Math Olympiad awardee conducts Russian styled Math Circles in Sundarbans

Souradip Das, a student of Cheenta Academy, has made remarkable achievements in the world of Mathematics Olympiad. Souradip has secured impressive ranks in prestigious competitions like IOQM 2024, RMO 2023, IOQM 2023, and the AMC (American Math Competition) 10-12. His consistent hard work and dedication have placed him among the top young mathematicians, and his success is a true reflection of his passion for the subject.

But Souradip's contributions go beyond his own achievements. He is also a key mentor in the Math Circle program at Cheenta Academy. This initiative aims to bring the world of mathematics to students from rural areas in the Sundarbans, a region where access to quality education can be limited. Souradip has been a guiding light for these students, helping them discover their love for mathematics and nurturing their skills.

As a mentor, Souradip not only teaches mathematical concepts but also inspires his students to think critically and approach problems with confidence. His dedication to the 'Math Circle' program has had a profound impact on many young minds, encouraging them to pursue their academic dreams despite the challenges they face.

In addition to his mentoring, Souradip has also created a Math Circle Diary, documenting his experiences and the journey of the program. This diary serves as an inspiring resource for both students and teachers, capturing the stories, struggles, and successes of the students in the Math Circle. Through this diary, Souradip shares valuable insights into the impact of the program, as well as the importance of making mathematics accessible to all students, no matter their background.

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Souradip’s story is an inspiring example of how talent, combined with a strong sense of community and responsibility, can make a real difference.

American Math Competition (AMC) 10 B - Problem and Solution

Problem 1

In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in line?

(A) 2021
(B) 2022
(C) 2023
(D) 2024
(E) 2025

Problem 2

What is $10!-7!\cdot 6!$

(A) -120
(B) 0
(C) 120
(D) 600
(E) 720

Problem 3

For how many integer values of $x$ is $|2 x| \leq 7 \pi$

(A) 16
(B) 17
(C) 19
(D) 20
(E) 21

Problem 4

Balls numbered $1,2,3, \ldots$ are deposited in 5 bins, labeled $A, B, C, D$, and $E$, using the following procedure. Ball 1 is deposited in bin $A$, and balls 2 and 3 are deposited in bin $B$. The next 3 balls are deposited in bin $C$, the next 4 in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin E. (For example, balls numbered $22,23, \ldots, 28$ are deposited in bin B at step 7 of this process.) In which bin is ball 2024 deposited?

(A) $A$
(B) $B$
(C) $C$
(D) $D$
(E) $E$

Problem 5

In the following expression, Melanie changed some of the plus signs to minus signs:

$$
1+3+5+7+\ldots+97+99
$$

When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?

(A) 14
(B) 15
(C) 16
(D) 17
(E) 18

Problem 6

A rectangle has integer length sides and an area of 2024. What is the least possible perimeter of the rectangle?

(A) 160
(B) 180
(C) 16
(D) 17
(E) 18

Problem 7

What is the remainder when $7^{2024}+7^{2025}+7^{2026}$ is divided by $19 ?$

(A) 0
(B) 1
(C) 7
(D) 11
(E) 18

Problem 8

Let $N$ be the product of all the positive integer divisors of 42 . What is the units digit of $N$ ?

(A) 0
(B) 2
(C) 4
(D) 6
(E) 8

Problem 9

Real numbers $a, b$, and $c$ have arithmetic mean 0 . The arithmetic mean of $a^2, b^2$, and $c^2$ is 10 . What is the arithmetic mean of $a b, a c$, and $b c$ ?

(A) -5
(B) $-\frac{10}{3}$
(C) $-\frac{10}{9}$
(D) 0
(E) $\frac{10}{9}$

Problem 10

Quadrilateral $A B C D$ is a parallelogram, and $E$ is the midpoint of the side $A D$. Let $F$ be the intersection of lines $E B$ and $A C$. What is the ratio of the area of quadrilateral $C D E F$ to the area of triangle $C F B$ ?

