Gauss Trick in ISI Entrance

Problem - Gauss Trick (ISI Entrance)

Let's learn Gauss Trick for ISI Entrance.

If k is an odd positive integer, prove that for any integer $ \mathbf{ n \ge 1 , 1^k + 2^k + \cdots + n^k } $ is divisible by $ \mathbf{ \frac {n(n+1)}{2} } $

Key Concepts

Gauss Trick

Factoring Binomial

Source

From I.S.I. Entrance and erstwhile Soviet Olympiad.

Test of Mathematics at 10+2 Level by East West Press, Subjective Problem 31

Challenges and Thrills of Pre College Mathematics

Try the first hint

Other useful links:-

Remaining hints, complete solution and related problems.

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Bijection Principle from I.S.I. Entrance

Try the problem

This problem of Bijection Principle is from B.Stat, B.Math Entrance.

How many natural numbers less that $ 10^8 $ are there, whose sum of digits equals 7?

Watch the first hint

Other useful links:-

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Primes and Polynomials from I.S.I. Entrance

Try the problem

This problem of primes and polynomials is from B.Stat, B.Math Entrance.

Let x and n be positive integers such that $ 1 + x + x^2 + … + x^{n-1} $ is a prime number. Then show that n is a prime number.

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Other Useful Links

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Gromov boundary

In mathematics, the Gromov boundary of a δ-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space. Conceptually, the Gromov boundary is the set of all points at infinity. For instance, the Gromov boundary of the real line is two points, corresponding to positive and negative infinity.

Suppose X is any set. It is, Suppose, we have defined a distance function or a metric on this set.

What is a distance function? 

It is a function d takes pairs of elements of X as input and gives away the distance (a nonnegative real number) between them as an output. This distance or metric d is usually computed by weird formulas. We only require the metric d to satisfy the following properties:

If you think closely, all of these properties mimic the notion of distance that we are familiar with. The last one is the triangular inequality.

Once we have defined a metric d on the set X, we can define some subsets of X as open sets. First, we define open balls. An open ball centered at ( x_o \in X ) and of radius ( \epsilon ) is the set of all points in X which are less then (\epsilon ) distance away from ( x_0 ). This distance is of course measured by the distance function that we defined earlier.

Sometimes we need to relax the notion of 'closed and bounded' sets. Imagine you are 'covering' a subset U with open sets. Intuitively speaking, think about the open sets as carpets. You are covering U with this collection of 'carpets' mean, that U is contained in the union of this collection of open sets (usually containing infinitely many 'carpets').

A set is said to be compact if whenever you can cover it with infinitely many carpets, you will be able to cover it using finitely many carpets of that collection. In the usual n-dimensional Euclidean space, closed and bounded is compact (this needs proof).

Collection of all open sets defines a topology on the set X (yes, the topology on X is just a collection of subsets of the set X, which have the designation of being 'open'; and this 'openness' is defined as above).

Now we have a set X, a distance function (metric) d and a topology induced by it. This apparatus (the set, along with metric and topology induced by it) is known as a metric space. Suppose that this metric space is proper. This means, that every closed and bounded set is compact (hence it has some similarity with the usual Euclidean space that we are familiar with).

Other useful links

Power Mean Inequality for Math Olympiad

What is Power Mean in Inequality?

The simplest example of power mean inequality is the arithmetic mean - geometric mean inequality. It says the following:

Arithmetic Mean is greater than or equal to Geometric Mean

Caution: All numbers must be non-negative.

Suppose $ a_1, a_2, ... , a_n $ be non-negative numbers. Then the two means are defined as follows:

Arithmetic Mean: $ \displaystyle{\frac{a_1 + a_2 + ... + a_n}{n}} $

Geometric Mean: $ \displaystyle{(a_1 \cdot a_2 \cdots a_n)^{\frac{1}{n}}} $

Try a problem

This problem is from Regional Math Olympiad, India.

Suppose a, b, c, d are positive numbers. Then show that $$ \displaystyle { \frac{a}{b} + \frac {b}{c} + \frac{c}{d} + \frac{d}{a} \geq 4 } $$

Regional Math Olympiad, India

Inequality (AM-GM)

6 out of 10

Secrets in Inequalities.

Use some hints

Notice that product of the fractions is 1. Can you use this fact to compute the geometric mean of the fractions?

The geometric mean of the fractions is $$ \displaystyle{(\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d} \cdot \frac{d}{a} )^{\frac{1}{4} }}$$

This is equal to $ 1^{\frac{1}{4}} = 1$

Hence the geometric mean of the fractions is 1!

Can you now finish the problem using Arithmetic Mean - Geometric Mean inequality?

Lets use the arithmetic mean - geometric mean inequality on the fractions.

$$ \displaystyle { \frac{\frac{a}{b} + \frac {b}{c} + \frac{c}{d} + \frac{d}{a}}{4} \\ \geq (\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d} \cdot \frac{d}{a} )^{\frac{1}{4} } } $$

But the geometric mean is 1 (right hand side is 1). Hence by cross multiplying we have the final result.

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Parity in Mathematics | Is the Sum Zero?

Understand the Problem

For each \( n \in \mathbb{N} \) let \( d_n \) denote the G.C.D. of n and (2019 - n). Find the value of \( d_1 + d_2 + ... + d_{2019} \).

First, try these problems.

  1. Show that G.C.D. of k and 0 is k for any positive integer k.
  2. Show rigorously that G.C.D. (a, b) = G.C.D. (a, a+b) for any non-negative integers a and b
  3. Can you find and prove a similar result with a negative sign?

