art of problem solving alcumus

Fun teaching resources & tips to help you teach math with confidence

Alcumus Online Learning {A Review}

I recently stumbled upon an online resource that looked useful, especially since it is completely free. It’s from the people at Art of Problem Solving and this particular resource is called Alcumus . I wasn’t sure what to expect, especially since I haven’t used any other AoPS materials, and I do not have the curriculum that it is meant to complement. But after playing with it and experimenting, I am happy to say that it seems to be an incredibly helpful and useful tool.

Please note : This is NOT a teaching tool or online class to learn the content. This interactive tool is meant to compliment their textbooks and courses. If you simply need extra practice, however, this could be very valuable.

One potential downside of this program may be that it is different from your math curriculum, so it may not cover everything you need, or it may go in a different order, etc. But from what I have seen, (and considering most upper level math curriculum cover essentially the same topics) I do not believe this would cause many problems. It would still be immensely helpful if you need extra practice and immediate feedback.

In my experience, and especially considering this is free (you just have to register), there are tons of benefits. Once registered, students can choose a focus “class” (anything from Pre-Algebra to Number Theory and Probability) and then within that focus, you can choose specific types of problems to work on. There are several things that I like about the program after trying out the problems. For one, when you get an answer right, it doesn’t just say, “Correct! Great job!” and move on. There is an explanation of why and how, and it often shows more than one way to go about it (which would be helpful for students who may have solved it the hard way, or got it right by simply guessing). I also appreciate that when an answer is wrong, students are given another chance AND their first answer is shown so that they don’t make the same mistake again.

Once students are showing proficiency in a topic, the problems begin to increase in difficulty. It seems to be very responsive to students’ abilities to make sure they can be successful, and then move on when they’re ready.

There are also “quests” or challenges presented to students to make it more fun and engaging, such as “get 7 new problems correct in a row,” as well as a “hall of fame” where recent successful students are showcased.

And for the parent or teacher, there is a reports page that not only shows which topics they have worked on and passed, but exactly what problems they have attempted and whether or not the student got them right.

Overall, I was very impressed with the program, and best of all, it is completely free ! Have you ever used this or had success with it? Let me know if you find it helpful!

~Math Geek Mama

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Calculus The Art of Problem Solving.pdf

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A comprehensive textbook covering single-variable calculus. Specific topics covered include limits, continuity, derivatives, integrals, power series, plane curves, and differential equations.

Calculus David Patrick Paperback (2nd edition) Text: 336 pages. A comprehensive textbook covering single-variable calculus. Specific topics covered include limits, continuity, derivatives, integrals, power series, plane curves, and differential equations.

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Alcumus Achievements

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align="center" |N/A |Fully Charged |Completely fill the XP Bonus Meter. |25 XP |Yes |- align="center" |N/A |Indecisive |Change your focus topic for five consecutive problems. |25 XP |Yes |- align="center" |N/A |James Bond | answer a problem correctly at 0:07 am PT| |Yes |- align="center" |N/A |Life, The Universe, and Everything |Input "42" as the answer to a problem. |42 XP |Yes |- align="center" |N/A |Perseverance |Correctly answer 10 problems on the second attempt without any incorrect second attempts in between. |50 XP |Yes |- align="center" |N/A | Richard |Input “richard” as the answer to a problem. |50 XP |Yes |- align="center" |N/A |Thanks For The Help! | | |Yes |- align="center" |N/A |Them Internets |Correctly answer a question with an answer equal to the last number of the player's IP address. |5 XP |Yes |- align="center" |N/A | Trisakaidekaphobia |On Friday, input "13" as the answer to a problem. |13 XP |Yes |- align="center" |N/A | Amnesia |Get a problem wrong that was previously solved correctly | |Yes |- align="center" |N/A |Aren't You A Bit Too Old For This |Master all but one prealgebra subject and the master it. |Yes |- align="center" |N/A |Caution: Hard Hat Area |Visit Alcumus while it was shut down for maintenance | Chicken? | Give up 100 problems |Yes |- align="center" |N/A | Double Trouble |Try the same wrong answer twice. |2 XP |Yes |- align="center" |N/A | Just Guessing |Get a problem wrong by answering 0 as your second guess | |Yes |- align="center" |N/A | Pick Your Poison |Miss your first problem after changing focus topics. |-1 XP |Yes |- align="center" |N/A | Rage Quit |Leave Alcumus for at least 12 hours after getting a problem wrong | -1XP |Yes |- align="center" |N/A | Superstitious |Input 7 to a problem that has the correct answer of 13 | |Yes |}

2 Swapnil Garg
3 Kevin Liu

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Problems Collection

This is a page where you can share the problems you made (try not to use past exams).

