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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 2, 2025
0 replies
AMC- IMO preparation
asyaela.   10
N 8 minutes ago by Squidget
I'm a ninth grader, and I recently attempted the AMC 12, getting 18 questions correct and leaving 7 empty. I started working on Olympiad math in November and currently dedicate about two hours per day to preparation. I'm feeling a bit demotivated, but if it's possible for me to reach IMO level, I'd be willing to put in more time. How realistic is it for me to get there, and how much study would it typically take?
10 replies
asyaela.
5 hours ago
Squidget
8 minutes ago
ABMC 2025 IN-PERSON Contest (April 5th)
ilovepizza2020   4
N 15 minutes ago by mynameisjefff
The 9th annual Acton-Boxborough Math Competition (ABMC) is quickly approaching! This year's ABMC will be held in-person at RJ Grey Junior High School, Acton, MA, on April 5th, 2025. The competition includes individual rounds and a team round, in which teams of 2-4 students participate. Anyone in grade 8 or below is welcome! You must register to compete. For more information about registration and the tentative schedule, please consult our website: https://abmathcompetitions.org/2025-contest/.

We offer prizes not only to top competitors; several of our sponsor prizes and educational awards are raffled among all in-person participants. Additionally, there are separate prizes for the top-scoring elementary schoolers.


For more information, visit https://abmathcompetitions.org/, especially the 2025 Competition page.
For the mailing list, visit https://abmathcompetitions.org/contact/.

Best,
ABMC Coordinators
4 replies
+2 w
ilovepizza2020
42 minutes ago
mynameisjefff
15 minutes ago
is this really supposed to be #13???
hgmium   3
N 16 minutes ago by hashbrown2009
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10A_Problems/Problem_13

I managed to do all the other geo problems for that year besides this one
misplaced?
3 replies
+2 w
hgmium
an hour ago
hashbrown2009
16 minutes ago
2025 ROSS Program
scls140511   13
N an hour ago by Bnn81351
Since the application has ended, are we now free to discuss the problems and stats? How do you think this year's problems are?
13 replies
scls140511
Yesterday at 2:36 AM
Bnn81351
an hour ago
No more topics!
Mock AMC J
Silverfalcon   39
N Jan 31, 2005 by Silverfalcon
This is a self-taking Mock AMC. I just made this so that many people in this site and I myself can have some extra mock AMC. I don't have an answer key so if you have a right answer, post here in the spoiler.

Many questions are came from old AHSMEs or even Mock AMC's. And other contests.

1. A lattice point is a point on a coordinate plane with integer values for x and y. How many lattice points lie on a circle centered at the origin with radius 25?
A) 40
B) 20
C) 10
D) 5
E) None of above

2. Find the unit digits of the sum:

$\sum^{2005}_{k=0} k^5$
A) 1
B) 3
C) 5
D) 7
E) 9

3. How many three element subsets of the set:

{88,95,99,132,166,173}

have the property that the sum of the three elements is even?
A) 6
B) 8
C) 10
D) 12
E) 24

4. If x and y are positive integesr such that $3x+4y=100$, then $x+y=$?
A) 10
B) 12
C) 14
D) 16
E) 18

5. Let a,a',b,b' be real numbers with $a,a' \neq 0$. The solution to $ax+b=0$ is less than the solution to $a'x+b' = 0$ if and only if:
A) a'b<ab'
B) ab'<a'b
C) ab<a'b'
D) b/a<b'/a'
E) b'/a'<b/a

6. In an h-meter race, Sunny is exactly d meters ahead of Windy when Sunny sportingly starts d meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finished the second race?
A) d/h
B) 0
C) $d^2/h$
D) $h^2/d$
E) $d^2/(h-d)$

7. Let $f$ be a function which $f(x/3) = x^2+x+1$. Find the sum of all values of $z$ for which $f(3z) = 7$.
A) -1/3
B) -1/9
C) 0
D) 5/9
E) 5/3

8. In an arithmetic sequence of complex numbers the first term is $3+4i$ and the second term is 7. Find the sum of the first 50 terms.
A) $5050-4700i$
B) $5000-7100i$
C) $5000+7100i$
D) $5050+7100i$
E) None of these

9. Two nonadjacent vertices of a rectangle are (4,3) and (-4,-3) and the coordinates of the other two vertices are integers. The number of such rectangles is:
A) 1
B) 2
C) 3
D) 4
E) 5

10. Inscribed in a circle is a quadrilateral having sides of lengths 25,39,52,and 60 taken consecutively. The diameter of the circle has length of:
A) 62
B) 63
C) 65
D) 66
E) 69

11. A wooden cube with edge length n units (where n is an integer > 2) is painted black all over. By slice parallel to its faces, the cube is cut into $n^3$ smaller cubes each of unit dege length. If the number of smaller of cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is n?
A) 5
B) 6
C) 7
D) 8
E) None of these

12. An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:

(a) the selection of four red marbles
(b) the selection of one white and three red marbles;
(c) the selection of one white, one blue, and two red marbles; and
(d) the selection of one marble of each color

What is the smallest number of marbles satisfying the given condition?
A) 19
B) 21
C) 46
D) 69
E) More than 69

13. In their base 10 representation, the integer $a$ consists of a sequence of 1985 eights and the integer $b$ contains a sequence of 1985 fives. What is the sum of digits of the base 10 representation of the integer $9ab$?
A) 15880
B) 17856
C) 17865
D) 17874
E) 19851

14. Let $D_n$ denote the number of diagonals plus the number of sides in a convex polygon with $n$ sides. Then:

$\frac {1}{D_4} + \frac {1}{D_5} + \frac {1}{D_6}+...\frac {1}{D_2004} = \frac {m}{n}$ where m and n are relatively prime integers. Find $m-n$.
A) -334
B) -330
C) -332
D) -328
E) None of the above

15. In a triangle ABC, angle C is a right angle and CB>CA. Point D is located on segment BC so that angle CAD is twice angle DAB. If AC/AD = 2/3, then CD/BD = m/n, where m and n are relatively prime integers. Find m+n.
A) 10
B) 14
C) 18
D) 22
E) 26

16. Evaluate:

$\frac {\sin 10 + \sin 20}{\cos 10 + \cos 20}$ where numbers are given in degrees.
A) $\tan 10 + \tan 20$
B) $\tan 30$
C) $\frac {1}{2} (\tan 10 + \tan 20)$
D) $\tan 15$
E) $\frac {1}{4} \tan 60$

17. Find the units digit in the decimal expansion of:

$(15 + \sqrt {220})^{19} + (15 + \sqrt {220})^{82}$.
A) 0
B) 2
C) 5
D) 9
E) None of the above

18. If John has 12 quarters to distribute among Akiba, Bobby, Crl, David, and Emanuel, in how many ways can he distribute the money given that Akia gets at least 50 cents?

19. If $x = \frac {-1 + i \sqrt 3}{2}$ and $y = \frac {-1 - i \sqrt 3}{2}$ where $i^2 = -1$, then which of the following is not correct?

A) $x^5+y^5 = -1$
B) $x^7+y^7 = -1$
C) $x^9+y^9 = -1$
D) $x^{11} + y^{11} = -1$
E) $x^{13} + y^{13} = -1$

20. A and C lie on a circle with radius $\sqrt {50}$. The point B is inside the circle such that <ABC = 90, AB = 6, BC = 2. Find OB.

A) 5
B) :sqrt: 26
C) :sqrt: 31
D) 11/2
E) 6

21. In triangle ABC, BC = 8 and AC = 10. If D is the midpoint of BC, and E is on AC such that EC = 3. Let EB and AD intersect at F. If DE:AF = m/n, where m and n are relatively prime integers, find m+n.
A) 28
B) 57
C) 100
D) 239
E) 87

22. How many elements in the 2002nd row of Pascal's Triangle (the one that begins with 1,2001,....) leaves remainder of 1 when divided by 3?
A) 23
B) 26
C) 28
D) 29
E) 667

23. TWo of the altitudes of the scalene triangle ABC have length 4 and 12. If the length of the third altitude is also integer, what is the biggest it can be?
A) 4
B) 5
C) 6
D) 7
E) None

24. Eight congruent equilateral triangles, each of different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like other)
A) 210
B) 560
C) 840
D) 1260
E) 1680

25. If the base 8 representation of a perfect square is ab3c, where a is not 0, then c is:

A) 0
B) 1
C) 3
D) 4
E) Not uniquely determined

Please note that this is problem thread AND solution thread. So, post your solution so people can see about it. Also, even you're not sure about answer, please post here.

Thanks and Enjoy the problems!

P.S. I have to try this too! :lol:
39 replies
Silverfalcon
Jan 23, 2005
Silverfalcon
Jan 31, 2005
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Silverfalcon
5006 posts
#1 • 2 Y
Y by Adventure10, Mango247
This is a self-taking Mock AMC. I just made this so that many people in this site and I myself can have some extra mock AMC. I don't have an answer key so if you have a right answer, post here in the spoiler.

Many questions are came from old AHSMEs or even Mock AMC's. And other contests.

1. A lattice point is a point on a coordinate plane with integer values for x and y. How many lattice points lie on a circle centered at the origin with radius 25?
A) 40
B) 20
C) 10
D) 5
E) None of above

2. Find the unit digits of the sum:

$\sum^{2005}_{k=0} k^5$
A) 1
B) 3
C) 5
D) 7
E) 9

3. How many three element subsets of the set:

{88,95,99,132,166,173}

have the property that the sum of the three elements is even?
A) 6
B) 8
C) 10
D) 12
E) 24

4. If x and y are positive integesr such that $3x+4y=100$, then $x+y=$?
A) 10
B) 12
C) 14
D) 16
E) 18

5. Let a,a',b,b' be real numbers with $a,a' \neq 0$. The solution to $ax+b=0$ is less than the solution to $a'x+b' = 0$ if and only if:
A) a'b<ab'
B) ab'<a'b
C) ab<a'b'
D) b/a<b'/a'
E) b'/a'<b/a

6. In an h-meter race, Sunny is exactly d meters ahead of Windy when Sunny sportingly starts d meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finished the second race?
A) d/h
B) 0
C) $d^2/h$
D) $h^2/d$
E) $d^2/(h-d)$

7. Let $f$ be a function which $f(x/3) = x^2+x+1$. Find the sum of all values of $z$ for which $f(3z) = 7$.
A) -1/3
B) -1/9
C) 0
D) 5/9
E) 5/3

8. In an arithmetic sequence of complex numbers the first term is $3+4i$ and the second term is 7. Find the sum of the first 50 terms.
A) $5050-4700i$
B) $5000-7100i$
C) $5000+7100i$
D) $5050+7100i$
E) None of these

9. Two nonadjacent vertices of a rectangle are (4,3) and (-4,-3) and the coordinates of the other two vertices are integers. The number of such rectangles is:
A) 1
B) 2
C) 3
D) 4
E) 5

10. Inscribed in a circle is a quadrilateral having sides of lengths 25,39,52,and 60 taken consecutively. The diameter of the circle has length of:
A) 62
B) 63
C) 65
D) 66
E) 69

11. A wooden cube with edge length n units (where n is an integer > 2) is painted black all over. By slice parallel to its faces, the cube is cut into $n^3$ smaller cubes each of unit dege length. If the number of smaller of cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is n?
A) 5
B) 6
C) 7
D) 8
E) None of these

12. An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:

(a) the selection of four red marbles
(b) the selection of one white and three red marbles;
(c) the selection of one white, one blue, and two red marbles; and
(d) the selection of one marble of each color

What is the smallest number of marbles satisfying the given condition?
A) 19
B) 21
C) 46
D) 69
E) More than 69

13. In their base 10 representation, the integer $a$ consists of a sequence of 1985 eights and the integer $b$ contains a sequence of 1985 fives. What is the sum of digits of the base 10 representation of the integer $9ab$?
A) 15880
B) 17856
C) 17865
D) 17874
E) 19851

14. Let $D_n$ denote the number of diagonals plus the number of sides in a convex polygon with $n$ sides. Then:

$\frac {1}{D_4} + \frac {1}{D_5} + \frac {1}{D_6}+...\frac {1}{D_2004} = \frac {m}{n}$ where m and n are relatively prime integers. Find $m-n$.
A) -334
B) -330
C) -332
D) -328
E) None of the above

15. In a triangle ABC, angle C is a right angle and CB>CA. Point D is located on segment BC so that angle CAD is twice angle DAB. If AC/AD = 2/3, then CD/BD = m/n, where m and n are relatively prime integers. Find m+n.
A) 10
B) 14
C) 18
D) 22
E) 26

16. Evaluate:

$\frac {\sin 10 + \sin 20}{\cos 10 + \cos 20}$ where numbers are given in degrees.
A) $\tan 10 + \tan 20$
B) $\tan 30$
C) $\frac {1}{2} (\tan 10 + \tan 20)$
D) $\tan 15$
E) $\frac {1}{4} \tan 60$

17. Find the units digit in the decimal expansion of:

$(15 + \sqrt {220})^{19} + (15 + \sqrt {220})^{82}$.
A) 0
B) 2
C) 5
D) 9
E) None of the above

18. If John has 12 quarters to distribute among Akiba, Bobby, Crl, David, and Emanuel, in how many ways can he distribute the money given that Akia gets at least 50 cents?

19. If $x = \frac {-1 + i \sqrt 3}{2}$ and $y = \frac {-1 - i \sqrt 3}{2}$ where $i^2 = -1$, then which of the following is not correct?

A) $x^5+y^5 = -1$
B) $x^7+y^7 = -1$
C) $x^9+y^9 = -1$
D) $x^{11} + y^{11} = -1$
E) $x^{13} + y^{13} = -1$

20. A and C lie on a circle with radius $\sqrt {50}$. The point B is inside the circle such that <ABC = 90, AB = 6, BC = 2. Find OB.

A) 5
B) :sqrt: 26
C) :sqrt: 31
D) 11/2
E) 6

21. In triangle ABC, BC = 8 and AC = 10. If D is the midpoint of BC, and E is on AC such that EC = 3. Let EB and AD intersect at F. If DE:AF = m/n, where m and n are relatively prime integers, find m+n.
A) 28
B) 57
C) 100
D) 239
E) 87

22. How many elements in the 2002nd row of Pascal's Triangle (the one that begins with 1,2001,....) leaves remainder of 1 when divided by 3?
A) 23
B) 26
C) 28
D) 29
E) 667

23. TWo of the altitudes of the scalene triangle ABC have length 4 and 12. If the length of the third altitude is also integer, what is the biggest it can be?
A) 4
B) 5
C) 6
D) 7
E) None

24. Eight congruent equilateral triangles, each of different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like other)
A) 210
B) 560
C) 840
D) 1260
E) 1680

25. If the base 8 representation of a perfect square is ab3c, where a is not 0, then c is:

A) 0
B) 1
C) 3
D) 4
E) Not uniquely determined

Please note that this is problem thread AND solution thread. So, post your solution so people can see about it. Also, even you're not sure about answer, please post here.

Thanks and Enjoy the problems!

P.S. I have to try this too! :lol:
Z K Y
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Silverfalcon
5006 posts
#2 • 2 Y
Y by Adventure10, Mango247
Number one: :D

My_answer
Z K Y
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Silverfalcon
5006 posts
#3 • 2 Y
Y by Adventure10, Mango247
Number 10

I'm not so sure about this so I'll give it a shot.

Um

Is there any better way or if I'm wrong, correct way?
Z K Y
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Silverfalcon
5006 posts
#4 • 2 Y
Y by Adventure10, Mango247
Number 7

My_ANSWER

I thought there was different answer on Harold Reiter's Page though.. ;)
Z K Y
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Silverfalcon
5006 posts
#5 • 2 Y
Y by Adventure10, Mango247
Click to reveal hidden text
Z K Y
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white_horse_king88
1779 posts
#6 • 2 Y
Y by Adventure10, Mango247
Silverfalcon wrote:
Click to reveal hidden text

Basically, if you line it up, you have the question: How many ways can I arrange 4 breakers/dividers, and 10 objects which I am dividing. That's just C(14,10).
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Silverfalcon
5006 posts
#7 • 2 Y
Y by Adventure10, Mango247
Hmm..