(A) $5: 4$
(B) $4: 3$
(C) $3: 2$
(D) $5: 3$
(E) $2: 1$

Problem 11

In the figure below $W X Y Z$ is a rectangle with $W X=4$ and $W Z=8$. Point $M$ lies $\overline{X Y}$, point $A$ lies on $\overline{Y Z}$, and $\angle W M A$ is a right angle. The areas of $\triangle W X M$ and $\triangle W A Z$ are equal. What is the area of $\triangle W M A$ ?

(A) 13
(B) 14
(C) 15
(D) 16
(E) 17

Problem 12

A group of 100 students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$, student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students?

(A) 9
(B) 10
(C) 12
(D) 51
(E) 100

Problem 13

Positive integers $x$ and $y$ satisfy the equation $\sqrt{x}+\sqrt{y}=\sqrt{1183}$. What is the minimum possible value of $x+y$.

(A) 585
(B) 595
(C) 623
(D) 700
(E) 791

Problem 14

A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x|+|y| \leq 8$. A target T is the region where $\left(x^2+y^2-25\right)^2 \leq 49$. A dart is thrown at a random point in B. The probability that the dart lands in $T$ can be expressed as $\frac{m}{n} \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

(A) 39
(B) 71
(C) 73
(D) 75
(E) 135

Problem 15

A list of 9 real numbers consists of $1,2.2,3.2,5.2,6.2,7$, as well as $x, y, z$ with $x \leq y \leq z$. The range of the list is 7 , and the mean and median are both positive integers. How many ordered triples $(x, y, z)$ are possible?

(A) 1
(B) 2
(C) 3
(D) 4
(E) infinitely many

Problem 16

Jerry likes to play with numbers. One day, he wrote all the integers from 1 to 2024 on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them with either their sum or their product. (For example, Jerry's first step might have been to erase $1,2,3$, and 5 , and then write either 11 , their sum, or 30 , their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the board were odd. What is the maximum possible number of integers on the board at that time?

(A) 1010
(B) 1011
(C) 1012
(D) 1013
(E) 1014

Problem 17

In a race among 5 snails, there is at most one tie, but that tie can involve any number of snails. For example, the result might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second; and Bruna is fifth. How many different results of the race are possible?

(A) 180
(B) 361
(C) 420
(D) 431
(E) 720

Problem 18

How many different remainders can result when the 100th power of an integer is divided by 125?

(A) 1
(B) 2
(C) 5
(D) 25
(E) 125

Problem 19

In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the 12 entries will be "Possible"?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 9

Problem 20

Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up?

(A) 60
(B) 72
(C) 90
(D) 108
(E) 120

Problem 21

Two straight pipes (circular cylinders), with radii 1 and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?

(A) $\frac{1}{9}$
(B) 1
(C) $\frac{10}{9}$
(D) $\frac{11}{9}$
(E) $\frac{19}{9}$

Problem 22

A group of 16 people will be partitioned into 4 indistinguishable 4-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^r M$, where $r$ and $M$ are positive integers and $M$ is not divisible by 3 . What is $r$ ?

(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

Problem 23

The Fibonacci numbers are defined by $F_1=1, F_2=1$, and $F_n=F_{n-1}+F_{n-2}$ for $n \geq 3$. What is

$$
\frac{F_2}{F_1}+\frac{F_4}{F_2}+\frac{F_6}{F_3}+\ldots+\frac{F_{20}}{F_{10}} ?
$$

(A) 318
(B) 319
(C) 320
(D) 321
(E) 322

Problem 24

Let

$$
P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}
$$

How many of the values $P(2022), P(2023), P(2024)$, and $P(2025)$ are integers?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Problem 25

Each of 27 bricks (right rectangular prisms) has dimensions $a \times b \times c$, where $a, b$ , and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $3 \times 3 \times 3$ block, as shown on the left below. A $28^{\text {th }}$ brick with the same dimensions is introduced, and these bricks are reconfigured into a $2 \times 2 \times 7$ block, shown on the right. The new block is 1 unit taller, 1 unit wider, and 1 unit deeper than the old one. What is $a+b+c$ ?

