice section show the same configuration with a different ratio.
Check your reasoning: Using 1/3 as the area ratio confuses length with area. Draw the labelled configuration before solving.
4. Combinatorics
Choose seven distinct integers from 1 through 12. Prove that two selected integers sum to 13.
Hint
Build six disjoint boxes with the desired sum.
Full solution
Partition the numbers into {1,12}, {2,11}, {3,10}, {4,9}, {5,8}, {6,7}. Seven choices into six boxes force a box containing two choices. Its numbers sum to 13.
Check your reasoning: Your boxes must cover every possible selected number, and each complete box must have the required property.
Mark the reasoning, then choose a route
For each problem, give yourself 0 if you have no valid route, 1 if the central idea is right but the explanation has a gap, or 2 if the argument is complete. This rubric is only for planning your study.
| Your work | Next step |
|---|---|
| 0–1 on a subject | Return to that subject in the practice plan; rewrite the solution and solve a related problem. |
| Mostly correct ideas, incomplete proofs | Study proof-writing; check domains, cases and the reason for each implication. |
| Complete solutions in all four subjects | Select an official first-stage national paper; compare its level and format before timing yourself. |
Do not add the scores into an IMO-readiness claim. Four short questions cannot measure performance on a full national selection test.