1 AMC styled
2 AIME styled
4.1 Problem 1
4.2 Solution 1
4.3 Problem 2
4.4 Solution 1 (Slow, probably official MAA)
4.5 Solution 2 (Fast)
4.6 Solution 3 (Faster)
4.7 Problem 3
4.8 Solution 1(Probably official MAA, lots of proofs)
4.9 Solution 2 (Fast, risky, no proofs)
4.10 Problem 4
4.11 Solution 1
4.12 Problem 5
4.13 Solution 1 (Euler's Totient Theorem)
4.14 Problem 6
4.15 Solution 1 (Recursion)
4.16 Problem 7
4.17 Solution 1 (Tedious Casework)
4.18 Solution 2 (Official)
4.19 Solution 3 (Official and Fastest)
4.20 Problem 8
4.21 Solution 1
4.22 Problem 9
4.23 Solution 1
4.24 Problem 10
4.25 Solution 1(Wordless endless bash)
4.26 Problem 11
4.27 Solution 1 (Analytic geo)
4.28.1 Solution 2a (Hard)
4.28.2 Solution 2b (Harder)

AIME styled

1. There is one and only one perfect square in the form

$(p^2+1)(q^2+1)-((pq)^2-pq+1)$

3.The fraction,

$\frac{ab+bc+ac}{(a+b+c)^2}$

Someone mind making a diagram for this?

$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n \cdot m^2+m \cdot n^2+2mn}+\lim_{x\rightarrow \infty} [\frac{x}{2}+x^2 [\frac{(1+\frac{1}{x})^{x}}{e}-1]]=\frac{p}{q}$

There is one and only one perfect square in the form

$(p^2+1)(q^2+1)-((pq)^2-pq+1)=p^2 \cdot q^2 +p^2+q^2+1-p^2 \cdot q^2 +pq-1=p^2+q^2+pq$

Solution 1 (Slow, probably official MAA)

$m^2=2^8+2^{11}+2^n$

Solution 2 (Fast)

$(a + b)^2$

Solution 3 (Faster)

$256 + 2048 + 2^n = 2304 + 2^n = m^2$

~ (also) cxsmi

The fraction,

Solution 1(Probably official MAA, lots of proofs)

$\text{max} (\frac{ab+bc+ac}{(a+b+c)^2})=\frac{1}{3}$

Proof: By the Triangle Inequality , we have

$a+b>c$

Add them together gives

$a^2+b^2+c^2<c(a+b)+a(b+c)+b(a+c)=2(ab+bc+ac)$

Solution 2 (Fast, risky, no proofs)

$a,b,c$

To make things even simpler, let

$a=\sqrt[3]{13}+\sqrt[3]{53}+\sqrt[3]{103}, b=\sqrt[3]{13 \cdot 53}+\sqrt[3]{13 \cdot 103}+\sqrt[3]{53 \cdot 103}, c=\sqrt[3]{13 \cdot 53 \cdot 103}+\frac{1}{3}$

Note: If you don't know Newton's Sums , you can also use Vieta's Formulas to bash.~ Ddk001

Solution 1 (Euler's Totient Theorem)

$2^4 \cdot 3^4 \cdot 7^4+2^7 \cdot 3^7 \cdot 5^6:$

where the last step of all 3 congruences hold by the Euler's Totient Theorem . Hence,

$x \equiv 1 \pmod{5}$

(ii) No bigger rings are on top of smaller rings.

