Free AMC 10 Sample Mock

Problems

1. Problem 1.
   A bag contains four green balls and five purple balls. Balls are randomly drawn one by one out of the bag until there are none left in the bag. What is the probability that the seventh ball drawn is green?
   (A)$\dfrac{1}{9}$(B)$\dfrac{4}{9}$(C)$\dfrac{5}{9}$(D)$\dfrac{2}{3}$(E)$\dfrac{7}{9}$
   Check answer
2. Problem 2.
   If $f(x)=2x+3$ and $f(g(x))=x$, what is the value of $g(5)$?
   (A)$1$(B)$2$(C)$3$(D)$4$(E)$5$
   Check answer
3. Problem 3.
   Two sides of a non-degenerate triangle are 5 and 9. If the third side is an integer, what is the maximum possible perimeter of the triangle?
   (A)$23$(B)$25$(C)$26$(D)$27$(E)$28$
   Check answer
4. Problem 4.
   At a local school, the ratio of teachers to students is initially $1:15$. After 2 teachers retire and 20 new students enroll, the new ratio of teachers to students becomes $1:20$. What is the total number of teachers and students currently at the school?
   (A)$192$(B)$200$(C)$210$(D)$220$(E)$240$
   Check answer
5. Problem 5.
   A right triangle has integer side lengths. If one of its legs has a length of 15, which of the following is NOT a possible perimeter for the triangle?
   (A)$40$(B)$60$(C)$80$(D)$90$(E)$240$
   Check answer
6. Problem 6.
   Four friends, $A$, $B$, $C$, and $D$, finish a race with no ties. $A$ says, "I did not finish first." $B$ says, "I finished third." $C$ says, "$A$ finished behind $D$." $D$ says, "I finished last." If exactly one person is lying, who finished first?
   (A)$A$(B)$B$(C)$C$(D)$D$(E)$\text{Cannot be determined}$
   Check answer
7. Problem 7.
   A sequence of 5 consecutive positive integers has the property that the sum of the squares of the three smallest integers equals the sum of the squares of the two largest integers. What is the smallest integer in this sequence?
   (A)$8$(B)$9$(C)$10$(D)$11$(E)$12$
   Check answer
8. Problem 8.
   A committee of $4$ people is to be chosen from a group of $6$ married couples. The committee must contain exactly one complete married couple. Furthermore, one person on the committee must be named President, and one person (different from the President) named Vice President. If the President and Vice President cannot be married to each other, how many such distinct committee structures can be formed?
   (A)$1920$(B)$2160$(C)$2400$(D)$2880$(E)$3120$
   Check answer
9. Problem 9.
   How many three-digit positive integers have the property that the product of their digits is exactly 24?
   (A)$15$(B)$18$(C)$21$(D)$24$(E)$27$
   Check answer
10. Problem 10.
    A ball is to be thrown into one of ten bins, numbered 1 through 10 in that order, standing in a row. The probability of the ball being thrown into bin $n$ is denoted by $p_n$. The probability the ball is thrown in the first or last bin is $p$ (so $p_1 = p_{10} = p$). The probability of the ball being thrown into bin $n$ for $1 < n < 10$ is given by $p_n = \dfrac{p_{n-1}+p_{n+1}}{2}$. What is $p$?
    (A)$\dfrac{1}{20}$(B)$\dfrac{1}{10}$(C)$\dfrac{1}{9}$(D)$\dfrac{2}{9}$(E)$\dfrac{1}{5}$
    Check answer
11. Problem 11.
    In right triangle $ABC$, the right angle is at $B$, leg $AB$ has length 6, and leg $BC$ has length 8. A square is inscribed in the triangle such that one vertex of the square coincides with $B$, and the opposite vertex of the square lies on the hypotenuse $AC$. What is the side length of the square?
    (A)$3$(B)$\dfrac{12}{5}$(C)$\dfrac{24}{7}$(D)$\dfrac{18}{5}$(E)$4$
    Check answer
12. Problem 12.
    A bag contains 8 tokens, numbered 1 through 8. Tokens are drawn one at a time without replacement, and a running sum of the drawn numbers is recorded. What is the probability that the running sum becomes a multiple of 3 for the very first time on exactly the fourth draw?
    (A)$\dfrac{11}{140}$(B)$\dfrac{3}{35}$(C)$\dfrac{13}{140}$(D)$\dfrac{1}{10}$(E)$\dfrac{3}{28}$
    Check answer
13. Problem 13.
    How many ordered pairs of positive integers $(x, y)$ satisfy the equation $xy - 3x + 4y = 2026$?
    (A)$2$(B)$4$(C)$6$(D)$8$(E)$12$
    Check answer
14. Problem 14.
    How many five-digit palindromes (numbers that read the same backward as forward, such as 25352) are divisible by 11?
    (A)$72$(B)$80$(C)$82$(D)$90$(E)$100$
    Check answer
15. Problem 15.
    Let $a$ and $b$ be positive integers such that $a, b \le 10000$ and

    $$\operatorname{lcm}(a, b) = 74 \cdot \operatorname{gcd}(a, b)$$

    What is the sum of all possible maximum powers of 17 that divide $ab$?
    (A)$6$(B)$8$(C)$10$(D)$12$(E)$14$
    Check answer
16. Problem 16.
    Let $P(x)$ be a polynomial of degree 4 such that $P(k) = \dfrac{k}{k+1}$ for the integers $k=0,1,2,3,4$. What is the value of $P(5)$?
    (A)$\dfrac{1}{2}$(B)$\dfrac{3}{5}$(C)$\dfrac{2}{3}$(D)$\dfrac{5}{6}$(E)$1$
    Check answer
17. Problem 17.
    What is the number of integers $n \in \{1, 2, 3, \dots, 2026\}$ such that $n^n+1$ is a multiple of 5?
    (A)$404$(B)$405$(C)$406$(D)$412$(E)$506$
    Check answer
18. Problem 18.
    A frog sits at one vertex of a regular cube. Every second, it randomly jumps to one of the three adjacent vertices, choosing each with a probability of $\dfrac{1}{3}$. What is the probability that the frog is exactly back at its starting vertex after exactly 8 seconds?
    (A)$\dfrac{182}{729}$(B)$\dfrac{547}{2187}$(C)$\dfrac{183}{729}$(D)$\dfrac{551}{2187}$(E)$\dfrac{61}{243}$
    Check answer
19. Problem 19.
    Let $ABCDEF$ be a regular hexagonal sheet of paper of side length 2. Points $C$ and $F$ are pinned to the ground while $A$ and $B$ are lifted until the trapezoid $ABCF$ is perpendicular to the trapezoid $CDEF$. Points $A$ and $E$ are connected, as are $B$ and $D$. The volume of the polyhedron contained within the faces $AEF$, $ABCD$, $BCD$, $ABCF$, and $CDEF$ can be written in the form $\dfrac{m\sqrt{p}}{q}$ where $p$ is not divisible by the square of any prime, and $m$ and $q$ are relatively prime positive integers. What is $m+p+q$?
    (A)$10$(B)$11$(C)$12$(D)$13$(E)$14$
    Check answer
20. Problem 20.
    Suppose there are 77 cupcakes and 22 cookies from which you must choose exactly 12 desserts consisting of any amount of cupcakes and cookies (even 12 of one dessert and 0 of the other is permitted). Let $N$ be the number of ways you can do this. What are the last two digits of $N$?
    (A)$16$(B)$36$(C)$56$(D)$76$(E)$96$
    Check answer
21. Problem 21.
    Three distinct positive integers $x$, $y$, and $z$ are randomly selected from the set $\{1, 2, 3, \dots, 50\}$. What is the probability that $x^3+y^3+z^3-3xyz=7(x+y+z)$?
    (A)$\dfrac{47}{9800}$(B)$\dfrac{47}{4900}$(C)$\dfrac{141}{9800}$(D)$\dfrac{94}{4900}$(E)$\dfrac{47}{2450}$
    Check answer
22. Problem 22.
    A right triangle $T$ on the Cartesian plane has vertices at $A(0,0)$, $B(21,0)$, and $C(0,28)$. A point $P$ is chosen uniformly at random from the set of all lattice points (points with integer coordinates) that lie strictly inside $T$. What is the probability that the area of triangle $PAB$ is an even integer?
    (A)$\dfrac{17}{89}$(B)$\dfrac{19}{89}$(C)$\dfrac{21}{89}$(D)$\dfrac{57}{269}$(E)$\dfrac{61}{269}$
    Check answer
23. Problem 23.
    Let $n$ be the least positive integer such that the equation

    $$\lfloor x^2+6 \rfloor = nx+1$$

    has the maximum possible number of solutions it can. What is the remainder when $n^{100}$ is divided by 13?
    (A)$1$(B)$3$(C)$4$(D)$7$(E)$9$
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24. Problem 24.
    Let $P(x)=x^4+ax^2+b$ be a polynomial with integer coefficients. It is given that $P(x)$ has exactly four distinct integer roots. If $P(25)=230400$, what is the sum of the absolute values of the four roots of $P(x)$?
    (A)$36$(B)$40$(C)$44$(D)$48$(E)$52$
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25. Problem 25.
    Let a square $S$ on the Cartesian plane have vertices $(0, 0)$, $(2, 0)$, $(2, 2)$, and $(0, 2)$, and let a square $T$ on the Cartesian plane have vertices $(2, 4)$, $(4, 4)$, $(4, 6)$, and $(2, 6)$. Square $S$ will undergo a sequence of four transformations, each being either a $45^\circ$, $90^\circ$, $135^\circ$, $180^\circ$, $225^\circ$, $270^\circ$, or $315^\circ$ rotation clockwise about the point $(0,4)$. What is the probability that at the end of this sequence of transformations, $S$ is mapped to $T$?
    (A)$\dfrac{285}{2401}$(B)$\dfrac{292}{2401}$(C)$\dfrac{300}{2401}$(D)$\dfrac{308}{2401}$(E)$\dfrac{315}{2401}$
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