OBMEP 2023

OBMEP 2023 – 1st phase – Level 1: complete resolution

Official material: complete proof · OBMEP solutions · collection of the year. The resolutions below follow the numbering of the official notebook. Open the test to follow the statements, alternatives and diagrams; The explanations here are copyright. Attempt each question before checking the answer. Template Questions Answers 1–5 C · A · D · B · C 6–10 E · C · A · B · C 11–15 D · A · A · B · E 16–20 B · D · E · B · E Question 1 — Cans that support other cans Start from the blue can and mark the two that depend on it. Repeat the marking on the following layers: two cans, then two and, finally, the top can. These dependencies need to be removed from the top down; Removing cans from outside this set does not help release the blue one. There are $2+2+2+1=7$ cans before it. Alternative C.

September 26, 2026 Read

OBMEP 2023 – 1st phase – Level 2: complete resolution

Official material: complete proof · OBMEP solutions · collection of the year. The resolutions below follow the numbering of the official notebook. Open the test to follow the statements, alternatives and diagrams; The explanations here are copyright. Attempt each question before checking the answer. Template Questions Answers 1–5 E · B · A · E · C 6–10 E · A · B · E · C 11–15 C · B · B · E · D 16–20 B · D · E · C · C Question 1 — Pins The bottom left and bottom right sheets must keep their respective pins: no other holds them. It is also necessary to keep at least one pin from the top left sheet and one from the top right. So that leaves at least four of the ten. Choose, on these two upper sheets, pins that also secure the overlapping sheets in the middle. This choice supports the six sheets using four pins. We can remove $10-4=6$ — E.

September 26, 2026 Read

OBMEP 2023 – 1st phase – Level 3: complete resolution

Official material: complete proof · OBMEP solutions · collection of the year. The resolutions below follow the numbering of the official notebook. Open the test to follow the statements, alternatives and diagrams; The explanations here are copyright. Attempt each question before checking the answer. Template Questions Answers 1–5 B · B · E · C · D 6–10 A · E · A · B · E 11–15 E · B · D · C · D 16–20 D · E · C · B · D Question 1 — Overlap And it was glued last. A is below it and above B; B covers C and D. Therefore the first two are C and D, the third is B, the fourth A and the fifth E. Alternative B.

September 26, 2026 Read

OBMEP 2023 – 2nd phase – Level 1: complete resolution

Official material: complete proof · OBMEP solutions · collection of the year. The resolutions below follow the numbering of the official notebook. Open the test to follow the statements, alternatives and diagrams; The explanations here are copyright. Attempt each question before checking the answer. Question 1 — Wagons, colors and flags a) The white cars have numbers $1,4,7,10,\ldots$; with a flag, they must also be pairs. The first one after 8 is 10.

September 26, 2026 Read

OBMEP 2023 – 2nd phase – Level 2: complete resolution

Official material: complete proof · OBMEP solutions · collection of the year. The resolutions below follow the numbering of the official notebook. Open the test to follow the statements, alternatives and diagrams; The explanations here are copyright. Attempt each question before checking the answer. Question 1 — Cards and multiples a) One possibility is $6,2,4,8,1,9,3$. Checking the pairs, we have divisibility in all of them: 6 by 2, 4 by 2, 8 by 4, 8 by 1, 9 by 1 and 9 by 3.

September 26, 2026 Read

OBMEP 2023 – 2nd phase – Level 3: complete resolution

Official material: complete proof · OBMEP solutions · collection of the year. The resolutions below follow the numbering of the official notebook. Open the test to follow the statements, alternatives and diagrams; The explanations here are copyright. Attempt each question before checking the answer. Question 1 — Cards and multiples a) One possibility is $6,2,4,8,1,9,3$. Checking the pairs, we have divisibility in all of them: 6 by 2, 4 by 2, 8 by 4, 8 by 1, 9 by 1 and 9 by 3.

September 26, 2026 Read