# Questions based on Wilson’s theorem for training for RMO

1(a) Find the remainder when $15!$ is divided by 17.
1(b) Find the remainder when $2(26!)$ is divided by 29.

2: Determine whether 17 is a prime by deciding if $16! \equiv -1 {\pmod 17}$

3: Arrange the integers 2,3,4, …, 21 in pairs a and b that satisfy $ab \equiv 1 {\pmod 23}$.

4: Show that $18! \equiv -1 {\pmod 437}$.

5a: Prove that an integer $n>1$ is prime if and only if $(n-2)! \equiv 1 {\pmod n}$.
5b: If n is a composite integer, show that $(n-1)! \equiv 0 {\pmod n}$, except when $n=4$.

6: Given a prime number p, establish the congruence $(p-1)! \equiv {p-1} {\pmod {1+2+3+\ldots + (p-1)}}$

7: If p is prime, prove that for any integer a, $p|a^{p}+(p-1)|a$ and $p|(p-1)!a^{p}+a$

8: Find two odd primes $p \leq 13$ for which the congruence $(p-1)! \equiv -1 {\pmod p^{2}}$ holds.

9: Using Wilson’s theorem, prove that for any odd prime p:
$1^{2}.3^{2}.5^{2}.\ldots (p-2)^{2} \equiv (-1)^{(p+1)/2} {\pmod p}$

10a: For a prime p of the form $4k+3$, prove that either

$(\frac{p-1}{2})! \equiv 1 {\pmod p}$ or $(\frac{p-1}{2})! \equiv -1 {\pmod p}$

10b: Use the part (a) to show that if $4k+3$ is prime, then the product of all the even integers less than p is congruent modulo p to either 1 or -1.

More later,
Nalin Pithwa.

# Eight digit bank identification number and other problems of elementary number theory

Question 1:

Consider the eight-digit bank identification number $a_{1}a_{2}\ldots a_{8}$, which is followed by a ninth check digit $a_{9}$ chosen to satisfy the congruence

$a_{9} \equiv 7a_{1} + 3a_{2} + 9a_{3} + 7a_{4} + 3a_{5} + 9a_{6} + 7a_{7} + 3a_{8} {\pmod {10}}$

(a) Obtain the check digits that should be appended to the two numbers 55382006 and 81372439.

(b) The bank identification number $237a_{4}18538$ has an illegitimate fourth digit. Determine the value of the obscured digit.

Question 2:

(a) Find an integer having the remainders 1,2,5,5 when divided by 2, 3, 6, 12 respectively (Yih-hing, died 717)

(b) Find an integer having the remainders 2,3,4,5 when divided by 3,4,5,6 respectively (Bhaskara, born 1114)

(c) Find an integer having remainders 3,11,15 when divided by 10, 13, 17, respectively (Regiomontanus, 1436-1476)

Question 3:

Question 3:

Let $t_{n}$ denote the nth triangular number. For which values of n does $t_{n}$ divide $t_{1}^{2} + t_{2}^{2} + \ldots + t_{n}^{2}$

Hint: Because $t_{1}^{2}+t_{2}^{2}+ \ldots + t_{n}^{2} = t_{n}(3n^{3}+12n^{2}+13n+2)/30$, it suffices to determine those n satisfying $3n^{3}+12n^{2}+13n+2 \equiv 0 {\pmod {2.3.5}}$

Question 4:

Find the solutions of the system of congruences:

$3x + 4y \equiv 5 {\pmod {13}}$
$2x + 5y \equiv 7 {\pmod {13}}$

Question 5:

Obtain the two incongruent solutions modulo 210 of the system

$2x \equiv 3 {\pmod 5}$
$4x \equiv 2 {\pmod 6}$
$3x \equiv 2 {\pmod 7}$

Question 6:

Use Fermat’s Little Theorem to verify that 17 divides $11^{104}+1$

Question 7:

(a) If $gcd(a,35)=1$, show that $a^{12} \equiv {\pmod {35}}$. Hint: From Fermat’s Little Theorem, $a^{6} \equiv 1 {\pmod 7}$ and $a^{4} \equiv 1 {\pmod 5}$

(b) If $gcd(a,42) =1$, show that $168=3.7.8$ divides $a^{6}-1$
(c) If $gcd(a,133)=gcd(b,133)=1$, show that $133| a^{18} - b^{18}$

Question 8:

Show that $561|2^{561}-1$ and $561|3^{561}-3$. Do there exist infinitely many composite numbers n with the property that $n|2^{n}-2$ and $n|3^{n}-3$?

Question 9:

Prove that any integer of the form $n = (6k+1)(12k+1)(18k+1)$ is an absolute pseudoprime if all three factors are prime; hence, $1729=7.13.19$ is an absolute pseudoprime.

Question 10:

Prove that the quadratic congruence $x^{2}+1 \equiv 0 {\pmod p}$, where p is an odd prime, has a solution if and only if $p \equiv {pmod 4}$.

Note: By quadratic congruence is meant a congruence of the form $ax^{2}+bx+c \equiv 0 {\pmod n}$ with $a \equiv 0 {\pmod n}$. This is the content of the above proof.

More later,
Nalin Pithwa.

# Pre RMO algebra : some tough problems

Question 1:

Find the cube root of $x^{3} -12x^{2} + 54x -112 + \frac{108}{x} - \frac{48}{x^{2}} + \frac{8}{x^{3}}$

Question 2:

Find the square root of $\frac{x}{y} + \frac{y}{x} +3 - 2\sqrt{\frac{x}{y}} -2\sqrt{\frac{y}{x}}$

Question 3:

Simplify (a):

$(\frac{x}{x-1} - \frac{1}{x+1}). \frac{x^{3}-1}{x^{6}+1}.\frac{(x-1)^{2}(x+1)^{2}+x^{2}}{x^{4}+x^{2}+1}$

Simplify (b):
$\{ \frac{a^{4}-y^{4}}{a^{2}-2ay+y^{2}} \div \frac{a^{2}+ay}{a-y} \} \times \{ \frac{a^{5}-a^{3}y^{2}}{a^{3}+y^{3}} \div \frac{a^{4}-2a^{3}y+a^{2}y^{2}}{a^{2}-ay+y^{2}}\}$

Question 4:

Solve : $\frac{3x}{11} + \frac{25}{x+4} = \frac{1}{3} (x+5)$

Question 5:

Solve the following simultaneous equations:

$2x^{2}-3y^{2}=23$ and $2xy - 3y^{2}=3$

Question 6:

Simplify (a):

$\frac{1- \frac{a^{2}}{(x+a)^{2}}}{(x+a)(x-a)} \div \frac{x(x+2a)}{(x^{2}-a^{2})(x+a)^{2}}$

Simplify (b):

$\frac{6x^{2}y^{2}}{m+n} \div \{\frac{3(m-n)x}{7(r+s)} \div \{ \frac{4(r-s)}{21xy^{2}} \div \frac{(r^{2}-s^{2})}{4(m^{2}-n^{2})}\} \}$

Question 7:

Find the HCF and LCM of the following algebraic expressions:

$20x^{4}+x^{2}-1$ and $25x^{4}+5x^{3} - x - 1$ and $25x^{4} -10x^{2} +1$

Question 8:

Simplify the following using two different approaches:

$\frac{5}{6- \frac{5}{6- \frac{5}{6-x}}} = x$

Question 9:

Solve the following simultaneous equations:

Slatex x^{2}y^{2} + 192 = 28xy\$ and $x+y=8$

Question 10:

If a, b, c are in HP, then show that

$(\frac{3}{a} + \frac{3}{b} - \frac{2}{c})(\frac{3}{c} + \frac{3}{b} - \frac{2}{a})+ \frac{9}{b^{2}}=\frac{25}{ac}$

Question 11:

if $a+b+c+d=2s$, prove that

$4(ab+cd)^{2} - (a^{2}+b^{2}-c^{2}-d^{2})^{2}= 16(s-a)(s-b)(s-c)(s-d)$

Question 12:

Determine the ratio $x:y:z$ if we know that

$\frac{x+z}{y} = \frac{z}{x} = \frac{x}{z-y}$

More later,
Nalin Pithwa

Those interested in such mathematical olympiads should refer to:

(I am a tutor for such mathematical olympiads).

# Elementary Number Theory, ISBN numbers and mathematics olympiads

Question 1:

The International Standard Book Number (ISBN) used in many libraries consists of nine digits $a_{1} a_{2}\ldots a_{9}$ followed by a tenth check digit $a_{10}$ (somewhat like Hamming codes), which satisfies

$a_{10} = \sum_{k=1}^{9}k a_{k} \pmod {11}$

Determine whether each of the ISBN’s below is correct.
(a) 0-07-232569-0 (USA)
(b) 91-7643-497-5 (Sweden)
(c) 1-56947-303-10 (UK)

Question 2:

When printing the ISBN $a_{1}a_{2}\ldots a_{9}$, two unequal digits were transposed. Show that the check digits detected this error.

Remark: Such codes are called error correcting codes and are fundamental to wireless communications including cell phone technologies.

More later,
Nalin Pithwa.

# Mathematics Olympiads: A curious calculation and its cute proof !!

Explain why the following calculations hold:

$1.9 + 2 =11$
$12.9 + 3 = 111$
$123.9 + 4 = 1111$
$1234.9 + 5 = 11111$
$12345.9 + 6 = 111111$
$123456.9 + 7 = 1111111$
$1234567.9 + 8 = 11111111$
$12345678.9 + 9 = 111111111$
$123456789.9 + 10 = 11111 11111$

Hint:

Show that $(10^{n-1}+2.10^{n-2}+3.10^{n-3}+ \ldots + n)(10-1) + (n+1)=\frac{10^{n+1}-1}{9}$

More later,
Nalin Pithwa