[/quote]
,
reals, and 
.
?
be positive real numbers such that 
.
knowing that the denominators are positive real numbers.
such that: Among every 
such that a positive integer power of 
equals to a positive integer power of 
.
be an acute-angled triangle with 
A circle passes through 
and 
and crosses the segments 
and 
again at 
and 
respectively. The circumcircles of 
and 
and 
The segments 
and 
intersect at 
Let 
and 
be the reflections of 
in the lines 
and 
respectively. Prove that the points 
and 
are concyclic.
be positive real numbers such that 
.
When is the equality valid?
be a center of circle which passes through vertices of quadrilateral 
,
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
cards on the table, and each card has a positive integer written on it. An odd number is written on exactly 
cards. Every minute, the following operation is performed: for all possible sets of 
cards on the table, the product of the numbers on these three cards is calculated, all the obtained results are summed, and this sum is written on a new card and placed on the table. A day later, it turns out that there is a card on the table, the number written on this card is divisible by 
Prove that one hour after the start of the process, there was a card on the table that the number written on that card is divisible by 
on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
and 
written on the board, erasing them, and writing the integer 
.
.
is called a winning number if there exists a sequence of operations such that, at the end, the only integer left on the board is 
and, for each of them, determine the minimum number of GCD operations Charlotte must use.
Y by MathRook7817, elasticwealth, RainbowJessa, DhruvJha, fake123, ihatemath123
Practice AMC 10A
1. Find the sum of the infinite geometric series 1 + 7/18 + 49/324 + …
A - 36/11, B - 9/22, C - 18/11, D - 18/7, E - 9/14
2. What is the first digit after the decimal point in the square root of 420?
A - 1, B - 2, C - 3, D - 4, E - 5
3. Caden’s calculator is broken and two of the digits are swapped for some reason. When he entered in 9 + 10, he got 21. What is the sum of the two digits that got swapped?
A - 2, B - 3, C - 4, D - 5, E - 6
4. Two circles with radiuses 47 and 96 intersect at two points A and B. Let P be the point 82% of the way from A to B. A line is drawn through P that intersects both circles twice. Let the four intersection points, from left to right be W, X, Y, and Z. Find (PW/PX)*(PY/PZ).
A - 50/5863, B - 47/96, C - 1, D - 96/47, E - 5863/50
5. Two dice are rolled, and the two numbers shown are a and b. How many possible values of ab are there?
A - 17, B - 18, C - 19, D - 20, E - 21
6. What is the largest positive integer that cannot be expressed in the form 6a + 9b + 4 + 20d, where a, b, and d are positive integers?
A - 29, B - 38, C - 43, D - 76, E - 82
7. What is the absolute difference of the probabilities of getting at least 6/10 on a 10-question true or false test and at least 3/5 on a 5-question true or false test?
A - 63/1024, B - 63/512, C - 63/256, D - 63/128, E - 0
8. How many arrangements of the letters in the word “sensor” are there such that the two vowels have an even number of letters (remember 0 is even) between them (including the original “sensor”)?
A - 72, B - 108, C - 144, D - 216, E - 432
9. Find the value of 0.9 * 0.97 + 0.5 * 0.1 * (0.5 * 0.97 + 0.5 * 0.2) rounded to the nearest tenth of a percent.
A - 89.9%, B - 90.0%, C - 90.1%, D - 90.2%, E - 90.3%
10. Two painters are painting a room. Painter 1 takes 52:36 to paint the room, and painter 2 takes 26:18 to paint the room. With these two painters working together, how long should the job take?
A - 9:16, B - 10:52, C - 17:32, D - 35:02, E - 39:44
11. Suppose that on the coordinate grid, the x-axis represents climate, and the y-axis represents landscape, where -1 <= x, y <= 1 and a higher number for either coordinate represents better conditions along that particular axis. Accordingly, the points (0, 0), (1, 1), (-1, 1), (-1, -1), and (1, -1) represent cities, plains, desert, snowy lands, and mountains, respectively. An area is classified as whichever point it is closest to. Suppose a theoretical new area is selected by picking a random point within the square bounded by plains, desert, snowy lands, and mountains as its vertices. What is the probability that it is a plains?
A - 1 - (1/4)pi, B - 1/5, C - (1/16)pi, D - 1/4, E - 1/8
12. Statistics show that people who work out n days a week have a (1/10)(n+2) chance of getting a 6-pack, and the number of people who exercise n days a week is directly proportional to 8 - n (Note that n can only be an integer from 0 to 7, inclusive). A random person is selected. Find the probability that they have a 6-pack.
