Published at - 27 June 2024

Backwards Trigonometric Substitutions: Integration is not always a simple, straightforward process. Sometimes, similar integrals require different solution strategies. For example, consider the four integrals

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Published at - 26 June 2024

Trigonometric Integral: Using Pythagorean Identities to set up a substitution.

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Published at - 22 April 2024

Problem [Russian 1995]: Let \( \, m\, \) and \( \, n\, \) be positive integers such that \[ \mbox{lcm}(m, n) +\gcd(m, n) = m + n. \] Prove that one of two numbers is divisible by the other.

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Published at - 16 April 2024

Problem [AHSME 1976]: If \( \, p\, \) and \( \, q\, \) are prime and \( \, x^2 - px + q =0\, \) has distinct positive integral roots, find \( \, p\, \) and \( \,q. \)

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Published at - 22 February 2024

Problem: A positive integer is written on each face of a cube. Each vertex is then assigned the product of the numbers written on the three intersecting the vertex. The sum of the numbers assigned to all the vertices is equal to 1001. Find the sum of the numbers written on the faces of the cube.

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Published at - 14 February 2024

Problem [AIME 1986]: What is the largest positive integer \( \, n\, \) for which \( \, n^3 + 100 \, \) is divisible by \( \, n + 10? \)

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Published at - 13 February 2024

Problem [HMMT 2004]: Given a sequence of six strictly increasing positive integers such that each number (besides the first) is a multiple of the one before it and the sum of all six numbers is 79, what is the largest number in the sequence?

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Published at - 12 February 2024

Problem [AMC10B 2004]: Let 1, 4, ... and 9, 16, ... be two arithmetic progressions. The set S is the union of the first 2004 terms of each sequence. How many distinct numbers are in S?

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Published at - 31 October 2023

Published at - 2 October 2023

Registration for math contests is open now. We are registering for the COMC (Canadian Open Math Challenge), CJMC (Canada Jay Mathematics Competition), Math Caribou Contest, CIMC (Canadian Intermediate Math Contest), CSMC (Canadian Senior Math Contest, and AMC 8.

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Published at - 6 July 2023

The 2020 Canadian Junior Mathematical Olympiad Problem: A purse contains a finite number of coins, each with distinct positive integer values. Is it possible that there are exactly 2020 ways to use coins from the purse to make the value 2020? Solution: It is possible.

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Published at - 28 February 2023

Problem: Let \( \, p \, \) be a prime number. Show that there are infinitely many positive integers \( \, n\, \) such that \( \, p\, \) divides \( \, 2^n - n. \)

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Published at - 30 January 2023

Problem: Find, with proof, all nonzero polynomials \( \, f(z)\, \) such that \[ f(z^2)+ f(z)f(z +1) = 0. \]

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Published at - 11 January 2023

Published at - 13 December 2022

Problem: If \( \, a \equiv b \pmod{n}, \, \) show that \( \, a^n\equiv b^n \pmod{n^2}.\, \) Is the converse true? Proof: From \( \, a\equiv b \pmod{n} \, \) is follows that \( \, a = b + qn\, \) for some integer \( \, q.\, \) By the binomial theorem we obtain \[ \begin{align} a^n - b^n = & (b + qn)^n - b^n \\ = &\binom{n}{1}b^{n -1} qn + \binom{n}{2}b^{n-2}q^2n^2 + \cdots + \binom{n}{n}q^nn^n \\ = & n^2\Bigl(b^{n -1}q + \binom{n}{2}b^{n -2}q^2 + \cdots + \binom{n}{n}q^nn^{n-2}\Bigr), \end{align} \] implying that \( \, a^n\equiv b^n\pmod{n^2}. \) The converse is not true because, for instance, \( \, 3^4 \equiv 1^4 \pmod{4^2}\, \)but \( \, 3\not\equiv 1 \pmod{4}. \)

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Published at - 12 December 2022

Problem: Find all positive integers \( \, n\, \) for which \( \, n! + 5\, \)is a perfect cube. Solution: The only answer is \( \, n = 5. \) One checks directly that \( \, n! + 5\, \) is not a perfect cube for \( \, n = 1, 2, 3, 4, 6, 7, 8, 9\, \) and that \( \, 5! + 5 \, \)is a perfect cube. If \( \, n! + 5\, \)were a perfect cube for \( \, n> 9,\, \) then, since it is a multiple of \( \, 5, \,\, \, n! + 5\, \) would be a multiple of 125. However, this is not true, since \( \, n!\, \) is a multiple of 125 for \( \, n > 9,\, \) but 5 is not. Thus the only positive integer with the desired property is \( \, n = 5. \)

