# Problems and Solutions SEAMO PAPER C 2020

The Southeast Asia Mathematical Olympiad (SEAMO) is an international Math Olympiad competition that originated in Singapore and was founded by Mr. Terry Chew in 2016 in 8 Southeast Asian Countries. Since then, it is growing its popularity around the world. In 2019 it was recognized by 18 countries. In 2020 total number of participating countries increased to 22, including students from Indonesia, Brazil, China, Newzealand, and Taiwan students enrolled in SEAMO 2022-23.

Problem and Solution SEAMO 2020 paper C. Soal ini bersumber dari seamo-official.org

1. A new operation is defined as below

$$π β π = π^π$$

Find the value of $$π$$ in $$π β 5 = 243$$
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

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2. Select 8 distinct numbers from 1 to 9 to fill in each blank below. What is the
greatest possible result?

$$[\;\;\;\; Γ·\;\;\;\; Γ\;\;\;\;(\;\;\;\; +\;\;\;\; )] β [\;\;\;\; Γ\;\;\;\; +\;\;\;\; β\;\;\;\; ]$$

(A) 129
(B) 131
(C) 133
(D) 143
(E) 145

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3. A cube has 8 vertices as shown. How many triangles can be formed by connecting any 3 vertices?

(A) 40
(B) 44
(C) 48
(D) 56
(E) None of the above

Banyak cara = $${8\choose 3}=\frac{8!}{5!.3!}=\frac{8.7.6.5!}{5!.6}= 8Γ7=56$$ cara

4. In 2018, Lucas invested $1000 for a period of 2 years. By the end of the 1st year, he saw his investment suffer an 18% loss. In the 2nd year, his investment showed a 25% gain on the remaining amount. Over the 2-year period, what was the percentage change in his investment? (A) 2.5% gain (B) 2.5% loss (C) 2.8% gain (D) 2.8% loss (E) No change Belum tersedia 5. $$π₯π΄πΈπ·$$ and $$π₯π΅πΉπΆ$$ are identical isosceles triangles. The area of $$π₯π΄πΈπ·$$ is $$202$$ ππΒ². Find the area of rectangle $$π΄π΅πΆπ·$$. (A) 340 ππΒ² (B) 360 ππΒ² (C) 380 ππΒ² (D) 400 ππΒ² (E) 404 ππΒ² Belum tersedia 6. Find the possible values of π such that the 101-digit number, as shown below, is divisible by 7. $$\underbrace{6 6 6 β― 6}_{\mbox{50}}m\underbrace{666…6}_{\mbox{50}}$$ (A) (1, 8) (B) (2 ,6) (C) (2, 9) (D) (5, 6) (E) (3, 7) Belum tersedia 7. In the 4 Γ 4 grid shown below, each line segment measures 1 unit in length. What is the longest distance (in units) a beetle can travel from Point π΄ to Point π΅, if each line segment can only be traversed once? (A) 24 (B) 26 (C) 28 (D) 30 (E) 31 Belum tersedia 8. How many triangles are there in the figure below? (A) 30 (B) 31 (C) 32 (D) 33 (E) 34 Segitiga kecilΒ ada 21 segitiga segitiga sedang ada 10 segitiga segitiga besar ada 3 segitiga Jadi total segitiga seluruhnya ada 34 segitiga 9. How many ways are there to choose two different numbers from the set {5, 6, 7, 8, β― , 15}, such that their sum is even? (A) 23 (B) 24 (C) 25 (D) 26 (E) None of the above $$S = \{5, 6, 7, 8, β¦, 15\}$$ $$A = \{5, 7, 9, 11, 13, 15\}$$ $$B = \{6, 8, 10, 12, 14\}$$ Dua buah bilangan jika dijumlahkan hasilnya adalah genap maka β’Β Memilih 2 bilangan ganjil Banyak cara adalah $${6\choose 2}=15$$ cara β’Β Memilih 2 bilangan genap Banyak cara adalah $${5\choose 2}=10$$ cara Jadi total cara seluruhnya adalah $$15 + 10 =25$$ cara 10. After Wattana spent$35 on a wallet, he and James had money in the ratio 3 βΆ 4. Wattana then received $220 and James spent$50. In the end, Wattana had twice as much money as James. How much money had Wattana at first?
(A) $225 (B)$227
(C) $231 (D)$234
(E) \$240

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11. If $$π΄ = 2^{248}$$ and $$π΅ = 3^{155}$$ , then
(A) $$π΄ = π΅$$
(B) $$π΄ β π΅ = 13$$
(C) $$π΅ β π΄ = 15$$
(D) $$π΄ > π΅$$
(E) $$π΅ > π΄$$

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12. Evaluate

$$\frac {(1+7)\times (1+\frac{7}{2})\times (1+\frac{7}{3})\times …\times (1+\frac{7}{9})}{(1+9)\times (1+\frac{9}{2})\times (1+\frac{9}{3})\times …\times (1+\frac{9}{7})}$$

