Integer Sided Obtuse angled triangles with perimeter 8

Let's discuss a problem based on Integer Sided Obtuse angled triangles with Perimeter.

Find the number of integer-sided isosceles obtuse-angled triangles with perimeter 2008. (Indian RMO 2008)

Discussion:

Let the three sides be a, a and b. Hence 2a + b = 2008 ... (i)

Using the triangular inequality we have 2a > b ...(ii)

Also using the cosine rule for finding sides of a triangle we note that $latex a^2 + a^2 - 2 a \cdot a \cos \theta = b^2 $ where $latex \theta $ is the angle between the two equal sides and obtuse (none of the equal angles can be obtuse as a triangle cannot have more than one obtuse angle). Since $latex \theta $ is obtuse $latex \cos \theta $ is negative hence we get the inequality $latex 2a^2 < b^2 $ which implies $latex \sqrt 2 a < b $. ... (iii)

Combining (i), (ii) and (iii) we have $latex \sqrt 2 a < 2008 - 2a < 2a $. Solving the inequalities we get 502 < a < 588.25 allowing total 86 values of a. Thus there are 86 such triangles.

Back to RMO 2008

Critical Ideas: Cosine rule for measuring sides of a triangle, Pythagorean Inequality

Some Useful Links:

RMO 2002 Problem 2 – Fermat’s Last Theorem as a guessing tool– Video

Our Math Olympiad Program

RMO 2008 | Regional Mathematics Olympiad Problem

In this post, there are problems from Regional Mathematics Olympiad, RMO 2008. Try out these problems.

  1. Let $ABC$ be an acute-angled triangle, let $D$, $F$ be the mid-points of $BC, AB$ respectively. Let the perpendicular from $F$ to $AC$ and the perpendicular at $B$ to $BC$ meet in $N$. Prove that $ND$ is equal to circum-radius of $ABC$.
    Discussion
  2. Prove that there exists two infinite sequences $ {\left \langle a_n \right \rangle}_{n \ge 1} $ and $ {\left \langle b_n \right \rangle}_{n \ge 1} $ of positive integers such that the following conditions holds simultaneously:
    1. $1<a_1<a_2<a_3<..... $;
    2. $a_n < b_n < a_n^{2} $, for all $n \ge 1 $;
    3. $a_{n}-1 $ divides $ b_{n}-1 $, for all $n \ge 1 $;
    4. $a_{n}^2-1 $ divides $b_{n}^2-1 $, for all $n \ge 1 $.
  3. Suppose a and b are real numbers such that the roots of the cubic equation $ax^3-x^2+bx+1=0 $ are all positive real numbers. Prove that i) $0<3ab<1 $ and ii) $ b \ge \sqrt{3} $.
  4. Find the number of all $6$-digits natural number such that the sum of their digits is $10$ and each of the digits $0,1,2,3$ occurs at least once in them.
  5. Three non-zero numbers a,b,c are said to be in harmonic progression if $ \frac{1}{a}+\frac{1}{c}=\frac{2}{b} $. Find all three term harmonic progressions $a,b,c$ of strictly increasing positive integers in which $a=20$ and $b$ divides $c$.
  6. Find the number of integer-sided isosceles obtuse-angled triangles with perimeter $2008$.
    Discussion

Some Useful Links:

RMO 2002 Problem 2 – Fermat’s Last Theorem as a guessing tool– Video

Our Math Olympiad Program

Crease of a square paper

A square sheet of paper ABCD is so folded that B falls on the mid-point of M of CD. Prove that the crease will divide BC in the ratio 5:3.

Discussion:

creaseAssuming the side of the square is 's'. Let a part of the crease be 'x' (hence the remaining part is 's-x'). We apply Pythagoras Theorem we solve for x:

$ x^2 + \frac {s^2 }{4} = (s-x)^2 $ implies $ x = \frac {3s}{8} $ and $ s-x = \frac {5s}{8} $

Hence the ratio is 5:3.

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RMO 1990 | Problems

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  1. Two boxes contain between them 65 balls of several different sizes. Each ball is white, black, red or yellow. If you take any five balls of the same colour, at least two of them will always be of the same size (radius). Prove that there are at least three balls which lie in the same box have the same colour and have the same size (radius).
  2. For all positive real numbers a, b, c prove that $latex \frac{a}{b+c} + \frac{b}{a+c}+ \frac{c}{a+b} \ge \frac{3}{2} $
  3. A square sheet of paper ABCD is so folded that B falls on the mid-point of M of CD. Prove that the crease will divide BC in the ratio 5:3.

    Discussion

  4. Find the remainder when $latex 2^{1990} $ is divided by 1990.
  5. P is any point inside a triangle ABC. The perimeter of the triangle AB +BC +CA=2s. Prove that s < AP + BP + CP < 2s.
  6. N is a 50 digit number (in a decimal scale). All digits except the $latex 26^{th} $ digit (from the left) are 1.If N is divisible by 13, find the $latex 26^{th} $ digit.
  7. A census-man on duty visited a house in which the lady inmates declined to reveal their individual ages, but said “We do not mind giving you the sum of the ages of any two ladies you may choose”. There upon the census man said, “In that case, please give me the sum of the ages of every possible pair of you”. They gave the sums as follows: 30, 33, 41, 58, 66, 69. The census-man took these figures and happily went away. How did he calculate the individual ages of the ladies from these figures?
  8. If the circumcenter and centroid of a triangle coincide, prove that the triangle must be equilateral.
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INMO 2012 | Problems

This post contains problems from Indian National Mathematics Olympiad, INMO 2012. Try them and share your solution in the comments.

Problem 1

Let ABCD be a quadrilateral inscribed in a circle. Suppose AB =$latex \sqrt {2 + \sqrt {2} } $ and AB subtends 1350 at the center of the circle. Find the maximum possible area of ABCD.

Problem 2

Let $latex p_1<p_2< p_3< p_4 $ and $latex q_1<q_2<q_3<q_4 $ be two sets of prime numbers such that $latex p_4 - p_1 = 8 $  and $latex q_4 - q_1 = 8 $ . Suppose $latex p_1>5 $ and $latex q_1>5 $ . Prove that 30 divides $latex p_1 - q_1 $.

Problem 3

Define a sequence $latex <fn(x)> $ n∈N of functions as $latex f_0(x )=1, f_1(x )=x $, $latex (f_n(x))^2 - 1 = f_{n-1} (x) f_{n+1} (x) $, for $latex n \ge 1 $ . Prove that each $latex f_n(x ) $ is a polynomial with integer coefficients.

Problem 4

Let ABC be a triangle. An interior point P of ABC is said to be good if we can find exactly 27 rays emanating from P intersecting the sides of the triangle ABC such that the triangle is divided by these rays into 27 smaller triangles of equal area. Determine the number of good points for a given triangle ABC.

Problem 5

Let ABC be an acute angled triangle. Let D, E, F be points on BC, CA, AB such that AD is the median, BE is the internal bisector and CF is the altitude. Suppose that angle FDE = angle C and angle DEF = angle A and angle EFD = angle B. Show that ABC is equilateral.

Problem 6

Let $latex f :Z \mapsto Z $be a function satisfying $latex f(0) \neq 0 $ , $latex f(1)=0 $ and

    1. $latex f(xy) + f(x)f(y) = f(x) + f(y) $,
    2. $latex (f(x-y) - f(0) ) f(x) f(y) = 0 $ for all x , $latex y in Z $ simultaneously.
      1. Find the set of all possible values of the function f.
      2. If $latex f(10) \neq 0 $ and $latex f(2) = 0 $, find the set of all integers n such that $latex f(n) \neq 0 $ .

Some Useful Links:

RMO 1990 Problems

INMO 2018 Problem 6 - Video

INMO 2011 | Problems

  1. This post contains problem from Indian National Mathematics Olympiad, INMO 2011. Try them out and share your solution in the comments.
  2.  
  3. Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that BD = CE = AF and ∠BDF = ∠CED = ∠AFE. Prove that ABC is equilateral.
  4. Call a natural number n faithful, if there exist natural numbers a < b < c such that a divides b, b divides c and n = a + b + c. (i) Show that all but a finite number of natural numbers are faithful. (ii) Find the sum of all natural numbers which are not faithful.
  5. Consider two polynomials $latex P(x)=a_nx^n+a_{n-1}x^{n-1}+......+a_1x+a_0=0 $ and $latex Q(x)=b_nx^n+b_{n-1}x^{n-1}+......+b_1x+b_0=0 $ with integer coefficients such that $latex a_n-b_n $ is a prime, $latex a_{n-1}=b_{n-1} $ and $latex a_nb_0-a_0b_n \neq 0 $. Suppose there exists a rational number r such that P(r) = Q(r) = 0. Prove that r is an integer.
  6. Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.
  7. Let ABCD be a quadrilateral inscribed in a circle $latex \Gamma $. Let E, F, G, H be the midpoints of the arcs AB, BC, CD, DA of the circle $latex \Gamma $. Suppose AC.BD = EG .FH. Prove that AC, BD, EG, FH are concurrent.
  8. Find all functions $latex f : R \mapsto R $ such that $latex f(x+y)f(x-y)=(f(x)+f(y))^2-4x^2f(y) $,  for all x, y ∈ R, where R denotes the set of all real numbers.

Some Useful Links:

INMO 2012 Problems

INMO 2018 Problem 6

Cyclic Group problem in NBHM M.Sc. 2013

This a problem from from National Board for Higher Mathematics (NBHM) 2013 based on Cyclic Group Problem for M.Sc Students.

Which of the following statements are true?

  1. Every group of order 11 is cyclic.
  2. Every group of order 111 is cyclic.
  3. Every group of order 1111 is cyclic.

Discussion:

  1. Every group of order 11 is cyclic. This is true. We know that (for example from Lagrange's Theorem) that a group of prime order is necessarily cyclic.
  2. Every group of order 111 is cyclic. This is true. $latex 111 = 3 \times 37 $. This is a favorite problem for any college level test maker. Order of this group is of the form $latex p \times q $ where p and q are primes. This group is isomorphic to $latex Z_p \times Z_q $ that is the direct product of two cyclic groups of prime order. We have this theorem:  Let G be a group of order pq, where p,q are prime, p<q, and p does not divide q−1. Then G is cyclic. The proof of this theorem directly follows from Sylow's Theorem.
  3. Every group of order 1111 is cyclic. This is true. $latex 1111 = 11 \times 101 $ and using the argument of the previous problem we conclude that it is cyclic.

Critical Ideas: Lagrange's Theorem, Sylow's Theorem, Classification of finite abelian groups

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INMO 2012 Problems

INMO 2018 Problem 6

NBHM M.Sc. 2013 Analysis Problems and Discussions

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pg 3pg 4

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NBHM M.Sc. 2013 Geometry Problems and Discussions

Section 3: Geometry

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Algebra || Analysis

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  1. Find the reflection of the point (2, 1) with respect to the line x=y in the xy-plane.
  2. Find the area of the circle in the xy-plane which has its centre at the point (1,2) and which has the line x=y as a tangent.
  3. Find the incentre of the triangle in the xy-plane whose sides are given by the lines x=0 , y=0 and $latex \frac {x}{3} + \frac {y}{4} = 1 $
  4. Let A and B be fixed points in a plane such that, the length of the line segment AB is d. Let the point P describe an ellipse by moving on the plane such that the sum of the lengths of the line segments PA and PB is a constant l. Express the length of the semi-major axis, a and the length of the semi-minor axis, b, of the ellipse in terms of d and l.
  5. Let A = $latex a_{ij} $ be a non-zero $latex 2 \times 2 $ symmetric matrix with real entries. Let  S = { $latex (x, y ) \in R^2 | a_{11} x^2 + 2 a_{12} xy + a_{22} y^2 = 0 $ }. Which of the following conditions imply that S is unbounded?
    1. det(A) > 0.
    2. det(A) = 0
    3. det(A) < 0.
  6. Let $latex A \in M_2 (R) $  define an invertible linear transformation on $latex R^2 $. Let T be a triangle with one of its vertices at the origin and of area a. What is the area of the triangle which is the image of T under this transformation'?
  7. Find the area of the ellipse whose equation in the xy plane is given by $latex 5 x^2 - 6 x y + 5 y^2 = 8 $.
  8. Let a, b and c be positive real numbers. Find the equation of the sphere which passes through the origin and through the points where the plane $latex \frac {x}{a} + \frac {y}{b} + \frac {z}{c} $ meets the coordinate axes.
  9. Consider the sphere $latex x^2 + y^2 + z^2 = r^2 $.  Let $latex (a, b, c ) \ne (0, 0, 0) $ be a point in the interior of this sphere. Write down the equation of the plane whose intersection with the sphere is a circle whose center is the point (a, b, c) .
  10. Find the area of the polygon whose vertices are represented by the eighth roots of unity.

