Solutions to Pathfinder for Olympiad Mathematics

Polynomials

This exercise contains a lot of tricky questions

BUYU 1

Solution---------

1

Find a fourth degree equation with rational coefficients, one of whose roots is, \( \sqrt{3} + \sqrt{7} \).

Solution

2

Find a polynomial equation of the lowest degree with rational coefficients whose one root is \( \sqrt[3]{2} + 3\sqrt[3]{4} \)

Solution

3

Form the equation of the lowest degree with rational coefficients which has \( 2+\sqrt{3} \) and \( 3+ \sqrt{2} \) as two of its roots.

Solution

4

Show that \( (x-1)^2 \) is a factor of \( x^n -nx + n -1 \).

Solution

5

If a, b, c, d, e are all zeroes of the polynomial \(6x^5+5x^4+4x^3+3x^2+2x+1\), find the value of \( (1+a) (1+b) (1+c) (1+d) (1+e) \).

Solution

6

If \( 1 , \alpha_1, \alpha_2 , \cdots , \alpha_{n-1} \) be the roots of the equation \( x^n -1 = 0 \) , \( n \in \mathbb{N} , n \geq 2 \) show that \( n = (1-\alpha_1)(1-\alpha_2)(1-\alpha_3)\cdots(1-\alpha_{n-1}) \)

Solution

7

If \( \alpha, \beta, \gamma, \delta \) be the roots of the equation \( x^4+ px^3 + qx^2 + rx + s = 0\), show that \( (1+\alpha^2)(1+\beta^2)(1+\gamma^2)(1+\delta^2)=(1-q+s)^2+(p-r)^2 \)

Solution

8

If \( f(x) = x^4 +ax^3 +bx^2 +cx +d \) is a polynomial such that f(1)=10,f(2)=20,f(3) = 30, find the value of \( \frac{f(12) + f(-8)}{10} \)

Solution

9

The polynomial \( x^{2k} +1+(x+1)^{2k} \) is not divisible by \( x^2 + x + 1\) . Find the value of \( k \in \mathbb{N} \)

Solution

10

Find a such that \( ax^{17} + bx^{16} +1\) is divisible by \( x^2 -x-1 \).

Solution

11

Find all polynomials P(x) with real coefficients such that \( (x-8)P(2x)=8(x-1)P(x) \).

Solution

12

Find all polynomial P(x) with real coefficients such that \( (x^2 - ax+18)P(x) -(x^2+3x)P(x-3)=0\)

Solution

13

Let \( (x-1)^3 \) divides \( (p(x) + 1)\) and \( (x+1)^3 \) divides \( (p(x)-1) \). Find the polynomial p(x) of degree 5.

Solution

BUYU 2

Solution---------

1

Find the rational roots of \( x^4 -4x^3 +6x^2 -4x+1 = 0\)

Solution

2

Solve the equation \( x^4 +10x^3 +35x^2 +50x^2 +24 = 0 \), if sum of two of its roots is equal to sum of the other two roots.

Solution

3

Find the rational roots of \( 6x^4 +x^3 -3x^2 -9x-4 = 0 \).

Solution

4

Find the rational roots of \( 6x^4 +35x^3 +62x^2 +35x +2 = 0\).

Solution

5

Given that the sum of two of the roots of \( 4x^3 +ax^2 -x+b = 0\) is zero, where \( a,b \in \mathbb{Q} \). Solve the equation for all values of a and b.

Solution

6

Find all a, b, such that the roots of \( x^3 + ax^2 + bx -8 = 0\) are real and in G.P.

Solution

7

Show that \( 2x^6 +12x^5 +30x^4 +60x^3 +80x^2 +30x +45 = 0 \) has no real roots.

Solution

8

Construct a polynomial equation, of the least degree with rational coefficients, one of whose roots is \( \sin 10^\circ \).

Solution

9

Construct a polynomial equation of the least degree with rational coefficients, one of whose roots is \( \sin 20^\circ \).

Solution

10

Construct a polynomial equation of the least degree, with rational coefficients, one of whose roots is (a) \( \cos 10^\circ \) (b) \( \cos 20^\circ \)

Solution

11

Construct a polynomial equation of the least degree, with rational coefficients, one of whose roots is (a) \( \tan 10^\circ \) (b) \( \tan 20^\circ \)

Solution

12

Construct a polynomial equation with rational coefficients, two of whose roots are \( \sin 10^\circ \) and \( \cos 20^\circ \)

Solution

13

If p, q, r are the roots of \( x^3 -6x^2 +3x +1 = 0\) , determine the possible values of \( p^2q + q^2r +r^2p\). Also find \( |(p-q)(q-r)(r-p)|\).

Solution

14

The product of two of the four roots of the quartic equation \( x^4 -18x^3 +kx^2 +200x -1984 = 0\) is -32. Determine the value of k.