This is kinda confusing.. :?
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beta
3001 posts
#8 • 2 Y
Y by Adventure10, Mango247
#15)

Click to reveal hidden text
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eryaman
1130 posts
#9 • 2 Y
Y by Adventure10, Mango247
I don't understand how there can be one single answer to number 4. Aren't there many different possibilities?
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beta
3001 posts
#10 • 2 Y
Y by Adventure10, Mango247
eryaman wrote:
I don't understand how there can be one single answer to number 4. Aren't there many different possibilities?

Not only that, in fact none of the answers can work!

4(x+y)-x=100, so 4(x+y)>100, x+y>25! Therefore none of the choices can be correct!
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TripleM
1587 posts
#11 • 2 Y
Y by Adventure10, Mango247
Silverfalcon wrote:
Number 7

My_ANSWER

I thought there was different answer on Harold Reiter's Page though.. ;)

Mistakes in your answer
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Silverfalcon
5006 posts
#12 • 2 Y
Y by Adventure10, Mango247
It should have been 13x+4y=100

I must've typed the question wrong.

The correct answer then would be D.

13(4)+4(12) = 100 = 100 :)
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Silverfalcon
5006 posts
#13 • 2 Y
Y by Adventure10, Mango247
Hehe...

I knew I was wrong because the quesiton had another way of solving it using Sum and Product Formula (only sum part this case) but for my equation, that formula didn't work.

Somehow, it worked though. :-D

Right way would be:

$81z^2+9z-6 = (9z+3)(9z-2)$ and this also gives me a right answer.

Hehe... I got the right answer by wrong method... :lol:
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Treething
2107 posts
#14 • 2 Y
Y by Adventure10, Mango247
1. A lattice point is a point on a coordinate plane with integer values for x and y. How many lattice points lie on a circle centered at the origin with radius 25?
A) 40
B) 20
C) 10
D) 5
E) None of above

answer

2. Find the unit digits of the sum:

$\sum^{2005}_{k=0} k^5$
A) 1
B) 3
C) 5
D) 7
E) 9

answer

3. How many three element subsets of the set:

{88,95,99,132,166,173}

have the property that the sum of the three elements is even?
A) 6
B) 8
C) 10
D) 12
E) 24

answer

4. If x and y are positive integesr such that $13x+4y=100$, then $x+y=$?
A) 10
B) 12
C) 14
D) 16
E) 18

answer

1 to 4
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joml88
6343 posts
#15 • 2 Y
Y by Adventure10, Mango247
Just wanted to point a non-plug and chug method for this one:

4. If $x$ and $y$ are positive integesr such that $13x+4y=100$, then $x+y=$?

Solution

If you haven't used mods very much this might not seem like a very natural approach. Plug and chug certainly works without excruciating pain in this case but when you get harder diophantine equations (it just means an equation where you are looking for integer solutions) you will almost have to use modular arithmetic.
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1234567890
1553 posts
#16 • 2 Y
Y by Adventure10, Mango247
How would you use mods?
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NightFlarer
316 posts
#17 • 2 Y
Y by Adventure10, Mango247
This probably isn't a good explanation, but here goes:

Say you have 3 + 2 = 5
We know 5 = 0 (mod 5), and since 3 + 2 = 5, we can say 3 + 2 = 0 (mod 5), which is true.
Thus, when considering a certain mod x, the left hand side and the right hand side of the original equation must be equal to the same y (mod x)

We have 13x + 4y = 100

I considered mod 13, which wasn't the best idea, but if you do that you get
13x or 0 (mod 13) + 4y (mod 13) = 100 (mod 13)

13x is always 0 mod 13, so just ignore it for now

So 4y (mod 13) = 100 or 9 (mod 13)

4y = 9 (mod 13)

I forgot my other steps so I'll just leave it at this...
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JBL
16123 posts
#18 • 2 Y
Y by Adventure10, Mango247
The smoothest thing to do at that point is to notice that 9 = -4 (mod 13), so 4y = -4 (mod 13) so y = -1 = 12 (mod 13). (Notice that we divide -- you can only ONLY ONLY ever do this when the thing you're dividing by is relatively prime to the mod. Luckily, 13 is prime.)
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tongos
53 posts
#19 • 2 Y
Y by Adventure10, Mango247
13x+4y=100

4(25)-4y

13x= 4(25-y)

25-y=k

13x=4k (13 is prime, so k=13 and x=4)

25-y=13, y=12, x=4

12+4=16
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tongos
53 posts
#20 • 2 Y
Y by Adventure10, Mango247
quick question.

