Problem: Let the sequence \( { a_n} _{n \ge 1 } \) be defined by \(a_n = \tan n \theta \) where \( \tan \theta = 2 \). Show that for all n \( a_n \) is a rational number which can be written with an odd denominator. Discussion: This is simple induction. The claim […]
Here, you will find all the questions of ISI Entrance Paper 2017 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems. Problem 1 : Let the sequence \( \{ a_n\} _{n \ge 1 } \) be defined by $$ a_n = \tan n \theta $$ […]
Here are the problems and their corresponding solutions from BStat Hons Objective Admission Test 2005. Try it yourself and then read the solutions.
Here, you will find all the questions of ISI Entrance Paper 2016 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems. Problem 1: In a sports tournament of $n$ players, each pair of players plays exactly one match against each other. There are no draws. […]
Try this problem from TOMATO Objective 288, useful for ISI BStat, BMath Entrance Exam based on finding big remainder in a small way. Problem: Tomato objective 288 The remainder R(x) obtained by dividing the polynomial [latex]x^{100}[/latex] by the polynomial [latex]x^2-3x+2[/latex] is (A) [latex]2^{100}-1[/latex] (B) [latex](2^{100}-1)x-(2^{99}-1)[/latex] (C) [latex]2^{100}x-3(2^{100})[/latex] (D) [latex](2^{100}-1)x+(2^{99}-1)[/latex] SOLUTION: (B) The the divisor is […]
Problem: If the roots of the equation ${(x-a)(x-b)}$+${(x-b)(x-c)}$+${(x-c)(x-a)}$=$0$, (where a,b,c are real numbers) are equal , then (A) $b^2-4ac=0$ (B) $a=b=c$ (C) a+b+c=0 (D) none of foregoing statements is correct Answer: $(B)$ ${(x-a)(x-b)}$+${(x-b)(x-c)}$+${(x-c)(x-a)}$=$0$ => $x^2-{(a+b)}x$+$ab+x^2-{(b+c)}x$+$bc+x^2-{(c+a)}x+ca$=$0$ => $3x^2-2{(a+b+c)}x$+$(ab+bc+ca)$=$0$ discriminant, of the equation is => $4{(a+b+c)^2}$-$4.3{(ab+bc+ca)}$=$0$ => $a^2+b^2+c^2+2(ab+bc+ca)$-$3(ab+bc+ca)$=$0$ => $a^2+b^2+c^2$-$(ab+bc+ca)$=$0$ => $a=b=c$ So, option (B) is correct.
Try this beautiful problem from TOMATO Objective no. 258 based on Real Roots of a Cubic Polynomial. Problem: Real Roots of a Cubic Polynomial Let a,b,c be distinct real numbers. Then the number of real solution of [latex](x-a)^3+(x-b)^3+(x-c)^3=0[/latex] is (A) 1 (B) 2 (C) 3 (D) depends on a,b,c Solution: Ans: (A) Let [latex]f(x)=(x-a)^3+(x-b)^3+(x-c)^3[/latex] [latex]=> f'(x)=3(x-a)^2+3(x-b)^2+3(x-c)^2=0[/latex] [latex]=> […]
Try this beautiful problem from TOMATO Objective no. 257 based on Roots of a Quintic Polynomial. Problem: Roots of a Quintic Polynomial The number of real roots of [latex] x^5+2x^3+x^2+2=0[/latex] is (A) 0 (B) 3 (C) 5 (D) 1 Solution: Answer: (D) [latex] x^5+2x^3+x^2+2=0[/latex] [latex] \implies x^3(x^2+2)+(x^2+2)=0[/latex] [latex] \implies (x^3+1)(x^2+2)=0[/latex] [latex] \implies (x+1)\bold{\underline{(x^2-x+1)(x^2+2)}}=0[/latex] The expression in underline doesn't have any […]
Problem: The number of terms in the expression of $latex [(a+3b)^2 (a-3b)^2]^2 $ A) 4; B) 5; C) 6; D) 7; Solution: $latex [(a+3b)^2 (a-3b)^2]^2 $ $latex = [\{(a+3b)(a-3b)\}^2]^2 $ $latex = \{ (a^2 -9b^2)^2\}^2 = (a^2 - 9b^2)^4 $ By Binomial Theorem, the given expression contains 5 terms (since $latex (x +y)^n $ has […]
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Interior Angle.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Smallest positive Integer.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Proper divisors.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on combinatorics in Tournament.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Algebraic value.
