Five contestants – P, Q, R, S and T – performed in the final round of a singing contest. Each contestant was scored by three judges, J1, J2 and J3, on a scale of 1 to 10 (whole numbers only). A contestant's total score is the sum of the three judges' scores. The table shows some of the scores; the missing ones are marked '?'.
| Contestant | J1 | J2 | J3 |
|---|---|---|---|
| P | 8 | ? | 9 |
| Q | ? | 9 | 8 |
| R | 9 | 8 | ? |
| S | 7 | ? | 7 |
| T | ? | 10 | ? |
It is also known that:
- Each judge gave five different scores to the five contestants, and these five scores were consecutive integers.
- No two contestants had the same total score.
- In the final ranking by total score, P was first, followed by Q, R, T and S in that order.
| Contestant | J1 | J2 | J3 |
|---|---|---|---|
| P | 8 | ? | 9 |
| Q | ? | 9 | 8 |
| R | 9 | 8 | ? |
| S | 7 | ? | 7 |
| T | ? | 10 | ? |
It is also known that:
- Each judge gave five different scores to the five contestants, and these five scores were consecutive integers.
- No two contestants had the same total score.
- In the final ranking by total score, P was first, followed by Q, R, T and S in that order.
Question 1
Single correct option
What score did J1 give to T?
- A. 5
- B. 4
- C. 10
- D. 6
Show answer and solution
Answer: (A) 5
Deduction:
- J2 already has 8, 9, 10 → J2's scores are 6–10, so P and S get 6 and 7 in some order.
- J3 has 7, 8, 9 → R and T get {5, 6} or {6, 10}; J1 has 7, 8, 9 → Q and T get {5, 6} or {6, 10}.
- Q < P: 17 + Q₁ < 17 + P₂ ≤ 24 → Q₁ ≤ 6. R < Q: 17 + R₃ < 17 + Q₁ → R₃ < Q₁ ≤ 6 → R₃ = 5, so J3 = 5–9 and T₃ = 6; then Q₁ = 6 (must exceed 5).
- P₂ > Q₁ = 6 → P₂ = 7, S₂ = 6. Totals: P 24, Q 23, R 22, S 20.
- T = T₁ + 10 + 6 must lie strictly between 20 and 22 → 21 → T₁ = 5 (so J1 = 5–9).
| Contestant | J1 | J2 | J3 | Total |
|---|---|---|---|---|
| P | 8 | 7 | 9 | 24 |
| Q | 6 | 9 | 8 | 23 |
| R | 9 | 8 | 5 | 22 |
| S | 7 | 6 | 7 | 20 |
| T | 5 | 10 | 6 | 21 |
- J2 already has 8, 9, 10 → J2's scores are 6–10, so P and S get 6 and 7 in some order.
- J3 has 7, 8, 9 → R and T get {5, 6} or {6, 10}; J1 has 7, 8, 9 → Q and T get {5, 6} or {6, 10}.
- Q < P: 17 + Q₁ < 17 + P₂ ≤ 24 → Q₁ ≤ 6. R < Q: 17 + R₃ < 17 + Q₁ → R₃ < Q₁ ≤ 6 → R₃ = 5, so J3 = 5–9 and T₃ = 6; then Q₁ = 6 (must exceed 5).
- P₂ > Q₁ = 6 → P₂ = 7, S₂ = 6. Totals: P 24, Q 23, R 22, S 20.
- T = T₁ + 10 + 6 must lie strictly between 20 and 22 → 21 → T₁ = 5 (so J1 = 5–9).
| Contestant | J1 | J2 | J3 | Total |
|---|---|---|---|---|
| P | 8 | 7 | 9 | 24 |
| Q | 6 | 9 | 8 | 23 |
| R | 9 | 8 | 5 | 22 |
| S | 7 | 6 | 7 | 20 |
| T | 5 | 10 | 6 | 21 |
Question 2
Numerical answer
What was Q's total score?
Show answer and solution
Answer: 23
Deduction:
- J2 already has 8, 9, 10 → J2's scores are 6–10, so P and S get 6 and 7 in some order.
- J3 has 7, 8, 9 → R and T get {5, 6} or {6, 10}; J1 has 7, 8, 9 → Q and T get {5, 6} or {6, 10}.
- Q < P: 17 + Q₁ < 17 + P₂ ≤ 24 → Q₁ ≤ 6. R < Q: 17 + R₃ < 17 + Q₁ → R₃ < Q₁ ≤ 6 → R₃ = 5, so J3 = 5–9 and T₃ = 6; then Q₁ = 6 (must exceed 5).
- P₂ > Q₁ = 6 → P₂ = 7, S₂ = 6. Totals: P 24, Q 23, R 22, S 20.
- T = T₁ + 10 + 6 must lie strictly between 20 and 22 → 21 → T₁ = 5 (so J1 = 5–9).
| Contestant | J1 | J2 | J3 | Total |
|---|---|---|---|---|
| P | 8 | 7 | 9 | 24 |
| Q | 6 | 9 | 8 | 23 |
| R | 9 | 8 | 5 | 22 |
| S | 7 | 6 | 7 | 20 |
| T | 5 | 10 | 6 | 21 |
Q = 6 + 9 + 8 = 23.
- J2 already has 8, 9, 10 → J2's scores are 6–10, so P and S get 6 and 7 in some order.
- J3 has 7, 8, 9 → R and T get {5, 6} or {6, 10}; J1 has 7, 8, 9 → Q and T get {5, 6} or {6, 10}.
- Q < P: 17 + Q₁ < 17 + P₂ ≤ 24 → Q₁ ≤ 6. R < Q: 17 + R₃ < 17 + Q₁ → R₃ < Q₁ ≤ 6 → R₃ = 5, so J3 = 5–9 and T₃ = 6; then Q₁ = 6 (must exceed 5).
- P₂ > Q₁ = 6 → P₂ = 7, S₂ = 6. Totals: P 24, Q 23, R 22, S 20.
- T = T₁ + 10 + 6 must lie strictly between 20 and 22 → 21 → T₁ = 5 (so J1 = 5–9).
| Contestant | J1 | J2 | J3 | Total |
|---|---|---|---|---|
| P | 8 | 7 | 9 | 24 |
| Q | 6 | 9 | 8 | 23 |
| R | 9 | 8 | 5 | 22 |
| S | 7 | 6 | 7 | 20 |
| T | 5 | 10 | 6 | 21 |
Q = 6 + 9 + 8 = 23.
Question 3
Single correct option
Which contestant received the same score from two different judges?
- A. S
- B. P
- C. T
- D. Q
Show answer and solution
Answer: (A) S
Deduction:
- J2 already has 8, 9, 10 → J2's scores are 6–10, so P and S get 6 and 7 in some order.
- J3 has 7, 8, 9 → R and T get {5, 6} or {6, 10}; J1 has 7, 8, 9 → Q and T get {5, 6} or {6, 10}.
- Q < P: 17 + Q₁ < 17 + P₂ ≤ 24 → Q₁ ≤ 6. R < Q: 17 + R₃ < 17 + Q₁ → R₃ < Q₁ ≤ 6 → R₃ = 5, so J3 = 5–9 and T₃ = 6; then Q₁ = 6 (must exceed 5).
- P₂ > Q₁ = 6 → P₂ = 7, S₂ = 6. Totals: P 24, Q 23, R 22, S 20.
- T = T₁ + 10 + 6 must lie strictly between 20 and 22 → 21 → T₁ = 5 (so J1 = 5–9).
| Contestant | J1 | J2 | J3 | Total |
|---|---|---|---|---|
| P | 8 | 7 | 9 | 24 |
| Q | 6 | 9 | 8 | 23 |
| R | 9 | 8 | 5 | 22 |
| S | 7 | 6 | 7 | 20 |
| T | 5 | 10 | 6 | 21 |
Only S (7 from J1 and 7 from J3). All other rows have three distinct scores.
- J2 already has 8, 9, 10 → J2's scores are 6–10, so P and S get 6 and 7 in some order.
- J3 has 7, 8, 9 → R and T get {5, 6} or {6, 10}; J1 has 7, 8, 9 → Q and T get {5, 6} or {6, 10}.