(A) 88
(B) 89
(C) 90
(D) 91
(E) 92

American Math Competition (AMC) 10 A 2024 - Problem and Solution

Problem 1

What is the value of $9901 \cdot 101-99 \cdot 10101 ?$

(A) 2
(B) 20
(C) 200
(D) 202
(E) 2020

Problem 2

A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T=a L+b G$, where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take 69 minutes to hike to the top if a trail is 1.5 miles long and ascends 800 feet, as well as if a trail is 1.2 miles long and ascends 1100 feet. How many minutes does the model estimates it will take to hike to the top if the trail is 4.2 miles long and ascends 4000 feet?

(A) 240
(B) 246
(C) 252
(D) 258
(E) 264

Problem 3

What is the sum of the digits of the smallest prime that can be written as a sum of 5 distinct primes?

(A) 5
(B) 7
(C) 9
(D) 10
(E) 13

Problem 4

The number 2024 is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?

(A) 20
(B) 21
(C) 22
(D) 23
(E) 24

Problem 5

What is the least value of $n$ such that $n$ ! is a multiple of 2024 ?

(A) 11
(B) 21
(C) 22
(D) 23
(E) 253

Problem 6

What is the minimum number of successive swaps of adjacent letters in the string $A B C D E F$ that are needed to change the string to $F E D C B A$ ? (For example, 3 swaps are required to change $A B C$ to $C B A$; one such sequence of swaps is $
A B C \rightarrow B A C \rightarrow B C A \rightarrow C B A .)$

(A) 6
(B) 10
(C) 12
(D) 15
(E) 24

Problem 7

The product of three integers is 60. What is the least possible positive sum of the three integers?

(A) 2
(B) 3
(C) 5
(D) 6
(E) 13

Problem 8

Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at $1: 00 P M$ and were able to pack 4,3 , and 3 packages, respectively, every 3 minutes. At some later time, Daria joined the group, and Daria was able to pack 5 packages every 4 minutes. Together, they finished packing 450 packages at exactly $2: 45 P M$. At what time did Daria join the group?

(A) $1: 25 \mathrm{PM}$
(B) $1: 35 \mathrm{PM}$
(C) $1: 45 \mathrm{PM}$
(D) 1:55 PM
(E) 2:05 PM

Problem 9

In how many ways can 6 juniors and 6 seniors form 3 disjoint teams of 4 people so that each team has 2 juniors and 2 seniors?

(A) 720
(B) 1350
(C) 2700
(D) 3280
(E) 8100

Problem 10

Consider the following operation. Given a positive integer $n$, if $n$ is a multiple of 3 , then you replace $n$ by $\frac{n}{3}$. If $n$ is not a multiple of 3 , then you replace $n$ by $n+10$. Then continue this process. For example, beginning with $n=4$, this procedure gives $4 \rightarrow 14 \rightarrow 24 \rightarrow 8 \rightarrow 18 \rightarrow 6 \rightarrow$

$ 2 \rightarrow 12 \rightarrow \cdots$. Suppose you start with $n=100$. What value results if you perform this operation exactly 100 times?

(A) 10
(B) 20
(C) 30
(D) 40
(E) 50

Problem 11

How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2-49}=m$ ?

(A) 1
(B) 2
(C) 3
(D) 4
(E) Infinitely many

Problem 12

Zelda played the Adventures of Math game on August 1 and scored 1700 points. She continued to play daily over the next 5 days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August 2 was $1700+80=1780$ points.) What was Zelda's average score in points over the 6 days?

(A) 1700
(B) 1702
(C) 1703
(D) 1713
(E) 1715

Problem 13

Two transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:

Of the 6 pairs of distinct transformations from this list, how many commute?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Problem 14

One side of an equilateral triangle of height 24 lies on line $\ell$. A circle of radius 12 is tangent to line $l$ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line $\ell$ can be written as $a \sqrt{b}-c \pi$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a+b+c$ ?

