Singapore Math Olympiad Past Years Questions- Number Theory (Senior)

Problem 01: (Year 2023, Problem 04)

If \(\log {\sqrt{2}} x=10-3 \log {\sqrt{2}} 10\), find \(x\).
(A) 0.32
(B) 0.032
(C) 0.0032
(D) 0.64
(E) 0.064

Problem 02: (Year 2023, Problem 08)

Find the smallest positive odd integer greater than 1 that is a factor of
\((2023)^{2023}+(2026)^{2026}+(2029)^{2029}\) .

Problem 03: (Year 2023, Problem 09)

Find the remainder of \(7^{2023}+9^{2023}\) when divided by \(64\) .

Problem 04: (Year 2023, Problem 10)

Let \(x, y, z>1\), and let \(A\) be a positive number such that \(\log x A=30, \log _y A=50\) and \(\log {x y}(A z)=150\). Find

\(\left(\log _A z\right)^2\).

Problem 05: (Year 2023, Problem 11)

Find the largest integer that is less than
\(\text { - } \frac{3^{10}-2^{10}}{10 !}\left(\frac{1}{1 ! 9 ! 2}+\frac{1}{2 ! 8 ! 2^2}+\frac{1}{3 ! 7 ! 2^3}+\cdots+\frac{1}{9 ! 1 ! 2^9}\right)^{-1}\) .

Here, \(n !=n \cdot(n-1) \cdots 3 \cdot 2 \cdot 1\). For example, \(5 !=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=120\).

Problem 06: (Year 2023, Problem 19)

Find the remainder when \(3^{2023}\) is divided by \(215\) .

Problem 07: (Year 2023, Problem 20)

Find the sum of the prime divisors of \(64000027\) .

Problem 08: (Year 2022, Problem 02)

Which of the following is true?
(A) \(\sqrt[6]{\frac{1}{333}}<\sqrt[3]{\frac{1}{18}}<\sqrt{\frac{1}{7}}\)
(B) \(\sqrt[3]{\frac{1}{18}}<\sqrt[6]{\frac{1}{333}}<\sqrt{\frac{1}{7}}\)
(C) \(\sqrt[3]{\frac{1}{18}}<\sqrt{\frac{1}{7}}<\sqrt[6]{\frac{1}{333}}\)
(D) \(\sqrt{\frac{1}{7}}<\sqrt[6]{\frac{1}{333}}<\sqrt[3]{\frac{1}{18}}\)
(E) None of the above.

Problem 09: (Year 2022, Problem 03)

Suppose \(\sqrt{\left(\log {377 \times 377} 2022\right)\left(\log {377} 2022\right)}=\log _k 2022\). Find \(k\).
(A) \(\sqrt{337}\)
(B) \(337^{\sqrt{2}}\)
(C) \(337 \sqrt{2}\)
(D) \(\sqrt{337}^{\sqrt{2}}\)
(E) \(\sqrt{337 \times 2}\)

Problem 10: (Year 2022, Problem 08)

How many positive integers less than or equal to 2022 cannot be expressed as
\(\lfloor 2 x+1\rfloor+\lfloor 5 x+1\rfloor\) for some real number \(x\) ? Here, \(\lfloor x\rfloor\) denotes the greatest integer less than or equal to \(x\). For example, \(\lfloor-2.1\rfloor=-3,\lfloor 3.9\rfloor=3\).

Problem 11: (Year 2022, Problem 14)

Let \(k=-1+\sqrt{2022^{1 / 5}-1}\), and let \(f(x)=\left(k^2+2 k+2\right)^{10 x}\). Find the value of \(\log _{2022} f(2022)\).

Problem 12: (Year 2022, Problem 15)

Find the smallest odd integer \(N\), where \(N>2022\), such that when \(1808,2022\) and \(N\) are each divided by a positive integer \(p\), where \(p>1\), they all leave the same remainder.

Problem 13: (Year 2022, Problem 19)

For some positive integer \(n\), the number \(n^3-3 n^2+3 n\) has a units digit of \(6\) . Find the product of the last two digits of the number \(7(n-1)^{12}+1\).

Problem 14: (Year 2022, Problem 23)

How many positive integers \(n\) do not satisfy the inequality \(n^{\frac{1}{3} \log _{20} n}>\sqrt{n}\) ?

Problem 15: (Year 2021, Problem 03)

Find the value of \(2021^{\left(\log {2021} 2020\right)\left(\log {2020} 2019\right)\left(\log _{2019} 2018\right)}\).
(A) 2018
(B) 2019
(C) 2020
(D) 2021
(E) None of the above

Problem 16: (Year 2021, Problem 05)

Select all the inequalities which hold for all real values of (x) and (y).

(i) \(x \leq x^2+y^2\),
(ii) \(x y \leq x^2+y^2\),
(iii) \(x-y \leq x^2+y^2\),
(iv) \(y+x y \leq x^2+y^2\),
(v) \(x+y-1 \leq x^2+y^2 \).
(A) (i)
(B) (i) and (iii)
(C) (iii) and (iv)
(D) (ii)
(E) (ii) and (v)

Problem 17: (Year 2021, Problem 06)

Let \(x\) be the integer such that \(x=5 \sqrt{2+4 \log _x 5}\). Determine the value of \(x\).