Send the written solution to support@cheenta.com

we will try to give you a feedback

Now, watch this video

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Euclidean Algorithm for Math Olympiad

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Get motivated... try this quiz

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Understand the problem

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Euclidean algorithm is a very important tool of mathematics. Learn more about it using the video and the problems.

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Tutorial Problems... try these before watching the video.

[/et_pb_text][et_pb_text _builder_version="4.0.7" text_font_size="18px" custom_padding="20px|30px|20px|30px|false|false" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset1"]1. Why is division by zero undefined?
2. Is the divisor greater than or less than the remainder?
3. Find two numbers which are very hard to prime factorize. How can you compute their GCD in a short amount to time?

You may send solutions to support@cheenta.com. Though we usually look into internal students work, we will try to give you some feedback.

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Now watch the discussion video

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Similar Problems

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Cheenta faculty team consists of Olympians, researchers and active mathematicians from leading universities of the world.
We specialize in Math Olympiad training. We have worked with brilliant kids from 4 continents since 2010.

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Geometry of Wilson’s Theorem

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Understand the problem

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Wilson’s Theorem from Number Theory is a beautiful classical result. Enjoy this beautiful geometric proof of the classical result. 

We used animations to bring the idea into life. 

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Now watch the discussion video

[/et_pb_text][et_pb_video src="https://youtu.be/4qbh7mC6YCY" _builder_version="4.1" hover_enabled="0"][/et_pb_video][et_pb_team_member name="Kazi Abu Rousan" position="Cheenta Creative Team" image_url="https://cheenta.com/wp-content/uploads/2020/01/Rousan.jpg" _builder_version="4.1"]Kazi is a student of Physics. He loves the relation between physics and mathematics. He is the creator of this beautiful video.[/et_pb_team_member][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Tutorial Problems... try these after watching the video.

[/et_pb_text][et_pb_text _builder_version="4.1" text_font_size="18px" custom_padding="20px|30px|20px|30px|false|false" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset1"]1. Write down a rigorous proof of Wilson’s Theorem. 2. Is the converse of Wilson’s Theorem true? Can you prove it?

You may send solutions to support@cheenta.com. Though we usually look into internal students' work, we will try to give you some feedback.

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Connected Program at Cheenta

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Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.[/et_pb_blurb][et_pb_button button_url="https://cheenta.com/matholympiad/" url_new_window="on" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" background_layout="dark" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3"][/et_pb_button][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Similar Problems

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Inversion and Ptolemy's Theorem

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Understand the problem

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Ptolemy's Theorem gives a relation between product of diagonals of a quadrilateral with the sum of the product of its opposite sides. However it can be proved using tools from inversive geometry (that Ptolemy probably did not know). Try this! 

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Now watch the discussion video

[/et_pb_text][et_pb_video src="https://www.youtube.com/watch?v=pw-lXWZt5wg" _builder_version="4.1"][/et_pb_video][et_pb_team_member name="Kazi Abu Rousan" position="Cheenta Creative Team" image_url="https://cheenta.com/wp-content/uploads/2020/01/Rousan.jpg" _builder_version="4.1"]Kazi is a student of Physics. He loves the relation between physics and mathematics. He is the creator of this beautiful video.[/et_pb_team_member][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Tutorial Problems... try these after watching the video.

[/et_pb_text][et_pb_text _builder_version="4.1" text_font_size="18px" custom_padding="20px|30px|20px|30px|false|false" hover_enabled="0" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset1"]1. How can you generalize Ptolemy's theorem if the quadrilateral is not cyclic?? 2. Suppose two points A, B, are inside a unit circle G. Invert A and B about G to A' and B' respectively. If the length of AB is 0.5, what is the length of A'B'? You may send solutions to support@cheenta.com. Though we usually look into internal students' work, we will try to give you some feedback. [/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

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Connected Program at Cheenta

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Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.[/et_pb_blurb][et_pb_button button_url="https://cheenta.com/matholympiad/" url_new_window="on" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" background_layout="dark" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3"][/et_pb_button][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Similar Problems

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Kites in Geometry | INMO 2020 Problem 1

Understand the problem

Let \( \Gamma_1 \) and \( \Gamma_2 \) be two circles with unequal radii, with centers \(  O_1 \) and \( O_2 \) respectively, in the plane intersecting in two distinct points A and B. Assume that the center of each of the circles \( \Gamma_1 \) and \( \Gamma_2 \) are outside each other. The tangent to \( \Gamma_ 1 \) at B intersects \( \Gamma_2 \) again at C, different from B; the tangent to \(   \Gamma_2 \) at B intersects \(  \Gamma_1 \) again in D different from B. The bisectors of \( \angle DAB \) and \( \angle CAB \) meet \( \Gamma_1 \) and \( \Gamma_2 \) again in X and Y, respectively. different from A. Let P and Q be the circumcenters of the triangles ACD and XAY, respectively. Prove that PQ is perpendicular bisector of the line segment \( O_1 O_2 \). 

Tutorial Problems... try these before watching the video.

1. Suppose \( P O_1 Q O_2 \) be a kite (that is \( PO_1 = PO_2 \)  and \(  QO_1 1 = QO_2 \). Show that PQ is perpendicular bisector of the other diagonal $ O_1 O_2 $.$.

2. Show that for any two circles intersecting each other at two distinct points, the common chord is bisected perpendicularly by the line joining the center.

You may send solutions to support@cheenta.com. Though we usually look into internal students work, we will try to give you some feedback.

Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year.

Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

Now watch the discussion video

Subscribe to Cheenta's youtube channel