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NBHM M.Sc 2013 Geometry Problems and Discussions

ISI BStat 2010 Solution and Strategy

NBHM M.Sc. 2013 Algebra Problems and Discussions

Section 1: Algebra

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NBHM

  1. Which of the following statements are true?
    1. Every group of order 11 is cyclic.
    2. Every group of order 111 is cyclic.
    3. Every group of order 1111 is cyclic.

      Discussion

  2. Let $latex S_n $  denote the symmetric group of order n, i.e. the group of all permutations of the n symbols (1, 2, ...  , n). Given two permutations $latex sigma $ and $latex tau $ in $latex S_n $, we define the product $latex sigma tau $ as their composition got by applying $latex sigma $ first and then applying $latex tau $ to the set {1,2, ... , n}, Write down the following permutation in $latex S_8 $ as the product of disjoint cycles:
    (1 4 3 8 7)(5 4 8).
  3. Write down all the permutations in $latex S_4 $ which are conjugate to the permutation (1 2)(3 4).
  4. Let R be a ring such that $latex x^2 = x $ for every $latex x \in R $. Which of the following statements are true?
    1. $latex x^n $ for every $latex n \in N $ and every $latex x \in R $
    2. x= -x for every $latex x \in R $
    3. R is a commutative ring.
  5. For a prime number p let $latex F_p $ denote the field consisting of 0, 1, 2, ... ,  p - 1 with addition and multiplication modulo p. Which of the following quotient rings are fields?
    1. $latex F_5 [x] / (x^2 + x + 1) $
    2. $latex F_2 [x] / (x^3 + x + 1) $
    3. $latex F_3 [x] / (x^3 + x + 1) $
  6. Let V be the subspace of $latex M_2 (R) $ consisting of all matrices with trace zero and such that all entries of the first row add up to zero. Write down a basis for V.
  7. Let  V subset of $latex M_n (R) $ be a subspace of all matrices such that the entries in every row add up to zero and the entries in every column also add up to 0. What is the dimension of V?
  8. Let T : $latex M_2 (R) $ --> $latex M_2 (R) $ be a linear transformation defined by $latex T(A) = 2A + 3A^T $ . Write down the matrix of this transformation with respect to the basis { $latex E_i , 1 \le i \ge 4 $ } where $latex E_1 $ = $latex \begin {bmatrix} 1 & 0 0 & 0 \end {bmatrix} $ , $latex E_2 $ = $latex \begin {bmatrix} 0 & 1 0 & 0 \end {bmatrix} $ , $latex E_3 $ = $latex \begin {bmatrix} 0 & 0 1 & 0 \end {bmatrix} $ , $latex E_4 $ = $latex \begin {bmatrix} 0 & 0 1 & 0 \end {bmatrix} $
  9. Find the values of $latex \alpha\in R $ such that the matrix $latex \begin {bmatrix} 3 & \alpha \alpha & 5 \end {bmatrix} $ has 2 as an eigenvalue.
  10. Let $latex A = (a_{ij} ) \in M_3 (R) $ be such that $latex a_{ij} = - a_{ji} $ for all $latex 1 \le i , j \le 3 $. If 31 is a eigenvalue of A find it's other eigenvalues.

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