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Published at - 9 December 2022

Problem (HMMT 2005): The number 27000001 has exactly four prime factors. Find their sum. Solution:

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Published at - 8 December 2022

Problem: Let \( \, p\, \) be a prime of the form \( \, 3k + 2\, \) that divides \( \, a^2 + ab + b^2\, \) for some integers \( \, a\, \) and \( \, b.\, \)Prove that \( \, a\, \) and \( \, b\, \) are both divisible by \( \, p. \) Proof:

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Published at - 6 December 2022

Problem (UK 1998): Let \( \, x, y, z \, \) be positive integers such that \[ \frac{1}{x}-\frac{1}{y} = \frac{1}{z}. \] Let \( \, h\, \) be the greatest common divisor of \( \, x, y, z.\, \)Prove that \( \, hxyz\, \)and \( \, h(y - x)\, \) are perfect squares.

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Published at - 5 December 2022

Problem: Compute the sum of all numbers of the form \( \, \frac{a}{b}\, \), where \( \, a\, \) and \( \, b\, \) are relatively prime positive divisors of 27000. Solution: Because \( \, 27000=2^33^35^3\, \) each \( \, \frac{a}{b}\, \) can be written in the form of \( \, 2^a3^b5^c, \, \) where \( \, a, b, c\, \) are integers in the interval \( \, [-3, 3].\, \) It follows that each \( \, \frac{a}{b}\, \)appears exactly once in the expansion of

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Published at - 2 December 2022

Problem (AMC 12A 2005): Call a number prime looking if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000? Solution:

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Published at - 1 December 2022

Problem (Australia 1999): Solve the following system of equations: \[ \begin{align} x + \lfloor y \rfloor + \{z\} = 200.0,\\ \{x\} + y + \lfloor z \rfloor = 190.1,\\ \lfloor x\rfloor + \{y\} + z = 178.8. \end{align} \] Solution: Because \( \, x = \lfloor x\rfloor + \{x\}\, \) for all real numbers \( \, x\, \) adding the three equations gives

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Published at - 29 November 2022

Problem (Euclid 2020): (a) For each positive real number x, define f(x) to be the number of prime numbers p that satisfy x ≤ p ≤ x + 10. What is the value of f(f(20))? (b) Determine all triples (x, y, z) of real numbers that satisfy the following system of equations: \[ (x - 1)(y -2) = 0\\ (x - 3)((z + 2) = 0 \\ x + yz = 0 \]

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Published at - 25 November 2022

Problem (Euclid Contest 2022): (a) Find the three ordered pairs of integers (a, b) with 1 < a < b and ab = 2022. (b) Suppose that c and d are integers with c>0 and d>0 and \( \, \frac{2c + 1}{2d + 1} = \frac{1}{17}.\, \) What is the smallest possible value of d? (c) Suppose that p, r and t are real numbers for which (px+r)(x+5) = x2 +3x+t \( \, (px + r)(x + 5) = x^2 + 3x + t \) is true for all real numbers x. Determine the value of t.

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Published at - 24 November 2022

Problem: Conor has a summer lawn-mowing business. Based on experience, Conor knows that \( \, P = -5x^2 + 200x - 1500\, \) models his profit, P, in dollars, where x is the amount, in dollars, charged per lawn. a) How much does he need to charge if he wants to break even? b) Howmuchdoesheneedtochargeifhewantstohaveaprofit of $500?

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Published at - 23 November 2022

Problem: Solve the equation \[ \sin x \cos y + \sin y\cos z + \sin z \cos x =\frac{3}{2} \] Solution: The equation is equivalent to

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Published at - 8 November 2022

Registration for AMC 8 (American Mathematics Contest 8) is open. The contest date is in the third week of January, 2023. To register, please call 416-996-2916 for more details.

