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Problem of the Day (June 27, 2024)
Published at - 27 June 2024
Problem of the Day (June 27, 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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Trigonometric Integral (June 26, 2024)
Published at - 26 June 2024
Trigonometric Integral (June 26, 2024)

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

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Problem of the Day (April 22, 2024)
Published at - 22 April 2024
Problem of the Day (April 22, 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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Problem of the Day (April 16, 2024)
Published at - 16 April 2024
Problem of the Day (April 16, 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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Problem of the Day (February 22, 2024)
Published at - 22 February 2024
Problem of the Day (February 22, 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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Problem of the Day (February 14, 2024—> Happy Valentine’s Day!)
Published at - 14 February 2024
Problem of the Day (February 14, 2024—> Happy Valentine’s Day!)

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

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Problem of the Day (February 13, 2024)
Published at - 13 February 2024
Problem of the Day (February 13, 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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Problem of the Day (February 12, 2024)
Published at - 12 February 2024
Problem of the Day (February 12, 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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Registration for math contests is open now.
Published at - 31 October 2023
Registration for math contests is open now.

Registration for math contests is open now.

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Registration for math contests is open now.
Published at - 2 October 2023
Registration for math contests is open now.

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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Problem of the Day (July 6, 2023)
Published at - 6 July 2023
Problem of the Day (July 6, 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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Problem of the Day (February 28, 2023)
Published at - 28 February 2023
Problem of the Day (February 28, 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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Problem of the Day (January 30, 2023)
Published at - 30 January 2023
Problem of the Day (January 30, 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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Registration for AMC 8 is open
Published at - 11 January 2023
Registration for AMC 8 is open

The AMC 8 is a 25-question, 40-minute, multiple choice examination in middle school mathematics designed to promote the development of problem-solving skills. The AMC 8 provides an opportunity for middle school students to develop positive attitudes towards analytical thinking and mathematics that can assist in future careers. The contest is run by the Mathematica Association of America.

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Problem of the Day (December 13, 2022)
Published at - 13 December 2022
Problem of the Day (December 13, 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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Problem of the Day (December 12, 20222)
Published at - 12 December 2022
Problem of the Day (December 12, 20222)

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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Problem of the Day (December 9, 2022)
Published at - 9 December 2022
Problem of the Day (December 9, 2022)

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

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Problem of the Day (December 8, 2022)
Published at - 8 December 2022
Problem of the Day (December 8, 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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Problem of the Day (December 6, 2022)
Published at - 6 December 2022
Problem of the Day (December 6, 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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Problem of the Day (December 5, 2022)
Published at - 5 December 2022
Problem of the Day (December 5, 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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Problem of the Day (December 2, 2022)
Published at - 2 December 2022
Problem of the Day (December 2, 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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Problem of the Day (December 1, 2022)
Published at - 1 December 2022
Problem of the Day (December 1, 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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Problem of the Day (November 30, 2022)
Published at - 29 November 2022
Problem of the Day (November 30, 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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Problem of the Day (November 25, 2022)
Published at - 25 November 2022
Problem of the Day (November 25, 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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Problem of the Day (November 24, 2022)
Published at - 24 November 2022
Problem of the Day (November 24, 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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Problem of the Day (November 23, 2022)
Published at - 23 November 2022
Problem of the Day (November 23, 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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Registration for AMC 8 is open
Published at - 8 November 2022
Registration for AMC 8 is open

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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Problem of the Day (September 16, 2022)
Published at - 16 September 2022
Problem of the Day (September 16, 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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University Math (Calculus I)
Published at - 13 September 2022
University Math (Calculus I)

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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Problem of the Day (September 13, 2022)
Published at - 13 September 2022
Problem of the Day (September 13, 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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Problem of the Day (September 12, 2022)
Published at - 12 September 2022
Problem of the Day (September 12, 2022)

