be a triangle with orthocenter 
and let 
be the second intersection of the circumcircle of triangle 
with the internal bisector of the angle 
.
be the circumcenter of triangle 
and 
the orthocenter of triangle 
.
is equal to the circumradius of triangle 
such that the following property holds: if we list the divisors of 
in increasing order as 
,![\[ d_2 - d_1 \leq d_3 - d_2 \leq \dots \leq d_k - d_{k-1}. \]](http://latex.artofproblemsolving.com/a/d/c/adc94af88b3579164aabba51bd021442133e364a.png)
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
be the incircle of a fixed equilateral triangle 
be a variable line that is tangent to 
and 
at points 
,
is chosen such that 
and 
.
be positive real numbers such that ![\(a+b+c=4\sqrt[3]{abc}\)](http://latex.artofproblemsolving.com/2/8/2/28214a1814762b71858db1543fae1a50212eaafa.png)
.![\[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]](http://latex.artofproblemsolving.com/7/0/4/7043a170ac5535465a5a1b4daccb1f1de3043280.png)
are multiples of 
or 
but not 
?
![\[222{,}222-22{,}222-2{,}222-222-22-2?\]](http://latex.artofproblemsolving.com/1/5/6/156c3662fe4c874a80e54887f76c302c13b9d14c.png)
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:
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
, then 
?
A) 10
B) 12
C) 14
D) 16
E) 18
5. Let a,a',b,b' be real numbers with
. The solution to 
is less than the solution to 
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)
E)
7. Let
be a function which 
. Find the sum of all values of 
for which 
.
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
and the second term is 7. Find the sum of the first 50 terms.
A)
B)
C)
D)
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
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
consists of a sequence of 1985 eights and the integer 
contains a sequence of 1985 fives. What is the sum of digits of the base 10 representation of the integer 
?
A) 15880
B) 17856
C) 17865
D) 17874
E) 19851
14. Let
denote the number of diagonals plus the number of sides in a convex polygon with 
sides. Then:
where m and n are relatively prime integers. Find 
.
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:
where numbers are given in degrees.
A)
B)
C)
D)
E)
17. Find the units digit in the decimal expansion of:
.
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
and 
where 
, then which of the following is not correct?
A)
B)
C)
D)
E)
20. A and C lie on a circle with radius
. 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!
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:
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
, then 
?A) 10
B) 12
C) 14
D) 16
E) 18
5. Let a,a',b,b' be real numbers with
. The solution to 
is less than the solution to 
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)
E)
7. Let
be a function which 
. Find the sum of all values of 
for which 
.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
and the second term is 7. Find the sum of the first 50 terms.A)
B)
C)
D)
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
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
consists of a sequence of 1985 eights and the integer 
contains a sequence of 1985 fives. What is the sum of digits of the base 10 representation of the integer 
?A) 15880
B) 17856
C) 17865
D) 17874
E) 19851
14. Let
denote the number of diagonals plus the number of sides in a convex polygon with 
sides. Then:
where m and n are relatively prime integers. Find 
.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:
where numbers are given in degrees.A)
B)
C)
D)
E)
17. Find the units digit in the decimal expansion of:
.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
and 
where 
, then which of the following is not correct?A)
B)
C)
D)
E)
20. A and C lie on a circle with radius
. 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!
which leads me that z=-1/9 and that's only possible value for z so it's -1/9.
by Pythagoras. Let AE be the angle bisector of DAC, E is on BC. by angle bisector theorem, ED=\frac{2\sqrt{5}}{5},
.
,
,
.
.
and this also gives me a right answer.
is obviously equal to 
which is obviously equal to 
. And since the units digit of 
,
. The units digit of the units digit of 1 + 32 + 243 + 1024 + 3125, which is 5.
,
and 
are positive integesr such that 
So 
doesn't work (x has to be positive) and x=8 makes the expression too big to start so it must be x=4. That makes y=12. Thus x+y=16.
?
,
,
and 
.
![\[1+\binom{3}{2}\binom{3}{1}=1+9=10\]](http://latex.artofproblemsolving.com/4/9/d/49d9f9aebae6473b8ba57f878c0f3263e3479e1e.png)
![\[\frac{-b}{a}<\frac{-b'}{a'}\Rightarrow\frac{b}{a}>\frac{b'}{a'}\]](http://latex.artofproblemsolving.com/7/3/5/7351b05ea7016fad9a179a2aea58b54ee786242e.png)
.
.![\[Re(S)=3+7+\cdots+(3+4\cdot49)=5050\]](http://latex.artofproblemsolving.com/4/9/c/49cd058d12c3b53b242c2f60b5c6773689b6935b.png)
![\[Im(S)=4+0+\cdots+(4-4\cdot49)=-4700\]](http://latex.artofproblemsolving.com/2/e/4/2e4fcd597ee5e473cdea24cc289906f1bccf595d.png)
![\[S=5050-4700i\]](http://latex.artofproblemsolving.com/a/3/0/a30c220a3bd255b83007b06590b6cc636a97ac72.png)
![\[\frac {\sin 10 + \sin 20}{\cos 10 + \cos 20}\\
= \frac {2 \cdot \sin \frac {(10+20)}{2} \cdot \cos \frac {(10-20)}{2}}{2 \cdot \cos \frac {(10+20)}{2} \cdot \cos \frac {(10-20)}{2}}\\
= \frac {\sin 15}{\cos 15} = \tan 15\]](http://latex.artofproblemsolving.com/7/c/5/7c52e62cda20b238e775985591d73b4cdd8b5a08.png)
![\[\text {So, our final answer is D.}\]](http://latex.artofproblemsolving.com/a/3/1/a318a51ee1c70e3f12d68526b555646cf1d187be.png)
![\[\displaystyle (15 + \sqrt {220})^{19} + (15 + \sqrt {220})^{82} \]](http://latex.artofproblemsolving.com/2/8/7/28750b246137d4891ce88283985e4fd3d4998cce.png)
![\[(15+\sqrt {220})^{19} \cdot [1+(15+\sqrt {220})^{63}]\]](http://latex.artofproblemsolving.com/b/4/a/b4a507b8483ff780ffd71b7e2a8cf9ef33b2905e.png)
.
,
.
.
.
,
satisfy the recursion 
.
and 
,
ends in 0. Since 
,
is small and positive, so 
.
,
.
.
,
.
.
.
in base 3, ![\[\binom{2001}{n}\equiv\binom{2}{a_6}\binom{2}{a_5}\binom{0}{a_4}\binom{2}{a_3}\binom{0}{a_2}\binom{1}{a_1}\binom{0}{a_0}\bmod3.\]](http://latex.artofproblemsolving.com/c/3/a/c3a2b8caab4f4f58dcc976ab922f0cab95910f27.png)
We just have to make that product equal 
.
are either 0 or 2, 
are each 0, and 
is either 0 or 1. There are therefore 16 ways in this case.
ways for this. Answer: 16+12=28.
.
.