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Published at - 16 September 2022

Problem: Find \( \, n\, \) such that \( \, 2^n ||3^{1024} - 1. \) Solution: Note that \( \, 2^{10} = 1024 \, \) and \( \, x^2 - y^2 = ( x - y)(x+ y).\, \) We have

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Published at - 13 September 2022

In this class, we help students grasp and understand the ideas and material of the course, fill gaps, and master their knowledge of the subject. We provide students with a strong foundation to do their best on their assignments.

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Published at - 13 September 2022

Problem (ARML 2003): Find the largest divisor of 1001001001 that does not exceed 10000. Solution: We have \[ 1001001001 = 1001\times 10^6 + 1001 = 1001\times (10^6 +1)= 7\times 11\times13\times(10^6 +1). \]

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Published at - 12 September 2022

Problem (Romania 1983): Let \( \, 0 Read More

Published at - 9 September 2022

Problem (HMMT 2002): Compute \[ \gcd(2002 + 2, 2002^2 +2, 2002^3 + 2, \ldots). \] Solution: Let \( \, g\, \) denote the desired greatest common divisor. Note that \( \, 2002^2 + 2 = 2002(2000 + 2) + 2 = 2000(2002 + 2) + 6. \, \) By Euclidean algorithm, we have

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Published at - 8 September 2022

Problem (Russia 2001): Find all primes \( \, p\, \) and \( \, q\, \) such that \( \, p + q = (p - q)^3. \) Solution: The only such primes are \( \, p = 5\, \) and \( q = 3. \) Because \( \, (p -q)^3 = P + q \ne 0, \,\, p \, \) and \( \, q\, \) are distinct and hence relatively prime.

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Published at - 6 September 2022

Problem: Let \( \, p\, \) be a prime number. Prove that \( \, p\, \)divides \( \, ab^p - ba^p\, \) for all integers \( \, a \, \) and \( \, b. \)

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Published at - 17 August 2022

In this class, we help students grasp and understand the ideas and material of the course, fill gaps, and master their knowledge of the subject. We provide students with a strong foundation to do their best on their assignments.

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Published at - 16 August 2022

Problem: Determine the limit of the following sequence. \[ \sqrt{3}. \sqrt{3\sqrt{3}}, \sqrt{3\sqrt{3{\sqrt{3}}}}, \sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}, \sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}}\ldots \] Solution:

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Published at - 16 August 2022

Pi School is a leader in academic tutoring and coaching, offering in-site and online sessions to students in grades 5-12 in Toronto. We provide result-oriented supplementary education through individualized lesson plans, in depth progress tracking and monitoring system, and effective group midterm and final exam review sessions. All of our in-home and exam prep sessions are delivered by our knowledgeable, caring, and passionate tutors who have extensive knowledge of the ON school curriculum and are committed to excellence.

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Published at - 11 August 2022

Problem: Solve the equation \[ \sqrt{x + a} + \sqrt{x + b} + \sqrt{x + c} = \sqrt{x + a + b -c}, \] where \( \, a, b, c\, \) are real numbers. Discuss the equation in terms of the values of the parameters. Solution: We distinguish two cases: 1) \( \, b = c.\, \) The equation becomes

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Published at - 10 August 2022

Problem: Solve the equation \[ \sqrt{x + \sqrt{4x + \sqrt{16x + \sqrt{\ldots + \sqrt{4^n x +3}}}}} =\sqrt{x} = 1. \] Solution: The equation is equivalent to

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Published at - 9 August 2022

Problem: Find the real solution to the following equation \[ (x + y)^2 = (x + 1)(y -1). \] Solution: Setting \( \, X = x +1 \, \) and \( \, Y = y -1 \, \) yields

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Published at - 9 August 2022

Problem: Solve the following equation in real numbers. \[ \sqrt{x} + \sqrt{y} + 2\sqrt{z - 2} + \sqrt{u} + \sqrt{v} = x + y + z + u + v. \] Solution: We can rewrite the equation in the following form

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Published at - 9 August 2022

Problem: Solve the following equation in complex numbers. \[ (x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360. \] Solution: Equation \( (x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360. \) is equivalent to

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Published at - 6 August 2022

Problem: A mechanic is reboring a 6-in-deep cylinder to fit a new piston. The machine they are using increases the cylinder’s radius one-three thousandth of an inch every minute. How rapidly is the cylinder volume increasing when the bore diameter is 3.8 inches?