(A) 72
(B) 73
(C) 74
(D) 75
(E) 76

Problem 15

Let $M$ be the greatest integer such that both $M+1213$ and $M+3773$ are perfect squares. What is the units digit of $M$ ?

(A) 1
(B) 2
(C) 3
(D) 6
(E) 8

Problem 16

All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length $A B$ ?

Problem 17

Two teams are in a best-two-out-of-three playoff: the teams will play at most 3 games, and the winner of the playoff is the first team to win 2 games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\frac{2}{3}$ chance of winning at home, and its probability of winning when playing away from home is $p$. Outcomes of the games are independent. The probability that Team A wins the playoff is $\frac{1}{2}$. Then $p$ can be written in the form $\frac{1}{2}(m-\sqrt{n})$, where $m$ and $n$ are positive integers. What is $m+n$ ?

(A) 10
(B) 11
(C) 12
(D) 13
(E) 14

Problem 18

There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base- $b$ integer $2024_b$ is divisible by 16 (where 16 is in base ten). What is the sum of the digits of $K$ ?

(A) 16
(B) 17
(C) 18
(D) 20
(E) 21

Problem 19

The first three terms of a geometric sequence are the integers $a, 720$, and $b$, where $a<720<b$. What is the sum of the digits of the least possible value of $b$ ?

(A) 9
(B) 12
(C) 16
(D) 18
(E) 21

Problem 20

Let $S$ be a subset of ${1,2,3, \ldots, 2024}$ such that the following two conditions hold:

What is the maximum possible number of elements in $S$ ?

(A) 436
(B) 506
(C) 608
(D) 654
(E) 675

Problem 21

The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length 5 . The numbers in positions $(5,5),(2,4),(4,3)$, and $(3,1)$ are $0,48,16$, and 12 , respectively. What number is in position $(1,2) ?$

(A) 19
(B) 24
(C) 29
(D) 34
(E) 39

Problem 22

Let $\mathcal{K}$ be the kite formed by joining two right triangles with legs 1 and $\sqrt{3}$ along a common hypotenuse. Eight copies of $\mathcal{K}$ are used to form the polygon shown below. What is the area of triangle $\triangle A B C$ ?

(A) $2+3 \sqrt{3}$
(B) $\frac{9}{2} \sqrt{3}$
(C) $\frac{10+8 \sqrt{3}}{3}$
(D) 8
(E) $5 \sqrt{3}$

Problem 23

Integers $a, b$, and $c$ satisfy $a b+c=100, b c+a=87$, and $c a+b=60$. What is $a b+b c+c a$ ?

(A) 212
(B) 247
(C) 258
(D) 276
(E) 284

Problem 24

A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^{+}, A^{-}, B^{+}, B^{-}, C^{+}$, and $C^{-}$is rolled. Suppose the bee occupies the point $(a, b, c)$. If the die shows $A^{+}$, then the bee moves to the point $(a+1, b, c)$ and if the die shows $A^{-}$, then the bee moves to the point $(a-1, b, c)$. Analogous moves are made with the other four outcomes. Suppose the bee starts at the point $(0,0,0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?

(A) $\frac{1}{54}$
(B) $\frac{7}{54}$
(C) $\frac{1}{6}$
(D) $\frac{5}{18}$
(E) $\frac{2}{5}$

Problem 25

The figure below shows a dotted grid 8 cells wide and 3 cells tall consisting of $1^{\prime \prime} \times 1^{\prime \prime}$ squares. Carl places 1 -inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks?

(A) 130
(B) 144
(C) 146
(D) 162
(E) 196

Three Resources for IOQM and Math Olympiads

IOQM and other Math Olympiads like American Math Competitions can be really challenging. It requires the student to go beyond the school curriculum and think about non-routine problems. In this post I will share three resources that can help you to systematically take up the preparation.