Solution

BUYU 3

Solution---------

1

If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 -(a+d)x+ad-bc = 0\), show that \( \alpha^3 \) and \( \beta^3 \) are the roots of \( y^2 -(a^3 +d^3 +3abc +3bcd)y +(ad-bc)^3 = 0\).

Solution

2

If \( a^3 +b^3 +c^3 = (a+b+c)^3 \), prove that \( a^5 + b^5 + c^5 = (a+b+c)^5\). Generalize your result.

Solution

3

If p, q and rare distinct roots of \( x^3 -x^2 +x-2 = 0\) find the value of \( p^3 +q^3 +r^3 \).

Solution

4

Find the sum of the 5th powers of the roots of the equation \(x^3 +3x+9 = 0\).

Solution

5

Find the sum of the fifth powers of the roots of the equation \( x^3 -7x^2 +4x -3 = 0\).

Solution

6

\( \alpha, \beta, \gamma \) are the roots of the equation \(x^3 -9x+9 = 0\). Find the value of \( \alpha^{-3} + \beta^{-3} + \gamma^{-3} \) and \( \alpha^{-5} + \beta^{-5} + \gamma^{-5} \).

Solution

7

Form the cubic equation whose roots are \( \alpha, \beta, \gamma \) such that

(i) \( \alpha + \beta + \gamma = 9 \)

(ii) \( \alpha^2 + \beta^2 + \gamma^2 = 29 \)

(iii) \( \alpha^3 + \beta^3 + \gamma^3 = 99\)

Hence, find the value of \( \alpha^4 + \beta^4 + \gamma^4 \)

Solution

8

If \( \alpha + \beta + \gamma = 4 \), \( \alpha^2 + \beta^2 + \gamma^2 = 7 \),

\( \alpha^3 + \beta^3 + \gamma^3 = 28\), find \( \alpha^4 + \beta^4 + \gamma^4 \) and \( \alpha ^5 + \beta^5 + \gamma^5 \).

Solution

9

Solve: \( x^3 + y^3 +z^3 = a^3, x^2 + y^2 + z^2 = a^2, x+y+z = a\) in terms of a.

Solution

10

If \( \alpha, \beta, \gamma \) be the roots of \( 2x^3 +x^2 +x+1 = 0 \), show that \[ \left ( \frac{1}{\beta^3}+\frac{1}{\gamma^3} -\frac{1}{\alpha^3} \right ) \left ( \frac{1}{\gamma^3}+\frac{1}{\alpha^3} -\frac{1}{\beta^3} \right ) \left ( \frac{1}{\alpha^3}+\frac{1}{\beta^3} -\frac{1}{\gamma^3}\right ) = 16 \]

Solution

11

Find \( x,y \in \mathbb{C} \) such that \( x^5+y^5=275, x+y=5 \).

Solution

12

Find real x such that \( \sqrt[4]{97-x} + \sqrt[4]{x} = 5\).

Solution

Number Theory

This exercise contains a lot of tricky questions

BUYU 1

Solution---------

1

Prove that \( (a-c) \mid (ab+cd) \) if and and only if \( (a-c) \mid (ad+bc) \)

Solution

2

Prove that \(6 \mid (a+b+c)\) if and only if \(6 \mid (a^3+b^3+c^3)\)

Solution

3

Prove that \( 641 \mid (2^{32} +1)\)

Solution

\( 641 = 640+1 = 5 \times 2^7 +1 \)

\( \therefore 5 \times 2^7 \equiv -1 (\text{mod} 641) \)

\( \implies (5 \times 2^7)^4 \equiv (-1)^4 (\text{mod} 641) \)

\( \implies 5^4 \times 2^{28} \equiv 1 (\text{mod} 641) \)

\( 5^4 \times 2^{28} -1 + 2^{32} +1 = 2^{28}\underbrace{(5^4 + 2^4)}_{=641} \)

\( \therefore 641 \mid (5^4 \times 2^{28} -1) \text{and} 641 \mid (5^4 \times 2^{28} -1 + 2^{32} +1) \)

\( \implies 641 \mid (2^{32} +1) \)

4

Find all natural numbers n, such that \( \frac{(n+1)^2}{n+7} \) is an integer. Find corresponding values of the expression also.

Solution

5

Prove that for natural number \( n\) , \( 1^n +8^n -3^n -6^n \) is divisible by 10.

Solution

6

Prove that \( 1^k + 2^k + 3^k + \cdots + n^k \) is divisible by by \( 1+2+3+\cdots + n\) when \(n\) is an integer and \(k\) is odd.

Solution

7

Prove that for any natural number \(n\) , the result of \( 1^{1987} +2^{1987} +\cdots + n^{1987} \) cannot be divided by \((n+2)\) without a remainder.

Solution

8

If \(a,m,n\) are positive integers with \( a>1\) and \( (a^m-1)\mid (a^n -1)\) then \( m \mid n\).

Solution

9

Let \(a,b\) be positive integers with \(b>2\). Show that \( (2^b-1) \nmid (2^a +1)\)

Solution

10

Let \(a,b,c,d\) be integers such that \(ad-bc>1\). Prove that there is at least one among \(a,b,c,d\) which is not divisible by \(ad-bc\).

Solution

BUYU 2

Solution---------

1

If \( a = qb +r\) where a,q,b and r are integers, then prove that \( (a,b) = (b,r) \)

Solution

2

If a,b are integers both greater than zero and d is their gcd, then prove that \( d = ax+by \) for some \( x,y \in \mathbb{Z} \).

Solution

3

Prove that \( \frac{12n+1}{30n+2} \) is irreducible for every positive integer n.

Solution

4

Prove that the expression \( \frac{63n+14}{42n+9} \) is irreducible for every positive integer n.

Solution

5

Show that \( \text{gcd}(n!+1,(n+1)!+1) = 1\) for any \( n \in \mathbb{N} \).

Solution

6

Prove that the expression \(2x+3y\) and \(9x+5y\) are divisible by 17 for the same set of integral values of x and y.

Solution

7

If x,y are integers and 17 divides both the expressions \( x^2-2xy+y^2-5x+7y\) and \(x^2-3xy+2y^2+x-y\), then prove that 17 divides \(xy-12x+15y\).

Solution

8

Find the least possible value of a+b, where a,b are positive integers such that 11 divides a+13b and 13 divides a+11b.

Solution

9

Show that if \( 13 \mid (n^2+3n+51) \) then \( 169 \mid (21n^2+89n+44\).

Solution

10

If \( \text{gcd}(a,b) = 1\), then prove that \( (a^2+b^2,ab) = 1\) and also prove that \( \text{gcd}(a+b,a^2-ab+b^2) =1 \text{ or } 3\).

Solution

11

If \( a,b \in \mathbb{N} \) and \( ab \mid (a^2 +b^2) \) then prove that \( a= b\).

Solution

12

Let a,b,c be positive integers such that \( a \mid b^2, b \mid c^2, c \mid a^2\). Prove that \( abc \mid (a+b+c)^7 \).

Solution

13

If \( \text{gcd}(a,b,c)=1\) and \( c = \frac{ab}{a-b} \) then prove that \(a-b\) is a perfect square.

Solution

14

Let m,n be positive integers such that \( 3m+n = 3 \text{lcm}[m,n]+\text{gcd}(m,n)\) prove that \( n \mid m\).

Solution

15

Let \(a_1,b_1,c_1 \in \mathbb{N} \). We define \( a_2 = \text{gcd}(b_1,c_1),b_2 = \text{gcd}(c_1,a_1), c_2 = \text{gcd}(a_1,b_1), a_3 = \text{lcm}(b_2,c_2),b_3 = \text{lcm}(c_2,a_2),c_3 = \text{lcm}(a_2,b_2)\). Show that \( \text{gcd}(b_3,c_3) = a_2\).

Solution

16

Find the minimum possible least common multiple of twenty(not necessarily distinct) natural numbers whose sum is 801.

Solution

17

Let \( m,n,l \in \mathbb{N} \) and \( \text{lcm}[m+l,m] = \text{lcm}[n+l,n]\) then prove that \( m=n\).

Solution

18

Find the set of all ordered pairs of integers (a,b) such that \( \text{gcd}(a,b) = 1\) and \( \frac{a}{b} + \frac{14b}{25a} \) is an integer.

Solution

19

Let \( \frac{a}{b} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots + \frac{1}{1319}\) such that gcd\((a,b) = 1\). Show that \( 1979 \mid a\).

Solution

20

Let \( \frac{a}{b} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{2002}\) such that gcd\((a,b) = 1\). Show that \( 2003 \mid a\).

Solution

21

Let \( \frac{a}{b} = 1 - \frac{1}{2} + \frac{1}{3} - \cdots + \frac{1}{67} \) such that gcd\( (a,b) = 1\). Show that \( 101 \mid a\).

Solution

22

Let \( m,n \in \mathbb{N}\) and n be an odd number then prove that gcd\( (2^n-1,2^m+1) = 1\).

Solution

23

For each positive integer n, define \( a_n = 20+n^2\) and \( d_n = \text{gcd}(a_n,a_{n+1})\). Find the set of all values that are taken by \( d_n\) and show by examples that each of these values are attained.

Solution

24

Let \( P(x) = x^3 +ax^2 +b\) and \( Q(x) = x^3 +bx+a\), where a,b are non-zero real numbers. Suppose that the roots of the equation \( P(x) = 0\) are the reciprocals of the roots of the equation \( Q(x) = 0\). Prove that a and b are integers. Find the greatest common divisor of \( P(2013! +1) \text{ and } Q(2013!+1) \).