(A) $$\frac{3}{4}$$
(B) $$\frac{2}{3}$$
(C) $$1$$
(D) $$\frac{4}{3}$$
(E) $$\frac{3}{2}$$

$$\frac {(1+7)\times (1+\frac{7}{2})\times (1+\frac{7}{3})\times …\times (1+\frac{7}{9})}{(1+9)\times (1+\frac{9}{2})\times (1+\frac{9}{3})\times …\times (1+\frac{9}{7})}$$

$$=\frac{8\times \frac{9}{2}\times \frac{10}{3}\times {11}{4}\times β¦\times {16}{9}}{10\times \frac{11}{2}\times {12}{3}\times {13}{4}\timesβ¦\times {16}{7}}$$

$$=\frac{\frac{8Γ9Γ10Γ11Γβ¦Γ16}{2Γ3Γ4Γβ¦Γ9}}{\frac{10Γ11Γ12Γ13Γβ¦Γ16}{2Γ3Γ4Γβ¦Γ7}}$$

$$=\frac{8Γ9Γ10Γ11Γβ¦Γ16}{2Γ3Γ4Γβ¦Γ9}Γ\frac{2Γ3Γ4Γβ¦Γ7}{10Γ11Γ12Γ13Γβ¦Γ16}$$

$$=\frac{8Γ9}{8Γ9}=1$$

13. Each letter represents a unique nonzero

$$π΄\;\; π΅\;\; πΆ$$
$$π·\;\; πΈ\;\; πΉ$$
$$—————-\;+$$
$$π»\;\; πΌ\;\; π½\;\;$$

What is the minimum value of $$(π» + πΌ + π½)$$Β ?
(A) 17
(B) 18
(C) 19
(D) 21
(E) None of the above

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14. Cindy used $$12$$ matchsticks to make a $$2\times 2$$ grid, $$24$$ matchsticks to make a $$3\times 3$$ grid and $$1300$$ matchsticks to make a $$π \times π$$ grid. What is the value of π?

(A) 20
(B) 22
(C) 24
(D) 25
(E) 26

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15. In the quadrant shown below,

(i) π΄ = 28 ππΒ²
(ii) π΄ = 30 ππΒ²
(iii) π΄ = π΅
(iv) π΄ > π΅
Which of the following statements are correct? Take $$π =\frac{22}{7}$$ .
(A) (i) and (iii)
(B) (ii) and (iii)
(C) (i) and (iv)
(D) (ii) and (iv)
(E) (iii) and (iv)

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16. $$Ξπ΄π΅πΆ$$ is an equilateral triangle. The shortest distance from Point $$π$$ to each side are $$4, 5$$ and $$6$$ units, respectively. Find the height of $$Ξπ΄π΅πΆ$$.

(A) 12
(B) 13
(C) 14
(D) 15
(E) 16

Luas segitiga sama sisi =$$\frac{1}{4}π ^2\sqrt 3$$

$$[π΄π΅πΆ]=[π΅ππ΄]+[π΄ππΆ]+[π΅ππΆ]$$
$$\frac{1}{4}π ^2\sqrt 3=\frac{1}{2}(5)(π )+\frac{1}{2}(6)(π )+\frac{1}{2}(4)(π )$$
$$\frac{1}{4}π ^2\sqrt 3=\frac{1}{2}(π )(5+6+4)=\frac{15}{2}π$$

Dengan menggunakan persamaan luas segitiga
$$[π΄π΅πΆ]=[π΄π΅πΆ]$$
$$\frac{1}{2}s.π‘=\frac{1}{4}π ^2\sqrt 3$$
$$\frac{1}{2}s.π‘=\frac{15}{2}s$$
$$π‘=15$$

Jadi tinggi segitiga $$ABC$$ adalah $$15$$

17. Find the value of

$$11 + 22 + 33 + β― + 1089 + 1100$$

(A) 55540
(B) 55550
(C) 55560
(D) 55580
(E) None of the above

\begin{align} 11+22+33+β―+1089+1100 & =11(1+2+3+β―+100)\\ &=11((1+100)1002)\\ &=11((101)1002)\\ &=11(5050)\\ &=55550 \end{align}

18. Find the value of $$β π$$ in $$Ξπ΄π΅πΆ$$ .

(The diagram is not drawn to scale.)
(A) 8Β°
(B) 9Β°
(C) 10Β°
(D) 12Β°
(E) 14Β°

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19. How many numbers are there in the range 1000 to 1999 where the ones digit is greater than the hundreds digit?
(A) 420
(B) 430
(C) 440
(D) 450
(E) None of the above

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20. A fair die is rolled twice. What is the probability that the sum of the two outcomes is prime?

(A) $$\frac{1}{3}$$
(B) $$\frac{5}{12}$$
(C) $$\frac{2}{5}$$
(D) $$\frac{7}{12}$$
(E) $$\frac{3}{5}$$

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21. The sum of ages of a group of people is 4476. The eldest person is β€79 years old. The youngest is β₯30 years old. No more than 3 people in the group are of the same age. What is the minimum number of people who are β₯60 years old?

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22. How many ways are there to form a 5 ππ Γ 3 ππ rectangle from squares of side lengths 1ππ, 2 ππ and 3 ππ ?

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23. Paul walked from π΄ to π΅ at a constant speed of 80π/πππ. At the same time,
Mary walked from π΅ to π΄ at a constant speed of 60π/πππ. They met 120 π away from the midpoint of π΄ and π΅. If Paul stopped for a break along the way, they would still have met 120 π away from the midpoint of π΄ and π΅. How long (in minutes) is the break?

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24. A 2020-digit number is written below. Find the remainder when it is divided by 9.

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 β―

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25. Pool is a game in which players use a cue pole to shoot balls into pockets located along the sides of a table. A 4 Γ 3 pool table has pockets located at π΄, π΅, πΆ and π· , as shown below. A ball shot from π΄ enters the pocket located at π΅ after a few bounces.

If lengths π΄π΅ = 2020 and π΅πΆ = 2019 , which pocket, π΄, π΅, πΆ or π· , will a ball shot in the same way from π΄ enter?

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