Books for IOQM and Math Olympiad

In this age of abundance, there is a lot of great books on Math Olympiads. The trick is to choose one relatively good book, and solve all problems from it, cover to cover. In fact we strongly suggest our students to focus on a single book before moving to other resources.

You may start with Challenge and Thrill of Pre-College Mathematics by V Krishnamurthy, C R Pranesachar, and K N Ranganathan. This is an excellent book that covers all the topics for IOQM standard. If you are in Grade 8 or below then use Mathematical Circles (Russian Experience) by Dmitry Fomin, Sergey Genkin, Ilia Itenberg.

Ofcourse there are plenty of other great books. Here are a few of them:

  1. Principles and Techniques in Combinatorics by Chen Chuan-Chon
  2. Number Theory: Structures, Examples, and Problem by Titu Andreescu, Dorin Andrica
  3. Euclidean Geometry in Mathematical Olympiads (EGMO) by Evan Chen
  4. Algebra
    • Secrets in Inequalities by Pham Kim Huang
    • Functional Equation by Venkatchala
    • An Introduction to Diophantine Equations: A Problem-Based Approach by Dorin Andrica, Ion Cucurezeanu, et al.
    • Complex Numbers from A to Z by by Titu Andreescu and Dorin Andrica
Software for IOQM and Math Olympiad

Cheenta has an excellent software for Math Olympiad, Physics Olympiad and ISI - CMI Entrance related problem solving. This tool provides students with sequence of problem. The software adapts with the child's performance and provides problems accordingly. It is available at https://panini8.com/

The software also lets students ask doubt problems and communicate with fellow learners using Latex. We have observed that children who use this problem solving software for 15 minutes every day experience remarkable improvement in their performance.

Start With a Problem

At Cheenta every learning begins with a

Non-Routine Problem

Try the problems with hints

but try them on your own!

Solve With Hints
Ask Doubts

Cheenta community has hundreds of teachers and students

Ask your doubts. Respond to others.

Learn about your strengths and weaknesses

Know your topicwise progress.

Check Topic Wise Progress
Live Classes for IOQM and Math Olympiad

Cheenta has outstanding program for IOQM and Math Olympiads. Typically the program includes three components.

  1. Compulsory Classes - twice a week
    • Concept Class
    • Homework Tutorial
  2. Optional Problem Solving Class - upto five times a week
  3. Optional 1-on-1 Class - for personalisation

Cheenta programs has been extremely successful in mathematical olympiads since 2010. For example in 2024, ten out of 78 students who qualified for Indian National Math Olympiad (from lakhs of Students) were from Cheenta.

Programs at CheentaDownload

AMC 10A 2002 Problem 15 | Prime Number

Try this beautiful Problem based on Number theory from AMC 10A, 2002 Problem 15.

Prime Number | AMC 10A 2021, Problem 15

Using the digits $1,2,3,4,5,6,7$, and 9 , form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?

Key Concepts

Arithmetic

Divisibility

Prime Number

Suggested Book | Source | Answer

Elementary Number Theory by David M. Burton.

AMC 10A 2002 Problem 15

190

Try with Hints

First try to find the probable digits for the unit place of the prime number.

The two digit prime number should end with $1, 3, 7, 9$ since it is prime and should not divisible by $2$ or $5$.

So now try to find which two digit primes will work here.

So, the primes should be $23, 41, 59, 67$.

Now find the sum of them.

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AMC 10A 2021 Problem 22 | System of Equations

Try this beautiful Problem based on System of Equations from AMC 10A, 2021 Problem 22.

System of Equations | AMC 10A 2021, Problem 22

Hiram's algebra notes are 50 pages long and are printed on 25 sheets of paper; the first sheet contains pages 1 and 2 , the second sheet contains pages 3 and 4 , and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly 19 . How many sheets were borrowed?

Key Concepts

Arithmetic Sequence

System of Equations

Algebra

Suggested Book | Source | Answer

Problem-Solving Strategies by Arthur Engel

AMC 10A 2021 Problem 22

13

Try with Hints

Let us assume that the roommate took sheets $a$ through $b$.
So, try to think what will be the changes in the page number?

