Found problems: 23
2022 CMWMC, R8
[u]Set 8[/u]
[b]p22.[/b] For monic quadratic polynomials $P = x^2 + ax + b$ and $Q = x^2 + cx + d$, where $1 \le a, b, c, d \le 10$ are integers, we say that $P$ and $Q$ are friends if there exists an integer $1 \le n \le 10$ such that $P(n) = Q(n)$. Find the total number of ordered pairs $(P, Q)$ of such quadratic polynomials that are friends.
[b]p23.[/b] A three-dimensional solid has six vertices and eight faces. Two of these faces are parallel equilateral triangles with side length $1$, $\vartriangle A_1A_2A_3$ and $\vartriangle B_1B_2B_3$. The other six faces are isosceles right triangles — $\vartriangle A_1B_2A_3$, $\vartriangle A_2B_3A_1$, $\vartriangle A_3B_1A_2$, $\vartriangle B_1A_2B_3$, $\vartriangle B_2A_3B_1$, $\vartriangle B_3A_1B_2$ — each with a right angle at the second vertex listed (so for instace $\vartriangle A_1B_2A_3$ has a right angle at $B_2$). Find the volume of this solid.
[b]p24.[/b] The digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ are each colored red, blue, or green. Find the number of colorings
such that any integer $ n \ge 2$ has that
(a) If $n$ is prime, then at least one digit of $n$ is not blue.
(b) If $n$ is composite, then at least one digit of $n$ is not green.
PS. You should use hide for answers.
2022 CMWMC, R3
[u]Set 3[/u]
[b]p7.[/b] On unit square $ABCD$, a point $P$ is selected on segment $CD$ such that $DP =\frac14$ . The segment $BP$ is drawn and its intersection with diagonal $AC$ is marked as $E$. What is the area of triangle $AEP$?
[b]p8.[/b] Five distinct points are arranged on a plane, creating ten pairs of distinct points. Seven pairs of points are distance $1$ apart, two pairs of points are distance $\sqrt3$ apart, and one pair of points is distance $2$ apart. Draw a line segment from one of these points to the midpoint of a pair of these points. What is the longest this line segment can be?
[b]p9.[/b] The inhabitants of Mars use a base $8$ system. Mandrew Mellon is competing in the annual Martian College Interesting Competition of Math (MCICM). The first question asks to compute the product of the base $8$ numerals $1245415_8$, $7563265_8$, and $ 6321473_8$. Mandrew correctly computed the product in his scratch work, but when he looked back he realized he smudged the middle digit. He knows that the product is $1014133027\blacksquare 27662041138$. What is the missing digit?
PS. You should use hide for answers.
2022 CMWMC, R2
[u]Set 2[/u]
[b]2.1[/b] What is the last digit of $2022 + 2022^{2022} + 2022^{(2022^{2022})}$?
[b]2.2[/b] Let $T$ be the answer to the previous problem. CMIMC executive members are trying to arrange desks for CMWMC. If they arrange the desks into rows of $5$ desks, they end up with $1$ left over. If they instead arrange the desks into rows of $7$ desks, they also end up with $1$ left over. If they instead arrange the desks into rows of $11$ desks, they end up with $T$ left over. What is the smallest possible (non-negative) number of desks they could have?
[b]2.3[/b] Let $T$ be the answer to the previous problem. Compute the largest value of $k$ such that $11^k$ divides $$T! = T(T - 1)(T - 2)...(2)(1).$$
PS. You should use hide for answers.
2022 CMWMC, R1
[u]Set 1
[/u]
[b]1.1[/b] Compute the number of real numbers x such that the sequence $x$, $x^2$, $x^3$,$ x^4$, $x^5$, $...$ eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence $1$, $2$, $3$, $4$, $5$, $6$, $5$, $6$, $5$, $6$, $...$ is eventually repeating with repeating block $5$, $6$.)
[b]1.2[/b] Let $T$ be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply $a$ by $b$, starts by multiplying $a$ by $b$, but then multiplies that product by b again, and then adds $b$ to the result. Nicole inputs the computation “$k \times k$” into the calculator for some real number $k$ and gets an answer of $10T$. If she instead used a working calculator, what answer should she have gotten?
[b]1.3[/b] Let $T$ be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as $x^2 + y^2$ for integers $x, y$ satisfying $\sqrt{T} \le x \le T$ and $\sqrt{T} \le y \le T$.
PS. You should use hide for answers.
2023 CMWMC, R2
[u]Set 2[/u]
[b]2.1[/b] A school has $50$ students and four teachers. Each student has exactly one teacher, such that two teachers have $10$ students each and the other two teachers have $15$ students each. You survey each student in the school, asking the number of classmates they have (not including themself or the teacher). What is the average of all $50$ responses?
[b]2.2[/b] Let $T$ be the answer from the previous problem. A ball is thrown straight up from the ground, reaching (maximum) height $T+1$. Then the ball bounces on the ground and rebounds to height $T-1$. The ball continues bouncing indefinitely, and the height of each bounce is $r$ times the height of the previous bounce for some constant $r$. What is the total vertical distance that the ball travels?
[b]2.3[/b] Let $T$ be the answer from the previous problem. The polynomial equation $$x^3 + x^2 - (T + 1)x + (T- 1) = 0$$
has one (integer) solution for x which does not depend on $T$ and two solutions for $x$ which do depend on $T$. Find the greatest solution for $x$ in this equation. (Hint: Find the independent solution for $x$ while you wait for $T$.)