- Q < P: 17 + Q₁ < 17 + P₂ ≤ 24 → Q₁ ≤ 6. R < Q: 17 + R₃ < 17 + Q₁ → R₃ < Q₁ ≤ 6 → R₃ = 5, so J3 = 5–9 and T₃ = 6; then Q₁ = 6 (must exceed 5).
- P₂ > Q₁ = 6 → P₂ = 7, S₂ = 6. Totals: P 24, Q 23, R 22, S 20.
- T = T₁ + 10 + 6 must lie strictly between 20 and 22 → 21 → T₁ = 5 (so J1 = 5–9).
| Contestant | J1 | J2 | J3 | Total |
|---|---|---|---|---|
| P | 8 | 7 | 9 | 24 |
| Q | 6 | 9 | 8 | 23 |
| R | 9 | 8 | 5 | 22 |
| S | 7 | 6 | 7 | 20 |
| T | 5 | 10 | 6 | 21 |
Only S (7 from J1 and 7 from J3). All other rows have three distinct scores.
Question 4
Single correct option
Suppose the organisers had instead ranked the contestants by the median of their three scores, breaking any tie in favour of the contestant with the higher J1 score. Who would then have been placed second?
- A. S
- B. R
- C. P
- D. Q
Show answer and solution
Answer: (C) P
Deduction:
- J2 already has 8, 9, 10 → J2's scores are 6–10, so P and S get 6 and 7 in some order.
- J3 has 7, 8, 9 → R and T get {5, 6} or {6, 10}; J1 has 7, 8, 9 → Q and T get {5, 6} or {6, 10}.
- Q < P: 17 + Q₁ < 17 + P₂ ≤ 24 → Q₁ ≤ 6. R < Q: 17 + R₃ < 17 + Q₁ → R₃ < Q₁ ≤ 6 → R₃ = 5, so J3 = 5–9 and T₃ = 6; then Q₁ = 6 (must exceed 5).
- P₂ > Q₁ = 6 → P₂ = 7, S₂ = 6. Totals: P 24, Q 23, R 22, S 20.
- T = T₁ + 10 + 6 must lie strictly between 20 and 22 → 21 → T₁ = 5 (so J1 = 5–9).
| Contestant | J1 | J2 | J3 | Total |
|---|---|---|---|---|
| P | 8 | 7 | 9 | 24 |
| Q | 6 | 9 | 8 | 23 |
| R | 9 | 8 | 5 | 22 |
| S | 7 | 6 | 7 | 20 |
| T | 5 | 10 | 6 | 21 |
Medians: P 8, Q 8, R 8, S 7, T 6. P, Q and R tie on 8; J1 scores are R 9, P 8, Q 6. So the order is R, P, Q, S, T and P is second.
- J2 already has 8, 9, 10 → J2's scores are 6–10, so P and S get 6 and 7 in some order.
- J3 has 7, 8, 9 → R and T get {5, 6} or {6, 10}; J1 has 7, 8, 9 → Q and T get {5, 6} or {6, 10}.
- Q < P: 17 + Q₁ < 17 + P₂ ≤ 24 → Q₁ ≤ 6. R < Q: 17 + R₃ < 17 + Q₁ → R₃ < Q₁ ≤ 6 → R₃ = 5, so J3 = 5–9 and T₃ = 6; then Q₁ = 6 (must exceed 5).
- P₂ > Q₁ = 6 → P₂ = 7, S₂ = 6. Totals: P 24, Q 23, R 22, S 20.
- T = T₁ + 10 + 6 must lie strictly between 20 and 22 → 21 → T₁ = 5 (so J1 = 5–9).
| Contestant | J1 | J2 | J3 | Total |
|---|---|---|---|---|
| P | 8 | 7 | 9 | 24 |
| Q | 6 | 9 | 8 | 23 |
| R | 9 | 8 | 5 | 22 |
| S | 7 | 6 | 7 | 20 |
| T | 5 | 10 | 6 | 21 |
Medians: P 8, Q 8, R 8, S 7, T 6. P, Q and R tie on 8; J1 scores are R 9, P 8, Q 6. So the order is R, P, Q, S, T and P is second.
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