So, page numbers $2 a-1$ through $2 b$. Because there are $(2 b-2 a+2)$ numbers.

Now apply the condition given there.

So we get, $\frac{(2 a-1+2 b)(2 b-2 a+2)}{2}$+$19(50-(2 b-2 a+2))$=$\frac{50 \cdot 51}{2}$

Now simplify this expression.

So , $2 a+2 b-39=25, b-a+1=13$

Now solve for $a, b$.

Find the number of pages using the values.

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AMC - AIME Program at Cheenta

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AMC 10A 2021 I Problem 20 | Enumeration

Try this beautiful Problem based on Enumeration appeared in AMC 10A 2021, Problem 20.

AMC 10A 2021 I Problem 20

In how many ways can the sequence $1$, $2$, $3$, $4$, $5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?

Key Concepts

Permutation

Enumeration

Combinatorics

Suggested Book | Source | Answer

An Excursion in Mathematics

AMC 10A 2021 Problem 20

32

Try with Hints

We have 5 numbers with us.

So, how many permutations we can have with those numbers?

So, $5!=120$ numbers can be made out of those $5$ numbers.

Now we have to remember that we are restricted with the following condition -

no three consecutive terms are increasing and no three consecutive terms are decreasing.

Now make a list of the numbers which are satisfying the condition given among all $120$ numbers we can have.

Now the list should be -

$13254$, $14253$, $14352$, $15243$, $15342$, $21435$, $21534$, $23154$, $24153$, $24351$, $25143$, $25341$
$31425$, $31524$, $32415$, $32514$, $34152$, $34251$, $35142$, $35241$, $41325$, $41523$, $42315$, $42513$,
$43512$, $45132$, $45231$, $51324$, $51423$, $52314$, $52413$, $53412$.

Count how many permutations are there?

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AMC 10A 2021 Problem 14 | Vieta's Formula

Try this beautiful Problem based on Vieta's Formula from AMC 10A, 2021 Problem 14.

Vieta's Formula | AMC 10A 2021, Problem 14

All the roots of the polynomial $z^{6}$-$10 z^{5}$+$A z^{4}$+$B z^{3}$+$C z^{2}$+$D z+16$ are positive integers, possibly repeated. What is the value of $B$ ?

Key Concepts

Vieta's Formula

Polynomial

Roots of the polynomial

Suggested Book | Source | Answer

Problem-Solving Strategies by Arthur Engel

AMC 10A 2021 Problem 14

-88

Try with Hints

Find out the degree of the given polynomial.

We know, Degree of polynomial= Number of roots of that polynomial.

Apply Vieta's Formula on the given polynomial.

By Vieta's Formula, the sum of the roots is 10 and product of the roots is 16.

Since there are 6 roots for this polynomial. By trial and check method find the roots.

The roots should be $2, 2, 2, 2, 1, 1$.

Now using the roots reconstruct the polynomial.

So the polynomial should be -

$(z-1)^{2}(z-2)^{4}$

$=(z^{2}-2 z+1)\\(z^{4}-8 z^{3}+24 z^{2}-32 z+16)$

Now equate it with the given polynomial to find the value of $B.$

Watch The Solution

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AMC - AIME Program at Cheenta

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ISI B.STAT PAPPER 2018 |SUBJECTIVE

Problem

Let $f$:$\mathbb{R} \rightarrow \mathbb{R}$ be a continous function such that for all$x \in \mathbb{R}$ and all $t\geq 0$

f(x)=f(ktx)
where $k>1$ is a fixed constant

Hint

Case-1


choose any 2 arbitary nos $x,y$ using the functional relationship prove that $f(x)=f(y)$

Case-2


when $x,y$ are of opposite signs then show that $$f(x)=f(\frac{x}{2})=f(\frac{x}{4})\dots$$
use continuity to show that $f(x)=f(0)$

Solution


Let us take any $2$ real nos $x$ and $y$.

