Free AMC 12 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.
   Let $p(x)$ be a polynomial $ax^2 + bx + c$ where $a$ is chosen from $\{3, 4, 5, 6\}$, $b$ is chosen from $\{2, 3, 4\}$, and $c$ is chosen from $\{2, 3, 4, 5, 6\}$. How many such polynomials $p(x)$ have at least one rational root?
   (A)$0$(B)$1$(C)$3$(D)$5$(E)$12$
   Check answer
5. Problem 5.
   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)$\text{Cannot be determined}$
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6. Problem 6.
   Three triangles can be joined such that one edge of each triangle coincides with one edge of each of the other triangles. What is true about the three triangles?
   (A)$\text{The sum of their smallest angles is } 180^\circ.$(B)$\text{Exactly one triangle is obtuse.}$(C)$\text{At least two triangles must be obtuse.}$(D)$\text{None of these is necessarily true.}$(E)$\text{Exactly two triangles are right triangles.}$
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7. Problem 7.
   A baker sells cookies in boxes of $a$ and $b$, where $a$ and $b$ are relatively prime positive integers and $a < b$. The largest number of cookies that cannot be purchased exactly using a combination of these boxes is $143$. What is the minimum possible value of $a+b$?
   (A)$27$(B)$28$(C)$32$(D)$42$(E)$53$
   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.
   Let $ABCDEFGH$ be a regular octagon with side length $2$. Let $X$ be the intersection of diagonals $AC$ and $BE$. What is the area of $\triangle ABX$?
   (A)$\sqrt{2}-1$(B)$2\sqrt{2}-2$(C)$2$(D)$2\sqrt{2}$(E)$4\sqrt{2}-4$
   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.
    The sum of all real numbers $x$ that satisfy the following equation:

    $$\log_{2x}(8) + \log_4(x^2) = \dfrac{11}{2}$$

    can be written in the form $\dfrac{m + \sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n + p$.
    (A)$64$(B)$66$(C)$68$(D)$70$(E)$72$
    Check answer
12. Problem 12.
    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
13. Problem 13.
    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
14. Problem 14.
    A parabola has its vertex at $(-3,-3)$. Its focus, a point on the parabola, and a point on the directrix the same distance from the point as the focus, can be connected to form an equilateral triangle of side length $3$. What is the positive difference between the roots of the equation represented by the parabola?
    (A)$3$(B)$6$(C)$9$(D)$12$(E)$15$
    Check answer
15. Problem 15.
    Let $f(k) = 1 + \left(1 - \dfrac{k}{50}\right)i$. Let $\arg(z)$ be the argument of the complex number $z$ in the range $[0, 2\pi)$. What is $\arg(f(0)f(1)f(2)\dots f(100))$?
    (A)$0$(B)$\dfrac{\pi}{4}$(C)$\dfrac{\pi}{2}$(D)$\pi$(E)$\dfrac{3\pi}{2}$
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16. Problem 16.
    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
17. Problem 17.
    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
18. Problem 18.
    A right circular hollow cone with radius $\sqrt{3}$ and height $3$, without a base, opens upwards, and a sphere of radius $2$ is placed inside the cone. A smaller cone is placed inside the section of the sphere, horizontally laying, bounded upwards by the base of the larger cone (if it had a base). The apex of this smaller cone is equidistant from the bottom of the sphere and a point on the circumference of the base of the larger cone. The base of this smaller cone is tangent to the base of the large cone and the sphere's surface below it. The volume of this small cone can be written as

    $$(\sqrt{3}-1)\pi \cdot \sqrt{m - n\sqrt{p}}$$

    where $m, n, p$ are positive integers such that $p$ is not divisible by the square of any prime. What is $m + n + p$?
    (A)$5$(B)$6$(C)$7$(D)$8$(E)$9$
    Check answer
19. Problem 19.
    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
20. Problem 20.
    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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21. Problem 21.
    A sequence of real numbers $\{x_n\}$ is defined by $x_1 = 2$, $x_2 = \sqrt{3} - 1$, and for all integers $n \ge 2$:

    $$x_{n+1} = \sqrt{3} x_n - x_{n-1}$$

    Evaluate the series:

    $$S = \sum_{n=1}^{23} \dfrac{1}{x_n x_{n+1}}$$
    (A)$\dfrac{1 - \sqrt{3}}{2}$(B)$\dfrac{2 - \sqrt{3}}{4}$(C)$\dfrac{1 - \sqrt{3}}{4}$(D)$\dfrac{\sqrt{3} - 1}{4}$(E)$\dfrac{1 + \sqrt{3}}{4}$
    Check answer
22. Problem 22.
    The sum

    $$\binom{2027}{0} + \binom{2027}{4} + \binom{2027}{8} + \dots + \binom{2027}{2024}$$

    can be written in the form $2^p - 2^q$ where $p$ and $q$ are positive integers. What is $p + q$?
    (A)$3031$(B)$3033$(C)$3035$(D)$3036$(E)$3037$
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23. Problem 23.
    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)$\dfrac{315}{2401}$
    Check answer
24. Problem 24.
    Let $r_1, r_2, \dots, r_{100}$ be a permutation of the 100 distinct roots of the equation $x^{100} - 1 = 0$. Consider the complex number $P$ defined by the nested conjugation expression:

    $$P = \overline{r_{100} \overline{r_{99} \overline{\dots \overline{r_2 \overline{r_1}}}}}$$

    Each root $r_k$ can be uniquely expressed in the form $e^{\frac{v_k \pi i}{50}}$, where $(v_1, v_2, \dots, v_{100})$ is a permutation of the integers $\{1, 2, \dots, 100\}$. Let $S$ be the sum of the exponents corresponding to the odd-indexed roots:

    $$S = v_1 + v_3 + v_5 + \dots + v_{99}$$

    Across all possible permutations of the roots of unity, the imaginary part of $P$ attains a maximum possible value. For exactly how many distinct values of $S$ is this maximum imaginary part achieved?
    (A)$49$(B)$50$(C)$99$(D)$124$(E)$100$
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25. Problem 25.
    Let $\mathcal{P}$ be a regular 2027-gon inscribed in a circle of radius 1. Let $A$ and $B$ be two vertices of $\mathcal{P}$ separated by exactly one intermediate vertex along the circumcircle. Let $x_1 = \dfrac{1}{4}(AB)^2$. For all integers $n \ge 1$, a sequence is defined by the recurrence relation:

    $$x_{n+1} = 4x_n(1 - x_n)$$

    Evaluate the sum:

    $$S = \sum_{n=1}^{1013} x_n$$
    (A)$\dfrac{1013}{2}$(B)$\dfrac{2025}{4}$(C)$\dfrac{1014}{2}$(D)$\dfrac{2027}{4}$(E)$\dfrac{2029}{4}$
    Check answer