PROBLEM Let $f (0,\infty)\rightarrow \mathbb{R}$ be a continous function such that for all $x \in (0,\infty)$ $f(x)=f(3x)$ Define $g(x)= \int_{x}^{3x} \frac{f(t)}{t}dt$ for $x \in (0,\infty)$ is a constant function HINT Use leibniz rule for differentiation under integral sign SOLUTION using leibniz rule for differentiation under integral sign we get $g'(x)=f(3x)-f(x)$ $\Rightarrow g'(x)=0$ [ Because f(3x)=f(x)] […]
PROBLEM Show that there are exactly $2$ numbers $a$ in the set $\{1,2,3\dots9400\}$ such that $a^2-a$ is divisible by $10000$ HINT Use Modular arithmetic and concepts of coprime numbers SOLUTION we know $10000=2^4*5^4$ In order for $10000$ to divide $a^2-a$ both $2^4$ and $5^4$ must divide $ a^2-a $ Write $a^2-a=a(a-1)$ Note that $a$ and […]
Author: Kazi Abu Rousan $\pi$ is not just a collection of random digits. $\pi$ is a journey; an experience; unless you try to see the natural poetry that exists in $\pi$, you will find it very difficult to learn. Today we will see a python code to find the value of $\pi $ up to […]
Author: Kazi Abu Rousan Pi is not merely the ubiquitous factor in high school geometry problems; it is stitched across the whole tapestry of mathematics, not just geometry’s little corner of it. $\pi$ is truly one of the most fascinating things exist in mathematics. It's not just there in geometry, but it's also there in pendulum, […]
To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people. A. Knauf from "Number theory, dynamical systems and statistical mechanics" This quote is indeed true. If you just think about the […]
Solve beautiful Number Theory problem from IMO 1959 with the help of Euclidean Algorithm and Division Lemma.
Problem 1: How many \(1 \times 1\) squares are in this diagram? (A) 16(B) 18(C) 20(D) 24(E) 25 Problem 2: What is half of 2020?(A) 20(B) 101(C) 110(D) 1001(E) 1010 Problem 3: What is the perimeter of this triangle? (A) 33 m(B) 34 m(C) 35 m(D) 36 m(E) 37 m Problem 4: I stepped on […]
Problem 1: What is the perimeter of this rhombus? (A) 20 cm(B) 24 cm(C) 28 cm(D) 32 cm(E) 36 cm Problem 2: The temperature in the mountains was \(4^{\circ} \mathrm{C}\) but dropped overnight by \(7^{\circ} \mathrm{C}\). What was the temperature in the morning?(A) \(3{ }^{\circ} \mathrm{C}\)(B) \(11^{\circ} \mathrm{C}\)(C) \(-3^{\circ} \mathrm{C}\)(D) \(-4^{\circ} \mathrm{C}\)(E) \(-11^{\circ} \mathrm{C}\) Problem […]
Problem 1: Kurt paved his courtyard in the pattern shown. How many \(1 \times 1\) pavers are in his courtyard? (A) 28(B) 30(C) 32(D) 34(E) 36 Problem 2: Which of the following expressions has the smallest value?(A) (3+2)(B) (3-2)(C) \(3 \times 2\)(D) \(3 \div 2\)(E) 32 Problem 3: The numbers on the top corners of […]
Books play a significant role in the preparation for the Singapore Mathematics Olympiad. In Cheenta we recommend a few books based on their age and grades that suit them. Books for Preliminary AMC Books for Advanced AMC
Problem 1: \(2021-1202=\) (A) 719(B) 723(C) 819(D) 823(E) 3223 Problem 2: What is the perimeter of this figure? (A) 28 units(B) 26 units(C) 24 units(D) 20 units(E) 21 units Problem 3: The area of this triangle is (A) \(10\) \(cm^2\) (B) \(12\) \(cm^2\) (C) \(12.5\) \(cm^2\)(D) \(15\) \(cm^2\) (E) \(16\) \(cm^2\) Problem 4: On the […]
Schedule of the Test: Registration Fee: The registration fee is $8.00 per participant (from SMS institutional member schools) per competition category; and $10.00 per participant (from non-institutional member schools). The participants' names and fees should be forwarded by the Head of the Department of Mathematics (of the competing school) to the Chairman of the Competition Committee in the prescribed […]
Problem 1: How many dots are in this pattern? (A) 20(B) 21(C) 22(D) 23(E) 24 Problem 2: What number is one hundred more than \(465 \) ? (A) 365(B) 455(C) 475(D) 565(E) 1465 Problem 3: What fraction of this rectangle is shaded? (A) \(\frac{1}{2}\)(B) \(\frac{1}{4}\)(C) \(\frac{1}{6}\)(D) \(\frac{1}{8}\)(E) \(\frac{1}{10}\) Problem 4: There were 17 dogs and […]
Problem 1: How many eggs are in these cartons? (A) 12(B) 15(C) 16(D) 18(E) 21 Problem 2: Which one of the following is the largest number? (A) 401(B) 410(C) 14(D) 140(E) 44 Problem 3: Which of the following is equal to 3 m? (A) 3 cm(B) 30 cm(C) 300 cm(D) 3000 cm(E) 36 cm Problem […]
Problem 1: What is double 4? (A) 2(B) 3(C) 8(D) 12(E) 24 Problem 2: Which pattern has exactly 10 dots? Problem 3: Which of the following is the same as 6 tens and 3 ones?(A) sixty-three(B) six and three(C) thirty-six(D) six hundred and three(E) sixty-one Problem 4: When I add 11 and another number, I […]