On number 14, what does d(2)004 mean, the one that is the last term of the series. ?????
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Treething
2107 posts
#21 • 2 Y
Y by Adventure10, Mango247
The main thing to notice in number 4 is that 13 and 4 are relatively prime.
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Mr.Ocax
341 posts
#22 • 2 Y
Y by Adventure10, Mango247
Um.. Number 18 doesn't have any answer choices and on number 14.. are you sure the answer isn't -335?? becuase you have -334 on there, but I'm not sure if its a typo or what. This is what i did on 14:

Click to reveal hidden text

So go figure.. maybe i did something wrong. Some help on 14 please??
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Silverfalcon
5006 posts
#23 • 2 Y
Y by Adventure10, Mango247
-335 is a correct answer.

For #18, I forgot.
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Silverfalcon
5006 posts
#24 • 2 Y
Y by Adventure10, Mango247
I'd like to get the answers for these ASAP since AMC is coming and if you can answer these 25 questions VERY EASILY, you're ready for most of AMC questions (-10 or -12).

I think exception of use of trig and log, this is harder than regular AMC and mock AMC's since it involves some hardest question on mock AMC.
Quote:
6. In an h-meter race, Sunny is exactly d meters ahead of Windy when Sunny sportingly starts d meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finished the second race?

Let's call r as the rate of Sunny. Then, (h+d)/r is the time for Sunny to finish race with d meters more. To find Windy's rate, we set proportion:

Windy's rate: r = (h-d): h

Then we get (h-d)r/h as Windy's rate. Since the time for d+ race is (h+d)/r, then:

$\frac {(h-d)r}{h} * \frac {(h+d)}{r}$

Which equals h - d^2/h so Windy's ahead by d^2/h.
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Silverfalcon
5006 posts
#25 • 2 Y
Y by Adventure10, Mango247
Few answers to the problems.

#13

Since it's in form of 9*a*b, we can try in simpler case:

9*8*5 = 360
9*88*55 = 43560
9*888*555 = 4435560

So, the sum of digits equals 9n then 1985 = 9*1985 = 17865

There was a constructive method that Mr. Crawford showed although I didn't quite get it.

If anyone can explain here, I'll be thankful.

#16

I'm not sure but can I do this?

$\frac {\sin 10}{\cos 10} + \frac {\sin 20}{\cos 20} = \tan 10 + \tan 20$?

Can anyone gives some helps for the rest?

Thanks!
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Agent_00pi
4 posts
#26 • 2 Y
Y by Adventure10, Mango247
In #20, is O the center of the circle?
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joml88
6343 posts
#27 • 2 Y
Y by Adventure10, Mango247
Silverfalcon....I'm disappointed. You should know that $\frac{a+b}{c+d}\neq \frac ac+\frac bd.$

Hint
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Silverfalcon
5006 posts
#28 • 2 Y
Y by Adventure10, Mango247
That's what I thought too.

Except I wasn't sure for "TRIG" part.

No wonder it came out so nice and easy..

Hmmm...
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d343seven
237 posts
#29 • 2 Y
Y by Adventure10, Mango247
Click to reveal hidden text
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sky9073
452 posts
#30 • 2 Y
Y by Adventure10, Mango247
#11. Click to reveal hidden text

#12. Click to reveal hidden text
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Silverfalcon
5006 posts
#31 • 2 Y
Y by Adventure10, Mango247
To make this mock AMC more useful, I'll now post the keys to the answers that has posted up here.

KEYS

Answers needed:

5,8,9,10,16,17,20,21,22,24,25.