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Published at - 4 August 2022

Problem: Sove the inequality \( \, x^3 - 2x^2 + 5x + 20 \ge 2x^2 + 14 x -16. \)

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Published at - 4 August 2022

Problem: If a 75-foot flagpole casts a shadow 43 ft long, to the nearest 10 minutes what is the angle of elevation of the sum from the tip of the shadow?

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Published at - 2 August 2022

Problem: An air conditioner is being purchased for the room shown below. The room has an open ceiling that follows the roof trusses. According to the US Department of Energy, to determine the cooling requirement of a room, use 2.5 BTUs per cubic foot of air space. To this value, add an additional 1000 for each window in the room. This room has 4 windows. Determine the number of BTUs needed to cool this room.

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Published at - 2 August 2022

Problem: Evaluate \( \, i^{2009}. \) Solution: We start by experimenting with smaller powers of \( \, i, \, \) to find a pattern.

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Published at - 2 August 2022

Problem: If \( \, \angle C = 26^{\circ}\, \) and \( \, r = 19, \, \) find \( \, x. \)

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Published at - 2 August 2022

Problem: A man climbs 213 meters up the side of a pyramid. Find that the angle of depression to his starting point is \( \, 52.6^{\circ}\, \) How high off of the ground is he?

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Published at - 31 July 2022

Problem: A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet/sec. How fast is the top of the ladder moving down the wall when the base of the ladder is 7, 15, and 24 feet from the wall?

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Published at - 28 July 2022

Problem: San Luis Obispo, California is 12 miles due north of Grover Beach. If Arroyo Grande is 4.6 miles due east of Grover Beach, what is the bearing of San Luis Obispo from Arroyo Grande? Solution:

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Published at - 28 July 2022

Problem: The length of the shadow of a building 34.09 m tall is 37.62 m. Find the angle of the elevation of the sun. Solution:

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Published at - 28 July 2022

Problem: Knowing that \( \, 2^{29}\, \)is a nine-digit number all of whose digits are distinct, without computing the actual number determine which of the ten digits is missing. Justify your answer.

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Published at - 28 July 2022

Problem: Compute the sum of all numbers of the form \( \, \frac{a}{b},\, \) where \( \, a\, \) and \( \, b\, \) are relatively prime positive divisors of 27000.

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Published at - 27 July 2022

Problem : Prove that each nonnegative integer can be represented in the form \( \, a^2 + b^2 - c^2, \, \) where \( \, a, b, and \,\, c\, \) are positive integers with \( \, a < b < c. \)

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Published at - 23 July 2022

Problem: The Ferry wheel has a 250 feet diameter and 14 feet above the ground. If \( \, \theta \, \) is the central angle formed as a rider moves from position \( \, P_0\, \) to position \( \, P_1\, \), find the rider’s height above the ground h when \( \, \theta \, \) is \( \, 45^{\circ}. \)

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Published at - 20 July 2022

Problem: A person standing at point A notices that the angle of elevation to the top of the antenna is \( \, 47^{\circ}30'.\, \) A second person standing 33.0 feet farther from the antenna than the person at A finds the angle of elevation to the top of the antenna to be \( \, 42^{\circ}10'.\, \) How far is the person at A from the base of the antenna?

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Published at - 20 July 2022

Problem: From a given point on the ground, the angle of elevation to the top of a tree is \( \, 36.7^{\circ}.\, \) From a second point, 50 feet back, the angle of elevation to the top of the tree is \( \, 22.2^{\circ}.\, \) Find the height of the tree to the nearest foot.

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Published at - 19 July 2022

Problem: A circle has its centre at C and a radius of 18 inches. If triangle ADC is a right triangle and \( \, \angle A = 35^{\circ}\, \). Find x, the distance from A to B.

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Published at - 19 July 2022

Problem: In the right triangle ABC, \( \, \angle A = 40^{\circ}\, \) and c = 12 cm. Find a, b, and \( \, \angle B. \) Solution:

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Published at - 17 July 2022

Problem: a) A 25-kg box is located 8 metres up a ramp inclined at an angle of 18°to the horizontal. Determine the work done by the force of gravity as the box slides to the bottom of the ramp. b) Determine the minimum force, acting at an angle of 40° to the horizontal, required to slide the box back up the ramp. (Ignore friction)

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Published at - 17 July 2022

Problem: Find the area of the triangle with vertices P(7, 2, -5), Q(9, -1, -6) and R(7, 3, -3). Solution: Start by finding the vectors that form two sides of this triangle. The area of the triangle is half of the area of the parallelogram having these vectors as sides.