Solution

25

If \( (a,b) = 1\) and \( x^a = y^b\) for some natural numbers a,b,x,y all greater than 1 then show that \( x = n^b \text{ and } y = n^a\) for some \(n > 1\).

Solution

26

Prove that \( \text{gcd}(k^a-1,k^b-1) = k^{\text{gcd}(a,b)} -1\) where \( k>1; k,a,b \in \mathbb{N} \).

Solution

BUYU 3

Solution---------

1

Show that \(4n^3+6n^2+4n+1\) is composite for n = 1, 2, 3...

Solution

2

Prove that \(512^3+ 675^3+720^3\) is not a prime number.

Solution

3

Prove that \(5^{12}+ 2^{10}\) is composite.

Solution

4

Show that \(3^{2008} + 4^{2009}\) can be written as a product of two integers each of which is greater than \(2009^{182}\).

Solution

5

Prove that if \(p\) and \(p^2+2\) are primes, then \(p^3+2\) is also a prime.

Solution

6

Prove that if \(2n + 1\) and \(3n+ 1\) are squares, then \(5n+ 3\) is not prime where, \( n \in \mathbb{N} \).

Solution

7

Find all distinct primes p, q such that \(p^2-2q^2=1\).

Solution

8

Find all integers n such that \( \left ( \frac{n^3 -1}{5} \right ) \) is prime.

Solution

9

Find all numbers p such that all six numbers \(p, p +2, p +6, p +8, p + 12, \text{ and } p+ 14\) are primes.

Solution

10

Prove that \( N = \frac{5^{125} -1}{5^{25}-1} \) is a composite number.

Solution

11

Find all primes p and q such that \(p^2+7pq + q^2\) is a square of an integer.

Solution

12

Find all triples (p, q, r) of primes such that \(pq = r + 1\) and \(2(p^2+ q^2)=r^2+1\)

Solution

13

Prove that, if a, b are prime numbers (a> b), each containing at least two digits, then \( (a^4 -b^4) \) is divisible by 240. Also prove that, 240 is the gcd of all the numbers which arise in this way.

Solution

14

Prove that there are infinitely many primes of the form 4n-1.

Solution

15

Prove that there are infinitely many primes of the form 6n - 1.

Solution

16

If \(ab=cd\) , prove that \(a^2+b^2+c^2+d^2 \) is composite.

Solution

17

Let \( m,n \in \mathbb{N} \) such that \(2m^2+m=3n^2+n\) , then prove that \(m -n\) and \(2m+2n+1\) are perfect squares. Also find all integral solution of \(2m^2 + m=3n^2+n\).

Solution

18

Let \( a,b,c,d \in \mathbb{N} \) and in strictly increasing order such that \(b^2-bd-d^2=a^2-ac-c^2\). Prove that \(ab + cd\) is not a prime number.

Solution

19

Let \((P_1, P_2, P_3,\cdots, P_n,\cdots)\) be a sequence of primes defined by \( P_1 = 2\) and for \( n \geq 1\) , \(P_{n+1}\) is the largest prime factor of \(P_1P_2 \cdots P_n +1\). (Thus \(p_2=3\), \(p_3=7\)). Prove that \(P_1 \neq 5\) for any n.

Solution

20

Let n be a positive integer and \( p_1,p_2,\cdots,p_n\), be n prime numbers all larger than 5 such that \( 6 \mid (p_1^2+p_2^2+\cdots+p_n^2)\). Prove that \( 6 \mid n\).

Solution

21

Prove that for \( n \geq 5\), \( p_{n+1}^3 < p_1p_2 \cdots p_n\) where \(p_i\) is the ith prime.

Solution

22

(a)If n is not a prime, prove that \(2^n -1\) is not a prime.

(b)Prove that if n> 1, \(a,n \in \mathbb{N}\) and \(a^n- 1\) is prime, then \(a = 2\) and n must be a prime.

(c)Show that if p is an odd prime then every prime divisor of \(2^p - 1\) is of the form 2kp+1 for some \(k \in \mathbb{N}\).

Solution

23

(a)If n has an odd divisor > 1, prove that \(2^n + 1\) is not prime.

(b)Prove that if \(a^n + 1\) is prime and a > 1, then a must be even and \(n = 2^k\) for some \(k \in \mathbb{N}_0\). Prove that \( F_0F_1 \cdots F_{n-1} +2 = F_n \).

(c)Show that every prime divisor of \( 2^{2^n} +1\) is of the form \( k2^{n+2} +1\) for some \( k \in \mathbb{N} \), for \( n > 1\).

(d)Prove that \( \text{gcd}(F_a,F_b) = 1\) and hence prove that there are infinitely many prime numbers.

Solution