Problem 18: (Year 2021, Problem 24)

Let \(n\) be a positive integer such that \(\frac{2021 n}{2021+n}\) is also a positive integer. Determine the smallest possible value of (n).

Problem 19: (Year 2021, Problem 25)

Determine the number of 5-digit numbers with the following properties:
(i) All the digits are non-zero;
(ii) The digits can be repeated;
(iii) The difference between consecutive digits is exactly 1 .

Problem 20: (Year 2020, Problem 02)

Which of the following is equal to \(\sqrt{5+\sqrt{3}}+\sqrt{5-\sqrt{3}}\) ?
(A) \(\sqrt{10-\sqrt{22}}\)
(B) \(\sqrt{10+\sqrt{22}}\)
(C) \(\sqrt{10-2 \sqrt{22}}\)
(D) \(\sqrt{10+2 \sqrt{22}}\)
(E) None of the above

Problem 21: (Year 2020, Problem 03)

Simplify
\(\log 8 5 \cdot\left(\log _5 3+\log {25} 9+\log _{125} 27\right)\) .
(A) \(\log _2 3\)
(B)\(\log _3 2\)
(C) \(\log _2 9\)
(D) \(\log _3 16\)
(E) \(\log _2 27\)

Problem 22: (Year 2020, Problem 04)

Let \(a=50^{\frac{1}{505}}, b=10^{\frac{1}{303}}\) and \(c=6^{\frac{1}{202}}\). Which of the following is true?
(A) \(a<b<c\)
(B) \(a<c<b\)
(C) \(b<a<c\)
(D) \(b<c<a\)
(E) \(c<b<a\)

Problem 23: (Year 2020, Problem 06)

Find the minimum possible value of \(|x-10|-|x-20|+|x-30|\), where \(x\) is any real number.

Problem 24: (Year 2020, Problem 19)

Let \(a, b\) be positive real numbers, where \(a>b\). Suppose there exists a real number (x) such that \(\left(\log _2 a x\right)\left(\log _2 b x\right)+25=0\). Find the minimum possible value of \(\frac{a}{b}\).

Problem 25: (Year 2020, Problem 21)

Find the smallest positive integer that is greater than the following expression:
(\(\sqrt{7}+\sqrt{5})^4\).

Problem 26: (Year 2020, Problem 24)

The digit sum of a number, say 987 , is the sum of its digits, \(9+8+7=24\). Let (A) be the digit sum of \(2020^{2021}\), and let (B) be the digit sum of (A). Find the digit sum of (B).

Problem 27: (Year 2020, Problem 25)

\(40=2 \times 2 \times 2 \times 5\) is a positive divisor of 1440 that is a product of 4 prime numbers. \(48=2 \times 2 \times 2 \times 2 \times 3\) is a positive divisor of 1440 that is a product of 5 prime numbers. Find the sum of all the positive divisors of 1440 that are products of an odd number of prime numbers.

Problem 28: (Year 2019, Problem 02)

Simplify
\[
(\sqrt{10}-\sqrt{2})^{\frac{1}{3}}(\sqrt{10}+\sqrt{2})^{\frac{7}{3}} .
\]
(A) \(24+4 \sqrt{5}\)
(B) \(24+6 \sqrt{5}\)
(C) \(24+8 \sqrt{5}\)
(D) \(24+10 \sqrt{5}\)
(E) \(24+12 \sqrt{5}\)

Problem 29: (Year 2019, Problem 03)

Let \(a=4^{3000}, b=6^{2500}\) and \(c=7^{2000}\). Which of the following statement is true?
(A) \(a<b<c\)
(B) \(a<c<b\)
(C) \(b<a<c\)
(D) \(c<a<b\)
(E) \(c<b<a\)

Problem 30: (Year 2019, Problem 04)

If \(\log _{21} 3=x\), express \(\log _7 9\) in terms of \(x\).
(A) \(\frac{2 x}{2-x}\)
(B) \(\frac{2 x}{1-x}\)
(C) \(\frac{2 x}{x-2}\)
(D) \(\frac{2 x}{x-1}\)
(E) \(\frac{x}{1-x}\)

Problem 31: (Year 2019, Problem 06)

Find the largest positive integer (n) such that (n+8) is a factor of \(n^3+13 n^2+40 n+40\).

Problem 32: (Year 2019, Problem 14)

Find the value of the following expression:
\[
\frac{2\left(1^2+2^2+3^2+\ldots+49^2+50^2\right)+(1 \times 2)+(2 \times 3)+(3 \times 4)+\ldots+(48 \times 49)+(49 \times 50)}{100} .
\]

Problem 33: (Year 2019, Problem 17)

Find the remainder when \(10^{43}\) is divided by \(126\) .