Case-1

$x$ and $y$ are of same sign . WLG $0<x<y$

Then$\frac{y}{x}>1$
so there is a no $t\geq 0$ such that
$\frac{y}{x}=k^t$
$f(y)=f(k^tx)=f(x)$ [using$f(x)=f(k^tx)$]

case-2

$x,y$ are of opposite sign. WLG $x<0<y$
Then $f(x)=f(k^tx)$

$\Rightarrow f(k^tx)=f(k^t2\frac{1}{2}x)$


$\Rightarrow f(k^t2\frac{1}{2}x)=f(k^tk^{log_k2}\frac{x}{2})$


$\Rightarrow f(k^tk^{log_k2}\frac{x}{2})=f(k^{t+log_k2}\frac{x}{2})$

$\Rightarrow f(k^{t+log_k2}\frac{x}{2})=f(\frac{x}{2})$


Using this logic repeatedly we get


$f(x)=f(\frac{x}{2})=f(\frac{x}{4})\dots =f(\frac{x}{2^n})$


Now $\frac{x}{2^n}\rightarrow0$ and $f$ is a continous function hence $\lim_{n\to\infty}f(\frac{x}{2^n})=f(0)$.


[Because we know if $f$ is a continous function and $x_n$ is a sequence that converges to $x$ then $\lim_{n\to\infty}f(x_n)=f(x)$]


using similar logic we can show that $f(y)=f(0)$ so $f(x)=f(y)$ for any $x,y\in \mathbb{R}$


Easy Guide to Prepare for MathCounts Competition 2021 - 2022

What is Mathcounts?

MATHCOUNTS is a national middle school mathematics contest held in different places in the U.S. states and territories. It is established in 1983, which provides engaging mathematics programs to the US middle school students of different ability levels to grow their confidence and improve the attitudes about mathematics and problem solving.

Who are the founders of Mathcounts?

This nation-wide math competition's founding sponsors are the CNA Foundation, the National Society of Professionals Engineers and the National Council of Teachers of Mathematics. The MATHCOUNTS foundation is a non-profitable organization which encourages students in grade 6-8 in all the US states and territories.

Who can appear for Mathcounts Competition?

Students of US in grades 6-8 are eligible to participate in MATHCOUNTS competitions.

The student has to attend a school located in a U.S. state or territory or it can be an overseas school that is affiliated with the U.S. Departments of Defense or State.

Maximum 12 NSCs from the same school can register for the Competition. Registration eligibility will be on a first-come, first-served basis.

Home schools and home school groups in agreement with the home school laws of the state in which they are located are entitled to join in MATHCOUNTS competitions in accordance with all other policies.

Mathcounts Competition Format 2021-22

The Mathcounts Competition Series has 4 levels—school, chapter, state and national. Each level contains four rounds- Sprint Round, Target Round, Team Round and Countdown Round.

Scoring system of the Examination

The score of a team is equal to the average of the total of its participants' individual scores plus twice the number of questions answered accurately on the team round. With the individual scores of a maximum of 46 each and team-round scores a maximum of 20, 66 is a perfect team score.

What is a good score for Mathcounts?

A perfect score is 46. It depends on the state or chapter. The ranking is determined by either raw individual score or by the results of the Countdown Round at the Chapter and State Levels.

Important dates for Mathcounts Competition, 2021- 22

Why is Mathcounts Competition important?

MATHCOUNTS enables the critical thinking and problem-solving skills essential for success. In an independent research, more than 85% of MATHCOUNTS students said their self-confidence in their STEM abilities enhanced after participating in the Competition Series. 

What does the winner of Mathcounts get?

Every year, the top team and the participants in the Countdown Round mostly win a trip to White House to meet the current president of the United States followed by a scholarship offering the sponsors.

How to prepare for Mathcounts Competition 2021-2022

Resources to prepare for Mathcounts : Curriculum at Cheenta

Preparation Tips

All the best!