#3, I get 10. Sets that are even:

{88,95,173}
{132,95,173}
{166,95,173}
{88,99,173}
{132,99,173}
{166,99,173}
{88,95,99}
{132,95,99}
{166,95,99}
{88,132,166}.

Which one's right?

And thanks again for the answers. More answers to fill in the spaces are appreciated!
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joml88
6343 posts
#32 • 2 Y
Y by Adventure10, Mango247
Okay here are the formulas I was talking about for that trig problem:

Formulas
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litoxe2718281828
276 posts
#33 • 2 Y
Y by Adventure10, Mango247
Silverfalcon wrote:
#3, I get 10. Sets that are even:

{88,95,173}
{132,95,173}
{166,95,173}
{88,99,173}
{132,99,173}
{166,99,173}
{88,95,99}
{132,95,99}
{166,95,99}
{88,132,166}.

Which one's right?

And thanks again for the answers. More answers to fill in the spaces are appreciated!

Click to reveal hidden text

#5
Click to reveal hidden text

#8

Click to reveal hidden text
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Silverfalcon
5006 posts
#34 • 2 Y
Y by Adventure10, Mango247
Thanks to joml88 for the formula. I think I can figure out this.

See if this is right.

\[\frac {\sin 10 + \sin 20}{\cos 10 + \cos 20}\\
= \frac {2 \cdot \sin \frac {(10+20)}{2} \cdot \cos \frac {(10-20)}{2}}{2 \cdot \cos \frac {(10+20)}{2} \cdot \cos \frac {(10-20)}{2}}\\
= \frac {\sin 15}{\cos 15} = \tan 15\]

\[\text {So, our final answer is D.}\]
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Silverfalcon
5006 posts
#35 • 2 Y
Y by Adventure10, Mango247
The question about the expansion:

Find the units digit of:

\[\displaystyle (15 + \sqrt {220})^{19} + (15 + \sqrt {220})^{82} \]

I noticed that this question is extremely hard since it was #30 on one of old AHSMEs on 80's but how do you do it?

Do you expand it? Use conjugates? Would this help?

\[(15+\sqrt {220})^{19} \cdot [1+(15+\sqrt {220})^{63}]\]

Applying binomial theorem is the key to this problem but in what way?

P.S. Rereading all these problems again, I noticed why there aren't many replies
> This is a set of DIFFICULT PROBLEMS!
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zabelman
1072 posts
#36 • 2 Y
Y by Adventure10, Mango247
Thanks for this great mock test! Not having the time to look at the whole thing, I worked only on the ones we needed answers for. Of course I was triple tasking when I did them, so my sols might be flawed. Let me know if I made any mistakes, or if there are problems I missed! And I think you should recheck problem 21.

9
Click to reveal hidden text
E

10
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16
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17
Click to reveal hidden text

20
Click to reveal hidden text

21
I think there's an error in this problem.

22
Click to reveal hidden text

24
Click to reveal hidden text

25
Click to reveal hidden text
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zabelman
1072 posts
#37 • 2 Y
Y by Adventure10, Mango247
Silverfalcon wrote:
This is a set of DIFFICULT PROBLEMS!

Indeed! I say this is a little harder than the hardest AMC would be. But it's still very good review! (Luckily, I've seen one or two of these problems before :D )
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litoxe2718281828
276 posts
#38 • 2 Y
Y by Adventure10, Mango247
Many thanks to Silverfalcon for this mock AMC!

Also, zabelman, your #17 solution is irresistably good. Puts me to shame. :blush:
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zabelman
1072 posts
#39 • 2 Y
Y by Adventure10, Mango247
litoxe2718281828 wrote:
Also, zabelman, your #17 solution is irresistably good.
Thank you! It just goes to show that simple conjugates can go a long way! :P

litoxe2718281828 wrote:
Puts me to shame. :blush:
Why should it do this? There's no shame in struggling with a difficult problem. That's what problem solving's all about! And indeed, that problem is difficult.
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Silverfalcon
5006 posts
#40 • 2 Y
Y by Adventure10, Mango247
Wow, new solution!

:) Thanks for the great replies!
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