A - 13/30, B - 17/30, C - 19/30, D - 23/30, E - 29/30
13. A factory must produce 3,000 items today. The manager of the factory initially calls over 25 employees, each producing 5 items per hour starting at 9 AM. However, he needs all of the items to be produced by 9 PM, and realizes that he must speed up the process. At 12 PM, the manager then encourages his employees to work faster by increasing their pay, in which they then all speed up to 6 items per hour. At 1 PM, the manager calls in 15 more employees which make 5 items per hour each. Unfortunately, at 3 PM, the AC stops working and the hot sun starts taking its toll, which slows every employee down by 2 items per hour. At 4 PM, the technician fixes the AC, and all employees return to producing 5 items per hour. At 5 PM, the manager calls in 30 more employees, which again make 5 items per hour. At 6 PM, he calls in 30 more employees. At 7 PM, he rewards all the pickers again, speeding them up to 6 items per hour. But at 8 PM, n employees suddenly crash out and stop working due to fatigue, and the rest all slow back down to 5 items per hour because they are tired. The manager does not have any more employees, so if too many of them drop out, he is screwed and will have to go overtime. Find the maximum value of n such that all of the items can still be produced on time, done no later than 9 PM.
A - 51, B - 52, C - 53, D - 54, E - 55
14. Find the number of positive integers n less than 69 such that the average of all the squares from 1^2 to n^2, inclusive, is an integer.
A - 11, B - 12, C - 23, D - 24, E - 48
15. Find the number of ordered pairs (a, b) of integers such that (a - b)^2 = 625 - 2ab.
A - 6, B - 10, C - 12, D - 16, E - 20
16. What is the 420th digit after the decimal point in the decimal expansion of 1/13?
A - 4, B - 5, C - 6, D - 7, E - 8
17. Two congruent right rectangular prisms stand near each other. Both have the same orientation and altitude. A plane that cuts both prisms into two pieces passes through the vertical axes of symmetry of both prisms and does not cross the bottom or top faces of either prism. Let the point that the plane crosses the axis of symmetry of the first prism be A, and the point that the plane crosses the axis of symmetry of the second prism be B. A is 81% of the way from the bottom face to the top face of the first prism, and B is 69% of the way from the bottom face to the top face of the second prism. What percent of the total volume of both prisms combined is above the plane?
A - 19%, B - 25%, C - 50%, D - 75%, E - 81%
18. What is the greatest number of positive integer factors an integer from 1 to 100 can have?
A - 10, B - 12, C - 14, D - 15, E - 16
19. On an analog clock, the minute hand makes one full revolution every hour, and the hour hand makes one full revolution every 12 hours. Both hands move at a constant rate. During which of the following time periods does the minute hand pass the hour hand?
A - 7:35 - 7:36, B - 7:36 - 7:37, C - 7:37 - 7:38, D - 7:38 - 7:39, E - 7:39 - 7:40
20. Find the smallest positive integer that is a leg in three different Pythagorean triples.
A - 12, B - 14, C - 15, D - 20, E - 21
21. How many axes of symmetry does the graph of (x^2)(y^2) = 69 have?
A - 2, B - 3, C - 4, D - 5, E - 6
22. Real numbers a, b, and c are chosen uniformly and at random from 0 to 3. Find the probability that a + b + c is less than 2.
A - 4/81, B - 8/81, C - 4/27, D - 8/27, E - 2/3
23. Let f(n) be the sum of the positive integer divisors of n. Find the sum of the digits of the smallest odd positive integer n such that f(n) is greater than 2n.
A - 15, B - 18, C - 21, D - 24, E - 27
24. Find the last three digits of 24^10.
A - 376, B - 576, C - 626, D - 876, E - 926
25. A basketball has a diameter of 9 inches, and the hoop has a diameter of 18 inches. Peter decides to pick up the basketball and make a throw. Given that Peter has a 1/4 chance of accidentally hitting the backboard and missing the shot, but if he doesn’t, he is guaranteed that the frontmost point of the basketball will be within 18 inches of the center of the hoop at the moment when a great circle of the basketball crosses the plane containing the rim. No part of the ball will extend behind the backboard at any point during the throw, and the rim is attached directly to the backboard. What is the probability that Peter makes the shot?
A - 3/128, B - 3/64, C - 3/32, D - 3/16, E - 3/8
1. Find the sum of the infinite geometric series 1 + 7/18 + 49/324 + …
A - 36/11, B - 9/22, C - 18/11, D - 18/7, E - 9/14
2. What is the first digit after the decimal point in the square root of 420?
A - 1, B - 2, C - 3, D - 4, E - 5
3. Caden’s calculator is broken and two of the digits are swapped for some reason. When he entered in 9 + 10, he got 21. What is the sum of the two digits that got swapped?