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Published at - 14 July 2022

Problem (APMO 1998): Show that for any positive integers \( \, a \, \) and \( \, b, \, \) the number \[ (36a + b)(a + 36b) \] cannot be a power of 2.

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Published at - 13 July 2022

Problem: Find all real numbers \( \, x\, \) such that \[ \frac{8^x + 27^x}{12^x + 18^x}=\frac{7}{6}. \]

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Published at - 12 July 2022

Problem: Solve the system of equations: \[ \begin{align*} \\& x + \frac{3x -y}{x^2 + y^2}=3 \\& y - \frac{x + 3y}{x^2 + y^2} = 0. \end{align*} \] Solution: We multiply the second equation by \( \, i\, \) and add it to the first equation

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Published at - 11 July 2022

Problem(ARML 1997): Find a triple of rational numbers \( \, (a, b, c)\, \) such that \[ \sqrt[3]{\sqrt[3]{2} -1} = \sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} \]

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Published at - 8 July 2022

Problem: A water tank has the shape of an inverted circular cone with base radius 2m and height 4m. If water is being pumped into the tank at a rate of 2 m3/min, find the rate at which the water level is rising when the water is 3 m deep.

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Published at - 8 July 2022

Problem (UK 1998): Let \( \, x, y, z\, \) be positive integers such that

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Published at - 8 July 2022

Pi School is an after school program that offers curricular and extra curricular enriched course

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Published at - 8 July 2022

Problem (UK 1998): Let \( \, x, y, z\, \) be positive integers such that \[ \frac{1}{x} - \frac{1}{y} = \frac{1}{z}. \] Let \( \, h\, \)be the greatest common divisor of \( \, x, y, z.\, \)Prove that \( \, hxyz\, \) and \( \, h(y - x)\, \) are perfect squares.

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Published at - 7 July 2022

Problem (AIME 1995): Let \( \, n =2^{31}3^{19}.\, \) How many positive integer divisors of \( \, n^2\, \) are less than \( \, n\, \) but do not divide \( \, n? \)

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Published at - 6 July 2022

Problem (Russia 1995): Let \( \, m\, \) and \( \, n\, \) be positive integers such that \[ lcm (m, n) +\gcd(m, n)= m + n. \] Prove that one of the two numbers is divisible by the other.

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Published at - 5 July 2022

Problem (AIME 1986): What is the largest positive integer \( \, n\, \) for which \( \, n^3 + 100 \, \) is divisible by \( \, n + 10? \)

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Published at - 5 July 2022

Problem: Let \( \, 1\le \alpha <\beta \, \) are real numbers. Show that there are integers \( \, m, n >1\, \) such that \( \, \alpha < \sqrt[n]{m} < \beta. \)

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Published at - 5 July 2022

Problem: Prove that \[ (4\cos^2 9^{\circ} - 3)(4\cos^2 27^{\circ} - 3) = \tan 9^{\circ}. \]

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Published at - 5 July 2022

Problem (AMC 10B 2004): Let 1, 4, ... and 9, 16, ... be two arithmetic progressions. The set S is the union of the first 2004 terms of each sequence. How many distinct numbers are in S?

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Published at - 2 July 2022

Problem: Evaluate the sum \[ S_n =\sum_{k = 1}^{n-1}\sin kx \cos(n - k)x. \]

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Published at - 2 July 2022

Problem: Prove that \[ \cos^2\frac{x}{3} + \cos^2\frac{x + 2\pi}{3} + \cos^2\frac{x + 4\pi}{3} = \frac{3}{4}\cos x \] for all \( \, x\in \mathbb{R}. \)

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Published at - 30 June 2022

Problem (AHSME 1976): If \( \, p\, \) and \( \, q\, \) are primes and \( \, x^2 -px + q = 0\, \)has distinct positive integral roots, find \( \, p\, \) and \( \, q. \)

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Published at - 30 June 2022

Problem: Let \( \, k\, \)be a integer. Prove that \( \, 3^{2^k} + 1\, \) is divisible by 2, but is not divisible by 4.

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Published at - 28 June 2022

Problem: Prove that \[ 3^{4^5} + 4^{5^6} \] is a product of two integers, each of which is larger than \( \, 10^{2002}. \)

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Published at - 27 June 2022

Problem: How many seven digit numbers that do not start nor end with 3 are there?

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Published at - 26 June 2022

Problem: Sole the next equation in real numbers. \[ \sqrt{x} + \sqrt{y} + 2\sqrt{z - 2} + \sqrt{u} + \sqrt{v} = x + y + z + u + v. \]

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Published at - 26 June 2022

Problem: Suppose that\( \, a, b, c, d \, \) are complex numbers such that \( \, a + b + c + d = 0.\, \) Prove that \[ a^3 + b^3 + c^3 +d^3 = 3(abc + bcd + cda + dab). \]

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Published at - 24 June 2022

Problem: Prove that if \( \, x, y, z, \, \)are nonzero real numbers with \( \, x + y + z =0, \, \) then \[ \frac{x^2 + y^2}{x + y} + \frac{y^2 + z^2}{y + z} + \frac{z^2 + x^2}{z + x} = \frac{x^3}{yz}+\frac{y^3}{xz}+\frac{z^3}{xy} \qquad (*) \]

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Published at - 23 June 2022

Problem: Given that the two vectors \( \, 8\overrightarrow{u} - \overrightarrow{v}, \quad 4\overrightarrow{u} + 3\overrightarrow{v}\, \)are perpendicular and \( \, |\overrightarrow{v}| = 2|\overrightarrow{u}|. \, \)Find the angle between the vectors \( \, \overrightarrow{u} \, \)and \( \, \overrightarrow{v}. \)

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Published at - 22 June 2022

Problem: Let , , and be positive integers with such that and . What is ? Solution: By adding the two equations we have \( \, 2a^2 + 2b^2 + 2c^2 – 2ab – 2ac – 2bc = 14. \) We can rearrange and factor it, \[ \begin{align*}(a^2 -2ab + b^2) + ( a^2 – 2ac + c^2) […]

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Published at - 21 June 2022

Problem (2015 AMC 12A Problems/Problem 18): The zeros of the function \( \, f(x) = x^2 - ax + 2a\, \) are integers. What is the sum of the possible values of ?

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Published at - 20 June 2022

Problem: In the diagram, ABCD is a trapezoid with BC parallel to AD and AB = CD. Point E is on AD so that BE is perpendicular to AD and point F is the point of intersection of AC with BE. If AF = FB and \( \, \angle (AFE) = 50^{\circ},\, \) what is the measure of \( \, \angle{ADC} ? \)

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Published at - 19 June 2022

Problem: In the diagram, triangle ABC is right-angled at B. MT is the perpendicular bisector of BC with M on BC and T on AC. If AT = AB, what is the size of \( \, \angle ABC? \)

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Published at - 18 June 2022

Problem: Evaluate \[ \frac{3}{1! + 2! + 3!} + \frac{4}{2! + 3! + 4!} + \ldots + \frac{2001}{1999! + 2000! + 2001!} \] Solution: Note that

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Published at - 17 June 2022

Problem: Suppose a store owner wants to make a 100 pound mixture of peanuts and cashews to sell for $4.30 per pound. If peanuts sell for $2.50 per pound and cashews sell for $7.00 per pound, how many pounds of each should be used?

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Published at - 16 June 2022

Problem: Find all complex numbers \( \, z\, \) such that \[ (3z + 1)(4z + 1)(6z +1)(12z + 1) = 2. \]

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Published at - 15 June 2022

Problem (AIME 1986): The polynomial \[ 1 - x + x^2 - x^3 + \ldots + x^{16} - x^{17} \] may be written in the form \[ a_0 + a_1y + a_2y^2 + \ldots + a_{16}y^{16} + a_{17}y^{17}, \] where \( \, y = x +1 \, \) and \( \, a_i \)s are constants. Find \( \, a_2. \)

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Published at - 14 June 2022

Problem (Canadian Mathematical Olympiad 2019): Let\( \, a\, \) and \( \, b\, \) be positive integers such that \( \, a + b^3\, \) is divisible by \( \, a^2 + 3ab + 3b^2 -1.\, \) Prove that \( \, a^2 + 3ab + 3b^2 -1\, \) is divisible by the cube of an integer greater than 1.