Try this beautiful problem from Probability based on dice from AMC-10A, 2011. You may use sequential hints to solve the problem
Try this beautiful problem from Geometry: Area of Region in a Circle from AMC-10A, 2011, Problem -18. You may use sequential hints to solve the problem.
Try this beautiful problem from Algebra based smallest positive value from PRMO 2019. You may use sequential hints to solve the problem.
Try this beautiful problem from combinatorics based on Regular Polygon from PRMO 2019. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Positive solution.
Junior Data Science Olympiad is suitable for students of grade 9 and above, interested in Data Science. Check out the resources for the Junior Data Science Olympiad in this post. Curriculum Algebra Trigonometry Coordinate Geometry Combinatorics Data Visualization Algebra AM, GM, and Cauchy Schwarz Inequality Rational Root Theorem, Remainder Theorem Roots of a polynomial Trigonometry […]
Mahalanobis Olympiad is suitable for College and University Students, interested in Statistics and Mathematics. Check out the resources for the Mahalanobis Olympiad in this post. Curriculum High School Mathematics Calculus and Linear Algebra Probability Statistics High School Mathematics Coordinate Geometry Trigonometry Complex Numbers Permutation and Combinatorics Calculus and Linear Algebra Pre Calculus One Variable Calculus […]
Bose Advanced Math Olympiad is suitable for College Students. Curriculum Linear Algebra Abstract Algebra Real Analysis Miscellaneous Linear Algebra Vector Space, basis and Dimension. Linear Transformation, rank-nullity. Matrix Algebra. Eigenvalues, Eigenvectors, Characteristic and Minimal polynomials. Diagonalizability. Inner Product space basic properties. Abstract Algebra Groups, Subgroups, Normal and Quotient groups. Homomorphisms, isomorphisms and automorphisms. Permutation group […]
NMTC 2010 Primary Stage 1 Question 1 $\mathrm{n}, \mathrm{a}$ are natural numbers each greater than 1 . If $a+a+a+a+\ldots+a=2010$, and there are $n$ terms on the left hand side, then the number of ordered pairs $(a, n)$ is NMTC 2019 Inter Stage 1 Question 17 The number of times the digit occurs in the result […]
NMTC 2019 Stage 1 Inter Question 5 The area of the curve enclosed by $|x-2 \sqrt{2}|+|y-\sqrt{5}|=2$ is : (A) 16(B) 12(C) 8(D) 4 NMTC 2019 Inter Stage 1 Question 11 In a rectangle $A B C D$, point $E$ lies on $B C$ such that $\frac{B E}{E C}=2$ and point $F$ lies on $C D$ […]
Watch the video to learn more about opportunities after Mathematical Olympiads in India, the United States and other countries.
Are you preparing for ISI MStat Entrance Exams? Here is the list of useful books for ISI MStat Entrance Exam based on the syllabus.
Answer Key (This is a work in progress, please proceed with caution. We are reviewing some of these answers). Problem 1 Three parallel lines $L_{1}, L_{2}, L_{3}$ are drawn in the plane such that the perpendicular distance between $L_{1}$ and $L_{2}$ is 3 and the perpendicular distance between $L_{2}$ and $L_{3}$ is also $3 .$ […]
Explore this beautiful book on problems useful for Math Olympiad, ISI CMI Entrance. It is written by three Russian authors. Title: Selected Problems and Theorems in Elementary Mathematics – Shklyarsky, Chentsov, Yaglom
Try these AMC 10 Geometry Questions and check your knowledge! Two right circular cones with vertices facing down as shown in the figure below contains the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius […]