Art of Problem Solving
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.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2025 AMC 10A Problems: Difference between revisions

← Older edit
Kayden3089 (talk | contribs)
editor
2 edits
Tag: Manual revert
 
(288 intermediate revisions by 86 users not shown)
Line 1: Line 1:
{{AMC10 Problems|year=2025|ab=A}}
{{AMC10 Problems|year=2025|ab=A}}


== Problem 1 ==
==Problem 1==
No cheating :)
Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at <imath>1{:}30</imath> traveling due north at a steady <imath>8</imath> miles per hour. Betsy leaves on her bicycle from the same point at <imath>2{:}30</imath>, traveling due east at a steady <imath>12</imath> miles per hour. At what time will they be exactly the same distance from their common starting point?


== Problem 2 ==
<imath>\textbf{(A) } 3{:}30 \qquad\textbf{(B) } 3{:}45 \qquad\textbf{(C) } 4{:}00 \qquad\textbf{(D) } 4{:}15 \qquad\textbf{(E) } 4{:}30</imath>
Dear Admin,


I am sorry for what I and my friend have done. We are truly sorry and won’t do this again.
[[2025 AMC 10A Problems/Problem 1|Solution]]


== Problem 3 ==
==Problem 2==
Someone deleted Question 15. Could you pweeze add it back.
My friend and I are extremely sorry for what we have done.


== Problem 4 ==
A box contains <imath>10</imath> pounds of a nut mix that is <imath>50</imath> percent peanuts, <imath>20</imath> percent cashews, and <imath>30</imath> percent almonds. A second nut mix containing <imath>20</imath> percent peanuts, <imath>40</imath> percent cashews, and <imath>40</imath> percent almonds is added to the box resulting in a new nut mix that is <imath>40</imath> percent peanuts. How many pounds of cashews are now in the box?
Solve the equation <math>4x+0x+5x+8x+5x=40585</math>.


== Problem 5 ==
<imath>\textbf{(A) } 3.5 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 4.5 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 6</imath>


== Problem 6 ==
[[2025 AMC 10A Problems/Problem 2|Solution]]


== Problem 7 ==
==Problem 3==
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length <imath>2025</imath>?


== Problem 8 ==
<imath>\textbf{(A) } 2025 \qquad\textbf{(B) } 2026 \qquad\textbf{(C) } 3012 \qquad\textbf{(D) } 3037 \qquad\textbf{(E) } 4050</imath>


== Problem 9 ==
[[2025 AMC 10A Problems/Problem 3|Solution]]


== Problem 10 ==
==Problem 4==


== Problem 11 ==
A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is <imath>15</imath>. Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from <imath>12</imath> to <imath>14</imath>. If Ash plays with the teachers, the average age on that team will decrease from <imath>55</imath> to <imath>52</imath>. How old is Ash?


== Problem 12 ==
<imath>\textbf{(A) } 28 \qquad\textbf{(B) } 29 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 32 \qquad\textbf{(E) } 33</imath>


== Problem 13 ==
[[2025 AMC 10A Problems/Problem 4|Solution]]


== Problem 14(34) ==
==Problem 5==


this is kim dokja. for 10 years kim dokja has been reading a book, which just ended today. with nothing else to read, he kickstarts the apocalypse. this is yoo joonghyuk, the novel’s protagonist, who also happens to be a bit of a psychopath. he greets kim dokja by grabbing him by the neck. let's be companions, says kim dokja. sure, says yoo joonghyuk, still holding him by the neck. to celebrate their new friendship, he throws kim dokja off a bridge into a large monster’s mouth but escapes death by making a trade deal with a supernatural creature kim dokja gathers a squad. ding ding ding. kim dokja’s sunfish depression sensor is tingling. yoo joonghyuk stop being depressed, says kim dokja while smacking his cheeks turns out kim dokja isn't the only person who reads. he gets invited to a rather sizable book club, except no one has finished reading the book. thanks to that, kim dokja gets his first kill so far. it's himself. yoo joonghyuk gets a debuff so kim dokja punches him to assert dominance. everyone claps. as a reward, he gets to sit on a big chair, which he destroys in anger. it's quiz time. yoo joonghyuk sees kim dokja as a companion, says yoo joonghyuk. c-nt, says kim dokja. kim dokja sacrifices himself for yoo joonghyuk, who develops anger management issues as a result. approximately 5 seconds later kim dokja sacrifices himself again and saves the city. after his death the people honor his noble memory by calling him ugly at his funeral. the squad has a very serious discussion about kim dokja’s kinks. after listening to this, kim dokja dies yoo joonghyuk grabs kim dokja by the neck and throws him into a large monster’s mouth, but this time with consent. also, kim dokja dies legend says that kim dokja is going to die, but for real this time. no worries, chuckles kim dokja, death is a social construct. yoo joonghyuk has a plan, he wants to sacrifice himself. hey, only i can do that, says kim dokja and sacrifices himself first to assert dominance kim dokja gets thrown into the garbage, where he belongs. kim dokja and yoo joonghyuk lead a coup d'etat while cosplaying as each other kim dokja and friends fight the sun kim dokja has a plan. he wants to sacrifice himself again. oh no, says his companions. oh yes, says kim dokja as he flies into a large squid’s mouth kim dokja gets deported and takes a holiday to the space-time continuum while his companions recover from ptsd yoo joonghyuk of the past tells kim dokja to kill yoo joonghyuk of the past so kim dokja makes yoo joonghyuk of the past eat dirt but accidentally turns one yoo joonghyuk into two yoo joonghyuks one of the yoo joonghyuks kills the other yoo joonghyuk and goes on to be yoo joonghyuk of the future who is also yoo joonghyuk of the present meanwhile yoo joonghyuk of the past who is a different yoo joonghyuk from these other two yoo joonghyuks is rather miffed about this whole thing and sends kim dokja back to his companions as an ugly squid everyone, especially yoo joonghyuk of the present, is also miffed at kim dokja for being a suicidal idiot and nobody forgives him kim dokja and friends fight the cast of percy jackson the squad discovers they are all part of kim dokja’s self-insert fanfiction. no one is too thrilled about this. yoo joonghyuk finds out that kim dokja once chose yoo joonghyuk of the past over yoo joonghyuk of the present and confronts kim dokja, feeling betrayed and wanting to ask why he would do this, and whether this world still held any meaning for him or if their companionship was all a lie. but, he says none of this. instead he says, "kim dokja i'm going to kill you" kim dokja had never wanted to betray or hurt yoo joonghyuk but wanted to save yoo joonghyuk of the past and present and reach an end where both yoo joonghyuks are happy. but, he says none of this. instead he says, "yoo joonghyuk im going to kill you" han sooyoung gets so fed up with their bullshit that she dies. 5 seconds later she decides it’s not worth it and comes back to life. meanwhile, yoo joonghyuk gets eaten by a wall kim dokja is feeling a bit itchy since he hasn’t sacrificed himself in a while. luckily, a great war is happening. it accidentally creates the apocalypse. kim dokja’s solution to fighting the apocalypse is to summon an even bigger apocalypse to fight the apocalypse but ends up getting dragged into the new apocalypse yoo joonghyuk of the past comes to kidnap kim dokja but is stopped by yoo joonghyuk of the present so yoo joonghyuk and yoo joonghyuk fight for custody but yoo joonghyuk wins and takes kim dokja away kim dokja wakes up surrounded by several small yoo joonghyuks and one large yoo joonghyuk. he passes out one of the small yoo joonghyuks turns into a dumpling and escapes with kim dokja. the squad is too busy writing fanfic about themselves to notice. kim dokja saves himself from becoming kim dokja the squid by becoming kim dokja the monkey. kim dokja the monkey fights evil with his monke brethren. yoo joonghyuk of the past and yoo joonghyuk of the present have a fight to see which yoo joonghyuk is the bigger tsundere. unsurprisingly, it’s yoo joonghyuk. after all that, kim dokja gets kidnapped again anyways kim dokja turns into a dumpling and wakes up surrounded by many small yoo joonghyuks, who all start kicking him. kim dokja finally returns to his squad. i’m back, says kim dokja. go fuck yourself, the squad replies. as retaliation, kim dokja gets kidnapped by his squad for a mandatory company vacation where failure to socialize is punishable by death. kim dokja turns into squid to sacrifice himself in order to un-sacrifice himself because he made a promise to stop sacrificing himself kim dokja and friends fight the marvel cinematic universe while yoo joonghyuk rides kim dokja like a surfboard. they meet the final boss. it’s a wall. they break the wall but must go through a different wall to get to the other side of the wall, where kim dokja meets kim dokja of the past. kim dokja wants to kill kim dokja but is stopped by yoo joonghyuk and yoo joonghyuk of the past who takes kim dokja of the past back to daycare. one kim dokja turns into two kim dokjas one half of kim dokja goes to aid yoo joonghyuk of the past and sends him off to become yoo joonghyuk of the future who is also yoo joonghyuk of the present. meanwhile the other half of kim dokja is with yoo joonghyuk of the present who wants to go to the past to save the half of kim dokja of the present and stop him from sacrificing himself this is kim dokja. for 10 years kim dokja has been reading a book, which just ended tod- hey wait a second han sooyoung who was previously many han sooyoungs had turned into two han sooyoungs one of whom was han sooyoung of the future who became han sooyoung of the past who then turned into two han sooyoungs to turn back into the one han sooyoung who later became the two han sooyoungs han sooyoung and friends play a dangerous game of tug-of-war. this gets tiring so they opt for a jigsaw puzzle instead, except nobody knows where the pieces are everyone needs therapy, including myself. on the bright side, at least they all have jobs except for yoo joonghyuk, who turns to terrorism and theft. yoo joonghyuk goes to space and has an existential crisis after he sorts his life together he helps han sooyoung brainwash innocents into plagiarizing her own book and with the power of imagination they successfully resurrect kim dokja and everyone claps
Consider the sequence of positive integers


== Problem 16 ==
<cmath> 1,2,1,2,3,2,1,2,3,4,3,2,1,2,3,4,5,4,3,2,1,2,3,4,5,6,5,4,3,2,1,2,\dots </cmath>


== Problem 17 ==
What is the 2025th term in this sequence?


== Problem 18 ==
<imath>\textbf{(A) } 5 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 44 \qquad\textbf{(E) } 45</imath>


== Problem 19 ==
[[2025 AMC 10A Problems/Problem 5|Solution]]


== Problem 20 ==
==Problem 6==


== Problem 21 ==
In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle 20°-angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?


== Problem 22 ==
<imath>\textbf{(A) } 80 \qquad\textbf{(B) } 90 \qquad\textbf{(C) } 100 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120</imath>


== Problem 23 ==
[[2025 AMC 10A Problems/Problem 6|Solution]]


== Problem 24 ==
==Problem 7==
Suppose <imath>a</imath> and <imath>b</imath> are real numbers. When the polynomial <imath>x^3+x^2+ax+b</imath> is divided by <imath>x-1</imath>, the remainder is <imath>4</imath>. When the polynomial is divided by <imath>x-2</imath>, the remainder is <imath>6</imath>. What is <imath>b-a</imath>?


== Problem 25 ==
<imath>\textbf{(A) } 14 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 17 \qquad\textbf{(E) } 18</imath>


If the answer to this problem is <math>x</math>, find <math>x</math>.
[[2025 AMC 10A Problems/Problem 7|Solution]]


== See Also ==
==Problem 8==
Agnes writes the following four statements on a blank piece of paper.
 
<imath>\bullet</imath> At least one of these statements is true.
 
<imath>\bullet</imath> At least two of these statements are true.
 
<imath>\bullet</imath> At least two of these statements are false.
 
<imath>\bullet</imath> At least one of these statements is false.
 
Each statement is either true or false. How many false statements did Agnes write on the paper?
 
<imath>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4</imath>
 
[[2025 AMC 10A Problems/Problem 8|Solution]]
 
==Problem 9==
Let <imath>f(x)=100x^3-300x^2+200x</imath>. For how many real numbers <imath>a</imath> does the graph of <imath>y=f(x-a)</imath> pass through the point <imath>(1,25)</imath>?
 
<imath>\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{more than } 4</imath>
 
[[2025 AMC 10A Problems/Problem 9|Solution]]
 
==Problem 10==
A semicircle has diameter <imath>\overline{AB}</imath> and chord <imath>\overline{CD}</imath> of length <imath>16</imath> parallel to <imath>\overline{AB}</imath>. A smaller semicircle
with diameter on <imath>\overline{AB}</imath> and tangent to <imath>\overline{CD}</imath> is cut from the larger semicircle, as shown below.
 
<asy>
import graph;
unitsize(14mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
pair A=(-3,0), B=(3,0);
fill(Arc((0,0),3,0,180)--cycle,palered);
fill(Arc((-1.125,0),0.75,0,180)--cycle,white);
draw(Arc((0,0),3,0,180),black);
draw(Arc((-1.125,0),0.75,0,180),black);
draw((-3,0) -- (-1.875,0),black);
draw((-0.375,0) -- (3,0),black);
draw((-2.895, 0.75) -- (2.895,0.75), black);
dot((-3,0));
dot((3,0));
dot((-2.925, 0.75));
dot((2.925, 0.75));
label("$16$",midpoint((-2.925, 0.75)--(2.925, 0.75)),N);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",(-2.925, 0.75),W);
label("$D$",(2.925, 0.75),E);
</asy>
 
 
What is the area of the resulting figure, shown shaded?
 
<imath>\textbf{(A) } 16\pi \qquad\textbf{(B) } 24\pi \qquad\textbf{(C) } 32\pi \qquad\textbf{(D) } 48\pi \qquad\textbf{(E) } 64\pi</imath>
 
[[2025 AMC 10A Problems/Problem 10|Solution]]
 
==Problem 11==
The sequence <imath>1,x,y,z</imath> is arithmetic. The sequence <imath>1,p,q,z</imath> is geometric. Both sequences are strictly increasing and contain only integers, and <imath>z</imath> is as small as possible. What is the value of <imath>x+y+z+p+q</imath>?
 
<imath>\textbf{(A) } 66 \qquad\textbf{(B) } 91 \qquad\textbf{(C) } 103 \qquad\textbf{(D) } 132 \qquad\textbf{(E) } 149</imath>
 
[[2025 AMC 10A Problems/Problem 11|Solution]]
 
==Problem 12==
Carlos uses a <imath>4</imath>-digit passcode to unlock his computer. In his passcode, exactly one digit is even, exactly one (possibly different) digit is prime, and no digit is <imath>0</imath>. How many <imath>4</imath>-digit passcodes satisfy these conditions?
 
<imath>\textbf{(A) } 176 \qquad\textbf{(B) } 192 \qquad\textbf{(C) } 432 \qquad\textbf{(D) } 464 \qquad\textbf{(E) } 608</imath>
 
[[2025 AMC 10A Problems/Problem 12|Solution]]
 
==Problem 13==
In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is <imath>k</imath>, where <imath>0 < k < 1.</imath> The spaces between squares are alternately shaded as shown in the figure (which is not necessarily drawn to scale).
<asy>
unitsize(1cm);
 
int n = 25;             
real s = 6;             
real ratio = 0.767;     
real a = s;             
 
for (int i = 0; i < n; ++i) {
  real b = a * ratio;                 
 
  // Draw current square
  draw(box((-a/2,-a/2),(a/2,a/2)));
 
  if (i % 2 == 1) { fill(box((-a/2,-a/2),(a/2,a/2)), gray(1)); } else
{
    fill(box((-a/2,-a/2),(a/2,a/2)), lightred); 
  }
 
  a = b; 
}
 
draw(box((-a/2,-a/2),(a/2,a/2)));
</asy>
 
The area of the shaded portion of the figure is <imath>64\%</imath> of the area of the original square. What is <imath>k</imath>?
 
<imath>\textbf{(A) } \frac{3}{5} \qquad\textbf{(B) } \frac{16}{25} \qquad\textbf{(C) } \frac{2}{3} \qquad\textbf{(D) } \frac{3}{4} \qquad\textbf{(E) } \frac{4}{5}</imath>
 
[[2025 AMC 10A Problems/Problem 13|Solution]]
 
==Problem 14==
 
Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?
 
<imath>\textbf{(A) } \frac{1}{6} \qquad\textbf{(B) } \frac{1}{5} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{3}{13} \qquad\textbf{(E) } \frac{1}{4}</imath>
 
[[2025 AMC 10A Problems/Problem 14|Solution]]
 
==Problem 15==
In the figure below, <imath>ABEF</imath> is a rectangle, <imath>\overline{AD}\perp\overline{DE}</imath>, <imath>AF=7</imath>, <imath>AB=1</imath>, and <imath>AD=5</imath>.
<asy>
unitsize(1cm);
pair A, B, C, D, E, F;
A = (5, 5);
B = (5.6, 4.2);
C = (5, 3.75);
D = (5, 0);
E = (0, 0);
F = (-0.6, 0.8);
fill(A--B--C--cycle, palered);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
label("$A$", A, N);
label("$B$", B, (1,0));
label("$C$", C, SE);
label("$D$", D, (1,0));
label("$E$", E, S);
label("$F$", F, W);
draw(A--D--E);
draw(A--B--E--F--A);
draw(rightanglemark(C, D, E));
</asy>
What is the area of <imath>\triangle ABC</imath>?
 
<imath>\textbf{(A) } \frac{3}{8} \qquad\textbf{(B) } \frac{4}{9} \qquad\textbf{(C) } \frac{1}{8}\sqrt{13} \qquad\textbf{(D) } \frac{7}{15} \qquad\textbf{(E) } \frac{1}{8}\sqrt{15}</imath>
 
[[2025 AMC 10A Problems/Problem 15|Solution]]
 
==Problem 16==
There are three jars. Each of three coins is placed in one of the three jars, chosen at random and independently of the placements of the other coins. What is the expected number of coins in a jar with the most coins?
 
<imath>\textbf{(A) } \frac{4}{3} \qquad\textbf{(B) } \frac{13}{9} \qquad\textbf{(C) } \frac{5}{3} \qquad\textbf{(D) } \frac{17}{9} \qquad\textbf{(E) } 2</imath>
 
[[2025 AMC 10A Problems/Problem 16|Solution]]
 
==Problem 17==
Let <imath>N</imath> be the unique positive integer such that dividing <imath>273436</imath> by <imath>N</imath> leaves a remainder of <imath>16</imath> and dividing <imath>272760</imath> by <imath>N</imath> leaves a remainder of <imath>15</imath>. What is the tens digit of <imath>N</imath>?
 
<imath>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4</imath>
 
[[2025 AMC 10A Problems/Problem 17|Solution]]
 
==Problem 18==
 
The <imath>\textit{harmonic\ mean}</imath> of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers in the collection. For example, the harmonic mean of 4, 4, and 5 is
 
<cmath>\frac{1}{\frac{1}{3}(\frac{1}{4}+\frac{1}{4}+\frac{1}{5})}=\frac{30}{7}</cmath>
 
What is the harmonic mean of all the real roots of the 4050th degree polynomial
 
<cmath> \prod_{k=1}^{2025} (kx^2-4x-3) = (x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)\dots (2025x^2-4x-3) ?</cmath>
<imath>\textbf{(A) } -\frac{5}{3} \qquad\textbf{(B) } -\frac{3}{2} \qquad\textbf{(C) } -\frac{6}{5} \qquad\textbf{(D) } -\frac{5}{6} \qquad\textbf{(E) } -\frac{2}{3}</imath>
 
[[2025 AMC 10A Problems/Problem 18|Solution]]
 
==Problem 19==
 
An array of numbers is constructed beginning with the numbers
<imath>-1\qquad3\qquad1</imath>in the top row. Each adjacent pair of numbers is summed to produce a number in the next row. Each row begins and ends with
<imath>-1</imath> and <imath>1</imath>, respectively.
 
<cmath>\large{-1}\qquad\large{3}\qquad\large{1}</cmath>
<cmath>\large{-1}\qquad\large{2}\qquad\large{4}\qquad\large{1}</cmath>
<cmath>\large{-1}\qquad\large{1}\qquad\large{6}\qquad\large{5}\qquad\large{1}</cmath>
If the process continues, one of the rows will sum to <imath>12,288</imath>. In that row, what is the third number from the left?
 
<imath>\textbf{(A) } -29 \qquad\textbf{(B) } -21 \qquad\textbf{(C) } -14 \qquad\textbf{(D) } -8 \qquad\textbf{(E) } -3</imath>
 
[[2025 AMC 10A Problems/Problem 19|Solution]]
 
==Problem 20==
 
A silo (right circular cylinder) with diameter 20 meters stands in a field. MacDonald is located 20 meters west and 15 meters south of the center of the silo. McGregor is located 20 meters east and <imath>g > 0</imath> meters south of the center of the silo. The line of sight between MacDonald and McGregor is tangent to the silo. The value of <imath>g</imath> can be written as <imath>\frac{a\sqrt{b}-c}{d}</imath>, where <imath>a,b,c,</imath> and <imath>d</imath> are positive integers, <imath>b</imath> is not divisible by the square of any prime, and <imath>d</imath> is relatively prime to the greatest common divisor of <imath>a</imath> and <imath>c</imath>. What is <imath>a+b+c+d</imath>?
 
<imath>\textbf{(A) } 119 \qquad\textbf{(B) } 120 \qquad\textbf{(C) } 121 \qquad\textbf{(D) } 122 \qquad\textbf{(E) } 123</imath>
 
[[2025 AMC 10A Problems/Problem 20|Solution]]
 
==Problem 21==
A set of numbers is called <imath>sum</imath>-<imath>free</imath> if whenever <imath>x</imath> and <imath>y</imath> are (not necessarily distinct) elements of the set, <imath>x+y</imath> is not an element of the set. For example, <imath>\{1,4,6\}</imath> and the empty set are sum-free, but <imath>\{2,4,5\}</imath> is not. What is the greatest possible number of elements in a sum-free subset of <imath>\{1,2,3,...,20\}</imath>?
 
<imath>\textbf{(A) } 8 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 10 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 12</imath>
 
[[2025 AMC 10A Problems/Problem 21|Solution]]
 
==Problem 22==
A circle of radius <imath>r</imath> is surrounded by three circles, whose radii are 1, 2, and 3, all externally tangent to the inner circle and externally tangent to each other, as shown in the diagram below.
 
<asy>
import olympiad;
size(260);
 
real r0 = 6/23.0;
real r1 = 1.0;
real r2 = 2.0;
real r3 = 3.0;
 
real d1 = r0 + r1;
real d2 = r0 + r2;
real d3 = r0 + r3;
 
real t1 = 0;
real t2 = 1.9857887796653;
real t3 = -2.0480149718113;
 
pair O  = (0,0);
pair C1 = d1*dir(degrees(t1));
pair C2 = d2*dir(degrees(t2));
pair C3 = d3*dir(degrees(t3));
 
draw(circle(O, r0), black+0.9);
draw(circle(C1, r1), black+0.9);
draw(circle(C2, r2), black+0.9);
draw(circle(C3, r3), black+0.9);
currentpicture.fit();
</asy>
 
What is <imath>r</imath>?
 
<imath>\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{6}{23}\qquad\textbf{(C) }\frac{3}{11}\qquad\textbf{(D) }\frac{5}{17}\qquad\textbf{(E) }\frac{3}{10}</imath>
 
[[2025 AMC 10A Problems/Problem 22|Solution]]
 
==Problem 23==
Triangle <imath>\triangle ABC</imath> has side lengths <imath>AB = 80</imath>, <imath>BC = 45</imath>, and <imath>AC = 75</imath>. The bisector of <imath>\angle B</imath> and the altitude to side <imath>\overline{AB}</imath> intersect at point <imath>P.</imath> What is <imath>BP</imath>?
 
<imath>\textbf{(A)}~18\qquad\textbf{(B)}~19\qquad\textbf{(C)}~20\qquad\textbf{(D)}~21\qquad\textbf{(E)}~22</imath>
 
[[2025 AMC 10A Problems/Problem 23|Solution]]
 
==Problem 24==
 
Call a positive integer <imath>\textit{fair}</imath> if no digit is used more than once, it has no 0s, and no digit is adjacent to two greater digits. For example, <imath>196</imath>, <imath>23</imath>, and <imath>12463</imath> are fair, but <imath>1546</imath>, <imath>320</imath>, and <imath>34321</imath> are not fair. How many fair positive integers are there?
<imath>\textbf{(A) } 511 \qquad\textbf{(B) } 2584 \qquad\textbf{(C) } 9841 \qquad\textbf{(D) } 17711 \qquad\textbf{(E) } 19682</imath>
 
[[2025 AMC 10A Problems/Problem 24|Solution]]
 
==Problem 25==
 
A point <imath>P</imath> is chosen at random inside square <imath>ABCD</imath>. The probability that <imath>\overline{AP}</imath> is neither the shortest nor the longest side of <imath>\triangle APB</imath> can be written as <imath>\frac{a + b \pi - c \sqrt{d}}{e}</imath>, where <imath>a, b, c, d, </imath> and <imath>e</imath> are positive integers, <imath>\text{gcd}(a, b, c, e) = 1</imath>, and <imath>d</imath> is not divisible by the square of a prime. What is <imath>a+b+c+d+e</imath>?
 
<imath>\textbf{(A) }25 \qquad \textbf{(B) }26 \qquad \textbf{(C) }27 \qquad \textbf{(D) }28 \qquad \textbf{(E)}  29 \qquad</imath>
 
[[2025 AMC 10A Problems/Problem 25|Solution]]
 
==See Also==
{{AMC10 box|year=2025|ab=A|before=[[2024 AMC 10B Problems]]|after=[[2025 AMC 10B Problems]]}}
{{AMC10 box|year=2025|ab=A|before=[[2024 AMC 10B Problems]]|after=[[2025 AMC 10B Problems]]}}
* [[AMC 10]]
* [[AMC 10]]

Latest revision as of 23:14, 9 November 2025

2025 AMC 10A (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Problem 1

Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at $1{:}30$ traveling due north at a steady $8$ miles per hour. Betsy leaves on her bicycle from the same point at $2{:}30$, traveling due east at a steady $12$ miles per hour. At what time will they be exactly the same distance from their common starting point?

$\textbf{(A) } 3{:}30 \qquad\textbf{(B) } 3{:}45 \qquad\textbf{(C) } 4{:}00 \qquad\textbf{(D) } 4{:}15 \qquad\textbf{(E) } 4{:}30$

Solution

Problem 2

A box contains $10$ pounds of a nut mix that is $50$ percent peanuts, $20$ percent cashews, and $30$ percent almonds. A second nut mix containing $20$ percent peanuts, $40$ percent cashews, and $40$ percent almonds is added to the box resulting in a new nut mix that is $40$ percent peanuts. How many pounds of cashews are now in the box?

$\textbf{(A) } 3.5 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 4.5 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 6$

Solution

Problem 3

How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length $2025$?

$\textbf{(A) } 2025 \qquad\textbf{(B) } 2026 \qquad\textbf{(C) } 3012 \qquad\textbf{(D) } 3037 \qquad\textbf{(E) } 4050$

Solution

Problem 4

A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is $15$. Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from $12$ to $14$. If Ash plays with the teachers, the average age on that team will decrease from $55$ to $52$. How old is Ash?

$\textbf{(A) } 28 \qquad\textbf{(B) } 29 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 32 \qquad\textbf{(E) } 33$

Solution

Problem 5

Consider the sequence of positive integers

\[1,2,1,2,3,2,1,2,3,4,3,2,1,2,3,4,5,4,3,2,1,2,3,4,5,6,5,4,3,2,1,2,\dots\]

What is the 2025th term in this sequence?

$\textbf{(A) } 5 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 44 \qquad\textbf{(E) } 45$

Solution

Problem 6

In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle 20°-angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?

$\textbf{(A) } 80 \qquad\textbf{(B) } 90 \qquad\textbf{(C) } 100 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120$

Solution

Problem 7

Suppose $a$ and $b$ are real numbers. When the polynomial $x^3+x^2+ax+b$ is divided by $x-1$, the remainder is $4$. When the polynomial is divided by $x-2$, the remainder is $6$. What is $b-a$?

$\textbf{(A) } 14 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 17 \qquad\textbf{(E) } 18$

Solution

Problem 8

Agnes writes the following four statements on a blank piece of paper.

$\bullet$ At least one of these statements is true.

$\bullet$ At least two of these statements are true.

$\bullet$ At least two of these statements are false.

$\bullet$ At least one of these statements is false.

Each statement is either true or false. How many false statements did Agnes write on the paper?

$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Solution

Problem 9

Let $f(x)=100x^3-300x^2+200x$. For how many real numbers $a$ does the graph of $y=f(x-a)$ pass through the point $(1,25)$?

$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{more than } 4$

Solution

Problem 10

A semicircle has diameter $\overline{AB}$ and chord $\overline{CD}$ of length $16$ parallel to $\overline{AB}$. A smaller semicircle with diameter on $\overline{AB}$ and tangent to $\overline{CD}$ is cut from the larger semicircle, as shown below.

[asy] import graph; unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(-3,0), B=(3,0); fill(Arc((0,0),3,0,180)--cycle,palered); fill(Arc((-1.125,0),0.75,0,180)--cycle,white); draw(Arc((0,0),3,0,180),black); draw(Arc((-1.125,0),0.75,0,180),black); draw((-3,0) -- (-1.875,0),black); draw((-0.375,0) -- (3,0),black); draw((-2.895, 0.75) -- (2.895,0.75), black); dot((-3,0)); dot((3,0)); dot((-2.925, 0.75)); dot((2.925, 0.75)); label("$16$",midpoint((-2.925, 0.75)--(2.925, 0.75)),N); label("$A$",A,S); label("$B$",B,S); label("$C$",(-2.925, 0.75),W); label("$D$",(2.925, 0.75),E); [/asy]


What is the area of the resulting figure, shown shaded?

$\textbf{(A) } 16\pi \qquad\textbf{(B) } 24\pi \qquad\textbf{(C) } 32\pi \qquad\textbf{(D) } 48\pi \qquad\textbf{(E) } 64\pi$

Solution

Problem 11

The sequence $1,x,y,z$ is arithmetic. The sequence $1,p,q,z$ is geometric. Both sequences are strictly increasing and contain only integers, and $z$ is as small as possible. What is the value of $x+y+z+p+q$?

$\textbf{(A) } 66 \qquad\textbf{(B) } 91 \qquad\textbf{(C) } 103 \qquad\textbf{(D) } 132 \qquad\textbf{(E) } 149$

Solution

Problem 12

Carlos uses a $4$-digit passcode to unlock his computer. In his passcode, exactly one digit is even, exactly one (possibly different) digit is prime, and no digit is $0$. How many $4$-digit passcodes satisfy these conditions?

$\textbf{(A) } 176 \qquad\textbf{(B) } 192 \qquad\textbf{(C) } 432 \qquad\textbf{(D) } 464 \qquad\textbf{(E) } 608$

Solution

Problem 13

In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is $k$, where $0 < k < 1.$ The spaces between squares are alternately shaded as shown in the figure (which is not necessarily drawn to scale). [asy] unitsize(1cm); int n = 25; real s = 6; real ratio = 0.767; real a = s; for (int i = 0; i < n; ++i) { real b = a * ratio; // Draw current square draw(box((-a/2,-a/2),(a/2,a/2))); if (i % 2 == 1) { fill(box((-a/2,-a/2),(a/2,a/2)), gray(1)); } else { fill(box((-a/2,-a/2),(a/2,a/2)), lightred); } a = b; } draw(box((-a/2,-a/2),(a/2,a/2))); [/asy]

The area of the shaded portion of the figure is $64\%$ of the area of the original square. What is $k$?

$\textbf{(A) } \frac{3}{5} \qquad\textbf{(B) } \frac{16}{25} \qquad\textbf{(C) } \frac{2}{3} \qquad\textbf{(D) } \frac{3}{4} \qquad\textbf{(E) } \frac{4}{5}$

Solution

Problem 14

Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?

$\textbf{(A) } \frac{1}{6} \qquad\textbf{(B) } \frac{1}{5} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{3}{13} \qquad\textbf{(E) } \frac{1}{4}$

Solution

Problem 15

In the figure below, $ABEF$ is a rectangle, $\overline{AD}\perp\overline{DE}$, $AF=7$, $AB=1$, and $AD=5$. [asy] unitsize(1cm); pair A, B, C, D, E, F; A = (5, 5); B = (5.6, 4.2); C = (5, 3.75); D = (5, 0); E = (0, 0); F = (-0.6, 0.8); fill(A--B--C--cycle, palered); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); label("$A$", A, N); label("$B$", B, (1,0)); label("$C$", C, SE); label("$D$", D, (1,0)); label("$E$", E, S); label("$F$", F, W); draw(A--D--E); draw(A--B--E--F--A); draw(rightanglemark(C, D, E)); [/asy] What is the area of $\triangle ABC$?

$\textbf{(A) } \frac{3}{8} \qquad\textbf{(B) } \frac{4}{9} \qquad\textbf{(C) } \frac{1}{8}\sqrt{13} \qquad\textbf{(D) } \frac{7}{15} \qquad\textbf{(E) } \frac{1}{8}\sqrt{15}$

Solution

Problem 16

There are three jars. Each of three coins is placed in one of the three jars, chosen at random and independently of the placements of the other coins. What is the expected number of coins in a jar with the most coins?

$\textbf{(A) } \frac{4}{3} \qquad\textbf{(B) } \frac{13}{9} \qquad\textbf{(C) } \frac{5}{3} \qquad\textbf{(D) } \frac{17}{9} \qquad\textbf{(E) } 2$

Solution

Problem 17

Let $N$ be the unique positive integer such that dividing $273436$ by $N$ leaves a remainder of $16$ and dividing $272760$ by $N$ leaves a remainder of $15$. What is the tens digit of $N$?

$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Solution

Problem 18

The $\textit{harmonic\ mean}$ of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers in the collection. For example, the harmonic mean of 4, 4, and 5 is

\[\frac{1}{\frac{1}{3}(\frac{1}{4}+\frac{1}{4}+\frac{1}{5})}=\frac{30}{7}\]

What is the harmonic mean of all the real roots of the 4050th degree polynomial

\[\prod_{k=1}^{2025} (kx^2-4x-3) = (x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)\dots (2025x^2-4x-3) ?\] $\textbf{(A) } -\frac{5}{3} \qquad\textbf{(B) } -\frac{3}{2} \qquad\textbf{(C) } -\frac{6}{5} \qquad\textbf{(D) } -\frac{5}{6} \qquad\textbf{(E) } -\frac{2}{3}$

Solution

Problem 19

An array of numbers is constructed beginning with the numbers $-1\qquad3\qquad1$in the top row. Each adjacent pair of numbers is summed to produce a number in the next row. Each row begins and ends with $-1$ and $1$, respectively.

\[\large{-1}\qquad\large{3}\qquad\large{1}\] \[\large{-1}\qquad\large{2}\qquad\large{4}\qquad\large{1}\] \[\large{-1}\qquad\large{1}\qquad\large{6}\qquad\large{5}\qquad\large{1}\] If the process continues, one of the rows will sum to $12,288$. In that row, what is the third number from the left?