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Published at - 13 June 2022

Problem (39th Canadian Mathematical Olympiad): Suppose that \( \, f\, \) is a real-valued function for which \[ f(xy) + f(y - x) \ge f(y + x) \] for all real numbers \( \, x \, \) and \( \, y. \) (a) Give a non constant polynomial that satisfies the condition. (b) Prove that \( \, f(x)\ge 0 \, \) for all real \( \, x. \) Solution:

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Published at - 12 June 2022

Problem: Find the real zeros of the polynomial \[ P_a(x) = (x^2 + 1)(x - 1)^2 - ax^2, \] where \( \, a \, \) is a given real number. Solution: We have

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Published at - 10 June 2022

Problem (IMO 1996): Let \( \, a, \, b, \, \) and \( \, c \, \) be positive real numbers such that \( \, abc =1. \) Prove that \[ \frac{ab}{a^5 + b^5 +ab} + \frac{bc}{b^5 + c^5 +bc} + \frac{ca}{c^5 + a^5 + ca} \le 1. \]

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Published at - 9 June 2022

Problem: Let \( \, m\, \) be a given real number. Find all complex numbers \( \, x \, \) such that \[ \Big(\frac{x}{x +1}\Big)^2 + \Big(\frac{x}{x -1}\Big)^2= m^2 + m. \]

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Published at - 8 June 2022

Problem (China 1992): Prove that \[ 16 < \sum_{k=1}^{80}\frac{1}{\sqrt{k}}< 17. \]

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Published at - 7 June 2022

Problem: A circle has its center at \( \, C\, \) and a radius of 18 inches. If triangle \( \, ADC\, \) is a right triangle and \( \, A = 35^{\circ}. \, \) Find \( \, x, \, \) the distance from \( \,A\, \) to \( \, B. \)

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Published at - 6 June 2022

Problem: Let \( \, a \, \) be an irrational number and let \( \, n \, \) be an integer greater than 1. Prove that \[ \Big(a + \sqrt{a^2 - 1}\Big)^{\frac{1}{n}} + \Big(a - \sqrt{a^2 -1}\Big)^{\frac{1}{n}} \] is an irrational number.

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Published at - 5 June 2022

Problem (Related Rate Problems, Grade 12 Calculus): Air is being pumped into a spherical balloon so that its volume increases at a rate of \( \, 100 \, cm^3/s . \, \) How fast is the radius of the balloon increasing when the diameter is 50 cm?

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Published at - 4 June 2022

Problem: Solve the equation: \[ \sqrt{x^2 + 4x + 4} = x^2 + 3x -6 \] Solution:

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Published at - 3 June 2022

Problem: Solve the equation \[ 2(2^x -1)x^2 + (2^{x^2} -2)x = 2^{x +1} -2 \] for real numbers \( \, x. \)

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Published at - 2 June 2022

Problem (Korean Mathematics competition 2001): Let \[ f(x) = \frac{2}{4^x + 2} \] for real numbers \( \, x.\, \) Evaluate \[ f\Big(\frac{1}{2001}\Big) + f\Big(\frac{2}{2001}\Big) + \dots + f\Big(\frac{2000}{2001}\Big). \]

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Published at - 1 June 2022

Problem (Korean Mathematics competition): Find all real numbers \( \, x \, \) satisfying the equation \[ 2^x + 3^x - 4^x + 6^x -9^x = 1. \]

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Published at - 1 June 2022

Problem: Let \( \, a, b, \, \) and \( \, c \, \)be distinct nonzero numbers such that \[ a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}. \] Prove that \( \, |abc| = 1. \)

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Published at - 18 May 2022

Summer camp at Pi School runs from July 4 to September 2, 2022.

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Published at - 14 May 2022

Pi School is an after school program that offers curricular and extra curricular enriched courses in math and science for students in grades 5 – 12. The purpose of our program is to develop and expand students’ understanding of Math and Science. We nurture students’ development by challenging them to be creative, critical and persistent problem solvers. We seek to develop an appreciation for the subject and promote joyful and effective education in mathematics and science.

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