A - 2, B - 3, C - 4, D - 5, E - 6
4. Two circles with radiuses 47 and 96 intersect at two points A and B. Let P be the point 82% of the way from A to B. A line is drawn through P that intersects both circles twice. Let the four intersection points, from left to right be W, X, Y, and Z. Find (PW/PX)*(PY/PZ).
A - 50/5863, B - 47/96, C - 1, D - 96/47, E - 5863/50
5. Two dice are rolled, and the two numbers shown are a and b. How many possible values of ab are there?
A - 17, B - 18, C - 19, D - 20, E - 21
6. What is the largest positive integer that cannot be expressed in the form 6a + 9b + 4 + 20d, where a, b, and d are positive integers?
A - 29, B - 38, C - 43, D - 76, E - 82
7. What is the absolute difference of the probabilities of getting at least 6/10 on a 10-question true or false test and at least 3/5 on a 5-question true or false test?
A - 63/1024, B - 63/512, C - 63/256, D - 63/128, E - 0
8. How many arrangements of the letters in the word “sensor” are there such that the two vowels have an even number of letters (remember 0 is even) between them (including the original “sensor”)?
A - 72, B - 108, C - 144, D - 216, E - 432
9. Find the value of 0.9 * 0.97 + 0.5 * 0.1 * (0.5 * 0.97 + 0.5 * 0.2) rounded to the nearest tenth of a percent.
A - 89.9%, B - 90.0%, C - 90.1%, D - 90.2%, E - 90.3%
10. Two painters are painting a room. Painter 1 takes 52:36 to paint the room, and painter 2 takes 26:18 to paint the room. With these two painters working together, how long should the job take?
A - 9:16, B - 10:52, C - 17:32, D - 35:02, E - 39:44
11. Suppose that on the coordinate grid, the x-axis represents climate, and the y-axis represents landscape, where -1 <= x, y <= 1 and a higher number for either coordinate represents better conditions along that particular axis. Accordingly, the points (0, 0), (1, 1), (-1, 1), (-1, -1), and (1, -1) represent cities, plains, desert, snowy lands, and mountains, respectively. An area is classified as whichever point it is closest to. Suppose a theoretical new area is selected by picking a random point within the square bounded by plains, desert, snowy lands, and mountains as its vertices. What is the probability that it is a plains?
A - 1 - (1/4)pi, B - 1/5, C - (1/16)pi, D - 1/4, E - 1/8
12. Statistics show that people who work out n days a week have a (1/10)(n+2) chance of getting a 6-pack, and the number of people who exercise n days a week is directly proportional to 8 - n (Note that n can only be an integer from 0 to 7, inclusive). A random person is selected. Find the probability that they have a 6-pack.
A - 13/30, B - 17/30, C - 19/30, D - 23/30, E - 29/30
13. A factory must produce 3,000 items today. The manager of the factory initially calls over 25 employees, each producing 5 items per hour starting at 9 AM. However, he needs all of the items to be produced by 9 PM, and realizes that he must speed up the process. At 12 PM, the manager then encourages his employees to work faster by increasing their pay, in which they then all speed up to 6 items per hour. At 1 PM, the manager calls in 15 more employees which make 5 items per hour each. Unfortunately, at 3 PM, the AC stops working and the hot sun starts taking its toll, which slows every employee down by 2 items per hour. At 4 PM, the technician fixes the AC, and all employees return to producing 5 items per hour. At 5 PM, the manager calls in 30 more employees, which again make 5 items per hour. At 6 PM, he calls in 30 more employees. At 7 PM, he rewards all the pickers again, speeding them up to 6 items per hour. But at 8 PM, n employees suddenly crash out and stop working due to fatigue, and the rest all slow back down to 5 items per hour because they are tired. The manager does not have any more employees, so if too many of them drop out, he is screwed and will have to go overtime. Find the maximum value of n such that all of the items can still be produced on time, done no later than 9 PM.
A - 51, B - 52, C - 53, D - 54, E - 55
14. Find the number of positive integers n less than 69 such that the average of all the squares from 1^2 to n^2, inclusive, is an integer.