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

Problem of the Day (September 8, 2022)
Published at - 9 September 2022
Problem of the Day (September 8, 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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Problem of the Day (September 7, 2022)
Published at - 8 September 2022
Problem of the Day (September 7, 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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Problem of the Day (September 6, 2022)
Published at - 6 September 2022
Problem of the Day (September 6, 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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University Math – Calculus I (August 17, 2022)
Published at - 17 August 2022
University Math – Calculus I (August 17, 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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Problem of the Day (August 15, 2022)
Published at - 16 August 2022
Problem of the Day (August 15, 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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Why Pi School? (August 16, 2022)
Published at - 16 August 2022
Why Pi School? (August 16, 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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Problem of the Day (August 9, 2022)
Published at - 11 August 2022
Problem of the Day (August 9, 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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Problem of the Day (August 8, 2022)
Published at - 10 August 2022
Problem of the Day (August 8, 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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Problem of the Day (August 7, 2022)
Published at - 9 August 2022
Problem of the Day (August 7, 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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Problem of the Day (August 6, 2022)
Published at - 9 August 2022
Problem of the Day (August 6, 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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The Problem of the Day (August 5, 2022)
Published at - 9 August 2022
The Problem of the Day (August 5, 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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Problem of the Day (August 4, 2022)
Published at - 6 August 2022
Problem of the Day (August 4, 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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Problem of the Day (August 3, 2022)
Published at - 4 August 2022
Problem of the Day (August 3, 2022)

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

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Problem of the Day (August 2, 2022)
Published at - 4 August 2022
Problem of the Day (August 2, 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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Problem of the Day (July 31, 2022)
Published at - 2 August 2022
Problem of the Day (July 31, 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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Problem of the Day (July 30, 2022)
Published at - 2 August 2022
Problem of the Day (July 30, 2022)

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

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Problem of the Day (July 29, 2022)
Published at - 2 August 2022
Problem of the Day (July 29, 2022)

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

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Problem of the Day (July 28, 2022)
Published at - 2 August 2022
Problem of the Day (July 28, 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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Problem of the Day (July 27, 2022)
Published at - 31 July 2022
Problem of the Day (July 27, 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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Problem of the Day (July 26, 2022)
Published at - 28 July 2022
Problem of the Day (July 26, 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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Problem of the Day (July 25, 2022)
Published at - 28 July 2022
Problem of the Day (July 25, 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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Problem of the Day (July 24, 2022)
Published at - 28 July 2022
Problem of the Day (July 24, 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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Problem of the Day (July 23, 2022)
Published at - 28 July 2022
Problem of the Day (July 23, 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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Problem of the Day (July 22, 2022)
Published at - 27 July 2022
Problem of the Day (July 22, 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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Problem of the Day (July 21, 2022)
Published at - 23 July 2022
Problem of the Day (July 21, 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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Problem of the Day (July 20, 2022)
Published at - 20 July 2022
Problem of the Day (July 20, 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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Problem of the Day (July 19, 2022)
Published at - 20 July 2022
Problem of the Day (July 19, 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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Problem of the Day (July 18, 2022)
Published at - 19 July 2022
Problem of the Day (July 18, 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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Problem of the Day (July 17, 2022)
Published at - 19 July 2022
Problem of the Day (July 17, 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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Problem of the Day (July 16, 2022)
Published at - 17 July 2022
Problem of the Day (July 16, 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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Problem of the Day (July 15, 2022)
Published at - 17 July 2022
Problem of the Day (July 15, 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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Problem of the Day (July 14, 2022)
Published at - 14 July 2022
Problem of the Day (July 14, 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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Problem of the Day (July 13, 2022)
Published at - 13 July 2022
Problem of the Day (July 13, 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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Problem of the Day (July 12, 2022)
Published at - 12 July 2022
Problem of the Day (July 12, 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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Problem of the Day (July 11, 2022)
Published at - 11 July 2022
Problem of the Day (July 11, 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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Problem of the Day (July 10, 2022)
Published at - 8 July 2022
Problem of the Day (July 10, 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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Problem of the Day (July 9, 2022)
Published at - 8 July 2022
Problem of the Day (July 9, 2022)

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

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Toronto Tutoring
Published at - 8 July 2022
Toronto Tutoring

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

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Problem of the Day (July 9, 2022)
Published at - 8 July 2022
Problem of the Day (July 9, 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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Problem of the Day (July 8, 2022)
Published at - 7 July 2022
Problem of the Day (July 8, 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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Problem of the Day (July 7, 2022)
Published at - 6 July 2022
Problem of the Day (July 7, 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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Problem of the Day (July 6, 2022)
Published at - 5 July 2022
Problem of the Day (July 6, 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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Problem of the Day (July 5, 2022)
Published at - 5 July 2022
Problem of the Day (July 5, 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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Problem of the Day (July 4, 2022)
Published at - 5 July 2022
Problem of the Day (July 4, 2022)