Solution 1 (Recursion)

$M_n$

Solution 1 (Tedious Casework)

$a>b$

In this case, we have

$\overline{ab}^2=a! +b!=(1+a \cdot (a-1) \cdot \dots \cdot (b+1)) \cdot b! \implies b!|\overline{ab}^2=(10a+b)^2$

In this case, we have that

$a! \equiv \overline{ab}^2-b! \equiv (10a+1)^2-1 \equiv 0 \pmod{10} \implies 10|a! \implies a \ge 5$

There is no apparent contradiction here, so we leave this as it is.

$a>b=2$

To simplify future calculations, note that

$a!=\overline{ab}^2-b!=(10a+1)^2-1=100a^2+20a=10a(10a+2)$

For this case, we must have

$(11a)^2=\overline{ab}^2=a!+b!=2a! \implies 11|a!$

Solution 2 (Official)

$100(10^{2})$

Hence unit digit of RHS is 0,1,2, 6 or 4. 0,2,4 and 6 are rejected as follows:-

1 . 2 can't be the unit digit of a perfect square.

3 . If 0 is the unit digit of LHS then 50 60 70 are the only cases (as one of the digits is greater than or equal to 5) that don't satisfy

$\boxed{008}$

Solution 3 (Official and Fastest)

$(mod 4)$

which is very easy to calculate and get 71 as the only possible solution to the problem and

$f(n)-n=0$

Now, notice that

$m!=(2+r_1)(2+r_2) \dots (2+r_{10000000010})$

Similarly, we have

$(1+r_1)(1+r_2) \dots (1+r_{10000000010})=\frac{f(-1)}{a}=-\frac{1}{a}$

Now we state a few claims :

$\Delta O’IO$

where the last equality holds by the Power of a Point Theorem .

$IJ= \frac{\sqrt{3}}{2} (IK+O’L)^2$

With this in mind, we see that

$2OJ=OO’=OI=OK+KI=OJ+GI=OJ+AC \implies OA=OJ=AC$

Here, we state another claim :

$BH$

Now, apply Ptolemy’s Theorem gives

$BH \cdot AC+BC \cdot AH=CH \cdot AB \implies BH \cdot AC+AC^2=3AC^2 \implies BH=2AC=2OA$

Solution 1(Wordless endless bash)

$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n \cdot m^2+m \cdot n^2+2mn}$

Solution 1 (Analytic geo)

$A=(0,0)$

Now, we see that

$\text{Slope} _ {AF}=\frac{b}{2a+2 \sqrt{a^2+b^2}}$

Solution 2 (Hard vector bash)

Solution 2a (hard).

$\overrightarrow{AF} \cdot \overrightarrow{BE}$

Solution 2b (Harder)

$\angle ACD=\angle ECD$

Now come the coordinates. Let

$B=(-a,-b)$

Here's the source for the problems:

1,2,3,4,5,6,8,9,10,11: Ddk001 , credits given to Ddk001

7: SANSKAR'S OG PROBLEMS , credits given to SANSGANKRSNGUPTA

Note: Problem 6 is based on the Tower of Hanoi Problem

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Art of Problem Solving Calculus Set
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Art of Problem Solving
Art of Problem Solving offers a wide variety of free resources for avid problem solvers, including hundreds of videos and interactive tools like Alcumus, our popular adaptive learning system. Alcumus It offers students a customized learning experience, adjusting to their performance to deliver problems that will challenge them appropriately.
What Is Alcumus, And Why We Called It That
Alcumus, sometimes also spelled 'Alkimos' or 'Alcimus,' was the father of Mentor, the character in Homer's Odyssey who looks out for Odysseus's son and guides him as he grows up. (Yup, Mentor in the Odyssey is where the word "mentor" comes from.) With this fancy pedigree, Alcumus the Learning Tool was born.
Alcumus
math training & tools Alcumus Videos For the Win! MATHCOUNTS Trainer AoPS Practice Contests AoPS Wiki LaTeX TeXeR MIT PRIMES/CrowdMath Keep Learning All Ten contests on aops Practice Math Contests USABO
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Art of Problem Solving textbooks have been used by outstanding students since 1993. The AoPS website launched in 2003, and its online community now has over one million users. ... Evan discovered Art of Problem Solving in middle school, after hearing about Alcumus. In 2014, he earned the second-highest score in the USA Mathematical Olympiad ...
Alcumus
Alcumus is a game that is designed to help students focus on a variety of subjects, prealgebra, algebra, number theory, counting and probability, geometry, and precalculus. Alcumus is free; however, you must have an AoPS account to play. For instructions on how to get one, visit the page Creating An Account. Users are able to earn achievements for their accounts. They are also able to receive ...
Alcumus Product Review
Alcumus Online Learning {A Review} By Bethany March 16, 2015. I recently stumbled upon an online resource that looked useful, especially since it is completely free. It's from the people at Art of Problem Solving and this particular resource is called Alcumus. I wasn't sure what to expect, especially since I haven't used any other AoPS ...
Art of Problem Solving
Calculus. The discovery of the branch of mathematics known as calculus was motivated by two classical problems: how to find the slope of the tangent line to a curve at a point and how to find the area bounded by a curve. What is surprising is that these two problems are fundamentally connected and, together with the notion of limits, can be ...
Art of Problem Solving
The Art of Problem Solving competition preparation books cover a variety of topics of interest to students of mathematics interested in competitive math. Art of Problem Solving Volume 1: the Basics; Art of Problem Solving Volume 2: and Beyond; Competition Math for Middle School; Other math books. AoPS sells numerous other math books at the AoPS ...
Art of Problem Solving
Art of Problem Solving is a website that provides rich resources and tools for students who love math and want to challenge themselves. You can find online courses, textbooks, videos, contests, and more on various topics, from prealgebra to calculus. You can also use Alcumus , a free adaptive learning system that gives you personalized problems and feedback.
AoPS Academy Virtual Campus
Advanced Online Math and Language Arts Courses for Grades 2-12. Starting at $50/week. Enroll Today. As seen in. Since 1993, Art of Problem Solving has helped train the next generation of intellectual leaders. Hundreds of thousands of our students have gone on to attend prestigious universities, win global math competitions, and achieve ...
PDF Calculus The Art of Problem Solving.pdf
The Art of Problem Solving Collection opensource Language English. Calculus. David Patrick. Paperback (2nd edition) Text: 336 pages. Solutions: 128 pages. A comprehensive textbook covering single-variable calculus. Specific topics covered include limits, continuity, derivatives, integrals, power series, plane curves, and differential equations.
AoPS Academy
Art of Problem Solving has been a leader in math education for high-performing students since 1993. We launched AoPS Academy in 2016 to bring our rigorous curriculum and expert instructors into classrooms around the United States. With campuses in 8 states (and growing!), our approach nurtures a love for complex problem solving, which is fully ...
Calculus
AoPS Academy parent to 16-year-old engineering major at UW and future cancer researcher. AoPS Academy is an ACS WASC Accredited School. Our calculus class goes well beyond mechanics and calculators, and provides students a rigorous understanding of the fundamental operations of calculus, delivering a first-year calculus course like those ...
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Art of Problem Solving currently operates brick-and-mortar campuses in seven US states. Art of Problem Solving AoPS Online. Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online . Books for Grades 5-12 Online Courses Beast Academy. Engaging math books and online learning for students ages 6-13. ...
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Here are a list of all of the alcumus achievements.Note: The list is incomplete. Please add onto it by adding correct information onto it. ... More Art Of Problem Solving Wikia. 1 Scott Wu; 2 Swapnil Garg; 3 2014 MATHCOUNTS National Competition; Explore properties. Fandom Muthead Fanatical Follow Us. Overview. What is Fandom? ...
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Art of Problem Solving AoPS Online. Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books ... math training & tools Alcumus Videos For the Win! MATHCOUNTS Trainer AoPS Practice Contests AoPS Wiki LaTeX TeXeR MIT PRIMES/CrowdMath Keep Learning All Ten.
Art of Problem Solving
Small live classes for advanced math and language arts learners in grades 2-12.
Art of Problem Solving
Welcome to the AoPS Wiki! The AoPS Wiki project is administered by the Art of Problem Solving for supporting educational content useful to avid math students. During AMC 10/12 testing week, the AoPS Wiki is in read-only mode. No edits can be made.
Art of Problem Solving
1. There is one and only one perfect square in the form. where and are prime. Find that perfect square. 2. and are positive integers. If , find . 3.The fraction, where and are side lengths of a triangle, lies in the interval , where and are rational numbers.