$\textbf{(A) } -29 \qquad\textbf{(B) } -21 \qquad\textbf{(C) } -14 \qquad\textbf{(D) } -8 \qquad\textbf{(E) } -3$

Solution

Problem 20

A silo (right circular cylinder) with diameter 20 meters stands in a field. MacDonald is located 20 meters west and 15 meters south of the center of the silo. McGregor is located 20 meters east and $g > 0$ meters south of the center of the silo. The line of sight between MacDonald and McGregor is tangent to the silo. The value of $g$ can be written as $\frac{a\sqrt{b}-c}{d}$, where $a,b,c,$ and $d$ are positive integers, $b$ is not divisible by the square of any prime, and $d$ is relatively prime to the greatest common divisor of $a$ and $c$. What is $a+b+c+d$?

$\textbf{(A) } 119 \qquad\textbf{(B) } 120 \qquad\textbf{(C) } 121 \qquad\textbf{(D) } 122 \qquad\textbf{(E) } 123$

Solution

Problem 21

A set of numbers is called $sum$-$free$ if whenever $x$ and $y$ are (not necessarily distinct) elements of the set, $x+y$ is not an element of the set. For example, $\{1,4,6\}$ and the empty set are sum-free, but $\{2,4,5\}$ is not. What is the greatest possible number of elements in a sum-free subset of $\{1,2,3,...,20\}$?

$\textbf{(A) } 8 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 10 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 12$

Solution

Problem 22

A circle of radius $r$ is surrounded by three circles, whose radii are 1, 2, and 3, all externally tangent to the inner circle and externally tangent to each other, as shown in the diagram below.

[asy] import olympiad; size(260); real r0 = 6/23.0; real r1 = 1.0; real r2 = 2.0; real r3 = 3.0; real d1 = r0 + r1; real d2 = r0 + r2; real d3 = r0 + r3; real t1 = 0; real t2 = 1.9857887796653; real t3 = -2.0480149718113; pair O = (0,0); pair C1 = d1*dir(degrees(t1)); pair C2 = d2*dir(degrees(t2)); pair C3 = d3*dir(degrees(t3)); draw(circle(O, r0), black+0.9); draw(circle(C1, r1), black+0.9); draw(circle(C2, r2), black+0.9); draw(circle(C3, r3), black+0.9); currentpicture.fit(); [/asy]

What is $r$?

$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{6}{23}\qquad\textbf{(C) }\frac{3}{11}\qquad\textbf{(D) }\frac{5}{17}\qquad\textbf{(E) }\frac{3}{10}$

Solution

Problem 23

Triangle $\triangle ABC$ has side lengths $AB = 80$, $BC = 45$, and $AC = 75$. The bisector of $\angle B$ and the altitude to side $\overline{AB}$ intersect at point $P.$ What is $BP$?

$\textbf{(A)}~18\qquad\textbf{(B)}~19\qquad\textbf{(C)}~20\qquad\textbf{(D)}~21\qquad\textbf{(E)}~22$

Solution

Problem 24

Call a positive integer $\textit{fair}$ if no digit is used more than once, it has no 0s, and no digit is adjacent to two greater digits. For example, $196$, $23$, and $12463$ are fair, but $1546$, $320$, and $34321$ are not fair. How many fair positive integers are there? $\textbf{(A) } 511 \qquad\textbf{(B) } 2584 \qquad\textbf{(C) } 9841 \qquad\textbf{(D) } 17711 \qquad\textbf{(E) } 19682$

Solution

Problem 25

A point $P$ is chosen at random inside square $ABCD$. The probability that $\overline{AP}$ is neither the shortest nor the longest side of $\triangle APB$ can be written as $\frac{a + b \pi - c \sqrt{d}}{e}$, where $a, b, c, d,$ and $e$ are positive integers, $\text{gcd}(a, b, c, e) = 1$, and $d$ is not divisible by the square of a prime. What is $a+b+c+d+e$?

$\textbf{(A) }25 \qquad \textbf{(B) }26 \qquad \textbf{(C) }27 \qquad \textbf{(D) }28 \qquad \textbf{(E)} 29 \qquad$

Solution

See Also

2025 AMC 10A (Problems • Answer Key • Resources)
Preceded by
2024 AMC 10B Problems 		Followed by
2025 AMC 10B Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America.

Retrieved from "https://wiki.aopstest.com/wiki/index.php?title=2025_AMC_10A_Problems&oldid=268219"