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

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Problem of the Day (July 3, 2022)
Published at - 5 July 2022
Problem of the Day (July 3, 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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Problem of the Day (July 2, 2022)
Published at - 2 July 2022
Problem of the Day (July 2, 2022)

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

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Problem of the Day (July 1, 2022)
Published at - 2 July 2022
Problem of the Day (July 1, 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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Problem of the Day (June 30, 2022)
Published at - 30 June 2022
Problem of the Day (June 30, 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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Problem of the Day (June 29, 2022)
Published at - 30 June 2022
Problem of the Day (June 29, 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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Problem of the Day (June 28, 2022)
Published at - 28 June 2022
Problem of the Day (June 28, 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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Problem of the Day (June 27, 2022)
Published at - 27 June 2022
Problem of the Day (June 27, 2022)

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

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Problem of the Day (June 26, 2022)
Published at - 26 June 2022
Problem of the Day (June 26, 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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Problem of the Day (June 25, 2022)
Published at - 26 June 2022
Problem of the Day (June 25, 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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Problem of the Day (June 24, 2022)
Published at - 24 June 2022
Problem of the Day (June 24, 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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Problem of the Day (June 23, 2022)
Published at - 23 June 2022
Problem of the Day (June 23, 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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Problem of the Day (June 22, 2022)
Published at - 22 June 2022
Problem of the Day (June 22, 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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Problem of the Day (June 21, 2022)
Published at - 21 June 2022
Problem of the Day (June 21, 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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Problem of the Day (June 20, 2022)
Published at - 20 June 2022
Problem of the Day (June 20, 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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Problem of the Day (June 19, 2022)
Published at - 19 June 2022
Problem of the Day (June 19, 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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Problem of the Day (June 18, 2022)
Published at - 18 June 2022
Problem of the Day (June 18, 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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Problem of the Day (June 17, 2022)
Published at - 17 June 2022
Problem of the Day (June 17, 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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Problem of the Day (June 16, 2022)
Published at - 16 June 2022
Problem of the Day (June 16, 2022)

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

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Problem of the Day (June 15, 2022)
Published at - 15 June 2022
Problem of the Day (June 15, 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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Problem of the Day (June 14, 2022)
Published at - 14 June 2022
Problem of the Day (June 14, 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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Problem of the Day (June 13, 2022)
Published at - 13 June 2022
Problem of the Day (June 13, 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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Problem of the Day (June 12, 2022)
Published at - 12 June 2022
Problem of the Day (June 12, 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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Problem of the Day (June 11, 2022)
Published at - 10 June 2022
Problem of the Day (June 11, 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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Problem of the Day (June 10, 2022)
Published at - 9 June 2022
Problem of the Day (June 10, 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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Problem of the Day (June 9, 2022)
Published at - 8 June 2022
Problem of the Day (June 9, 2022)

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

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Problem of the Day (June 8, 2022)
Published at - 7 June 2022
Problem of the Day (June 8, 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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Problem of the Day (June 7, 2022)
Published at - 6 June 2022
Problem of the Day (June 7, 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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Problem of the Day (June 6, 2022)
Published at - 5 June 2022
Problem of the Day (June 6, 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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Problem of the Day (June 5, 2022)
Published at - 4 June 2022
Problem of the Day (June 5, 2022)

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

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Problem of the Day (June 4, 2022)
Published at - 3 June 2022
Problem of the Day (June 4, 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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Problem of the Day (June 3, 2022)
Published at - 2 June 2022
Problem of the Day (June 3, 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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Problem of the Day (June 2, 2022)
Published at - 1 June 2022
Problem of the Day (June 2, 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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Problem of the Day (June 1, 2022)
Published at - 1 June 2022
Problem of the Day (June 1, 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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Registration for summer Camp at Pi School will open soon.
Published at - 18 May 2022
Registration for summer Camp at Pi School will open soon.

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

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Toronto Tutoring
Published at - 14 May 2022
Toronto Tutoring

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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