A - 11, B - 12, C - 23, D - 24, E - 48
15. Find the number of ordered pairs (a, b) of integers such that (a - b)^2 = 625 - 2ab.
A - 6, B - 10, C - 12, D - 16, E - 20
16. What is the 420th digit after the decimal point in the decimal expansion of 1/13?
A - 4, B - 5, C - 6, D - 7, E - 8
17. Two congruent right rectangular prisms stand near each other. Both have the same orientation and altitude. A plane that cuts both prisms into two pieces passes through the vertical axes of symmetry of both prisms and does not cross the bottom or top faces of either prism. Let the point that the plane crosses the axis of symmetry of the first prism be A, and the point that the plane crosses the axis of symmetry of the second prism be B. A is 81% of the way from the bottom face to the top face of the first prism, and B is 69% of the way from the bottom face to the top face of the second prism. What percent of the total volume of both prisms combined is above the plane?
A - 19%, B - 25%, C - 50%, D - 75%, E - 81%
18. What is the greatest number of positive integer factors an integer from 1 to 100 can have?
A - 10, B - 12, C - 14, D - 15, E - 16
19. On an analog clock, the minute hand makes one full revolution every hour, and the hour hand makes one full revolution every 12 hours. Both hands move at a constant rate. During which of the following time periods does the minute hand pass the hour hand?
A - 7:35 - 7:36, B - 7:36 - 7:37, C - 7:37 - 7:38, D - 7:38 - 7:39, E - 7:39 - 7:40
20. Find the smallest positive integer that is a leg in three different Pythagorean triples.
A - 12, B - 14, C - 15, D - 20, E - 21
21. How many axes of symmetry does the graph of (x^2)(y^2) = 69 have?
A - 2, B - 3, C - 4, D - 5, E - 6
22. Real numbers a, b, and c are chosen uniformly and at random from 0 to 3. Find the probability that a + b + c is less than 2.
A - 4/81, B - 8/81, C - 4/27, D - 8/27, E - 2/3
23. Let f(n) be the sum of the positive integer divisors of n. Find the sum of the digits of the smallest odd positive integer n such that f(n) is greater than 2n.
A - 15, B - 18, C - 21, D - 24, E - 27
24. Find the last three digits of 24^10.
A - 376, B - 576, C - 626, D - 876, E - 926
25. A basketball has a diameter of 9 inches, and the hoop has a diameter of 18 inches. Peter decides to pick up the basketball and make a throw. Given that Peter has a 1/4 chance of accidentally hitting the backboard and missing the shot, but if he doesn’t, he is guaranteed that the frontmost point of the basketball will be within 18 inches of the center of the hoop at the moment when a great circle of the basketball crosses the plane containing the rim. No part of the ball will extend behind the backboard at any point during the throw, and the rim is attached directly to the backboard. What is the probability that Peter makes the shot?
A - 3/128, B - 3/64, C - 3/32, D - 3/16, E - 3/8
This post has been edited 5 times. Last edited by freddyfazbear, Mar 30, 2025, 3:42 AM
Note that 
,
We can calculate that 
,
:![\[
f(x)=\sqrt{x},\quad a=400,\quad h=20.
\]](http://latex.artofproblemsolving.com/7/4/a/74a6f8b25f79d071104062bd3fd8e2464323e754.png)
![\[
f(a)=\sqrt{400}=20,\quad
f'(x)=\frac1{2\sqrt{x}}\;\Rightarrow\;f'(400)=\frac1{40},\quad
f''(x)=-\frac1{4x^{3/2}}\;\Rightarrow\;f''(400)=-\frac1{32000}.
\]](http://latex.artofproblemsolving.com/7/3/f/73f2846f462f793513962dcc5cf25ed66d1f1a0a.png)
![\[
f(420)=f(400+20)\approx f(400)+f'(400)\,20+\frac{f''(400)\,(20)^2}{2}
=20+\frac1{40}\cdot20+\frac{-\frac1{32000}\cdot400}{2}
=20.49375.
\]](http://latex.artofproblemsolving.com/2/8/d/28dfa3f4e5d1ee1631d7fd52ea1b438b2c1fde0c.png)
is indeed the first digit after the decimal point.![\[
R_2=\frac{f^{(3)}(\xi)}{3!}\,h^3,\quad400<\xi<420,
\quad f^{(3)}(x)=\frac{3}{8}x^{-5/2}.
\]](http://latex.artofproblemsolving.com/f/6/9/f69503ef5dbf0274ea3b2947ffaf8212b44288ba.png)
So:![\[
R_2=\frac{\frac{3}{8}\,\xi^{-5/2}}{6}(20)^3
=\frac{500}{\xi^{5/2}},
\quad |R_2|\le\frac{500}{400^{5/2}}=\frac{500}{20^5}=0.00015625.
\]](http://latex.artofproblemsolving.com/d/e/d/ded238b9428994753ed08eb9031afc183f1964b2.png)
So, the first digit after the decimal point is indeed 
it is on the radical axis of the two circles, so it's power to each of them is equal