PS. You should use hide for answers.
2023 CMWMC, R6
[b]p16.[/b] Let $P(x)$ be a quadratic such that $P(-2) = 10$, $P(0) = 5$, $P(3) = 0$. Then, find the sum of the coefficients of the polynomial equal to $P(x)P(-x)$.
[b]p17.[/b] Suppose that $a < b < c < d$ are positive integers such that the pairwise differences of $a, b, c, d$ are all distinct, and $a + b + c + d$ is divisible by $2023$. Find the least possible value of $d$.
[b]p18.[/b] Consider a right rectangular prism with bases $ABCD$ and $A'B'C'D'$ and other edges $AA'$, $BB'$, $CC'$ and $DD'$. Suppose $AB = 1$, $AD = 2$, and $AA' = 1$.
$\bullet$ Let $X$ be the plane passing through $A$, $C'$, and the midpoint of $BB'$.
$\bullet$ Let $Y$ be the plane passing through $D$, $B'$, and the midpoint of $CC'$.
Then the intersection of $X$, $Y$ , and the prism is a line segment of length $\ell$. Find $\ell$.
PS. You should use hide for answers.
2023 CMWMC, R4
[u]Set 4 [/u]
[b]4.1[/b] Triangle $T$ has side lengths $1$, $2$, and $\sqrt7$. It turns out that one can arrange three copies of triangle $T$ to form two equilateral triangles, one inside the other, as shown below. Compute the ratio of the area of the outer equilaterial triangle to the area of the inner equilateral triangle.
[img]https://cdn.artofproblemsolving.com/attachments/0/a/4a3bcf4762b97501a9575fc6972e234ffa648b.png[/img]
[b]4.2[/b] Let $T$ be the answer from the previous problem. The diagram below features two concentric circles of radius $1$ and $T$ (not necessarily to scale). Four equally spaced points are chosen on the smaller circle, and rays are drawn from these points to the larger circle such that all of the rays are tangent to the smaller circle and no two rays intersect. If the area of the shaded region can be expressed as $k\pi$ for some integer $k$, find $k$.
[img]https://cdn.artofproblemsolving.com/attachments/a/5/168d1aa812210fd9d60a3bb4a768e8272742d7.png[/img]
[b]4.3[/b] Let $T$ be the answer from the previous problem. $T^2$ congruent squares are arranged in the configuration below (shown for $T = 3$), where the squares are tilted in alternating fashion such that they form congruent rhombuses between them. If all of the rhombuses have long diagonal twice the length of their short diagonal, compute the ratio of the total area of all of the rhombuses to the total area of all of the squares. (Hint: Rather than waiting for $T$, consider small cases and try to find a general formula in terms of $T$, such a formula does exist.)
[img]https://cdn.artofproblemsolving.com/attachments/1/d/56ef60c47592fa979bfedd782e5385e7d139eb.png[/img]
PS. You should use hide for answers.
2022 CMWMC, R5
[u]Set 5[/u]
[b]p13.[/b] An equiangular $12$-gon has side lengths that alternate between $2$ and $\sqrt3$. Find the area of the circumscribed circle of this $12$-gon.
[b]p14.[/b] For positive integers $n$, let $\sigma(n)$ denote the number of positive integer factors of $n$. Then $\sigma(17280) = \sigma (2^7 \cdot 3^3 \cdot 5)= 64$. Let $S$ be the set of factors $k$ of $17280$ such that $\sigma(k) = 32$. If $p$ is the product of the elements of $S$, find $\sigma(p)$.
[b]p15.[/b] How many odd $3$-digit numbers have exactly four $1$’s in their binary (base $2$) representation? For example, $225_{10} = 11100001_2$ would be valid.
PS. You should use hide for answers.
2023 CMWMC, R7
[b]p19.[/b] Sequences $a_n$ and $b_n$ of positive integers satisfy the following properties:
(1) $a_1 = b_1 = 1$
(2) $a_5 = 6, b_5 \ge 7$
(3) Both sequences are strictly increasing
(4) In each sequence, the difference between consecutive terms is either $1$ or $2$
(5) $\sum^5_{n=1}na_n =\sum^5_{n=1}nb_n = S$
Compute $S$.
[b]p20.[/b] Let $A$, $B$, and $C$ be points lying on a line in that order such that $AB = 4$ and $BC = 2$. Let $I$ be the circle centered at B passing through $C$, and let $D$ and $E$ be distinct points on $I$ such that $AD$ and $AE$ are tangent to $I$. Let $J$ be the circle centered at $C$ passing through $D$, and let $F$ and $G$ be distinct points on $J$ such that $AF$ and $AG$ are tangent to $J$ and $DG < DF$. Compute the area of quadrilateral $DEFG$.
[b]p21.[/b] Twain is walking randomly on a number line. They start at $0$, and flip a fair coin $10$ times. Every time the coin lands heads, they increase their position by 1, and every time the coin lands tails, they decrease their position by $1$. What is the probability that at some point the absolute value of their position is at least $3$?
PS. You should use hide for answers.
2023 CMWMC, R3
[b]p7.[/b] Let $A, B, C$, and $D$ be equally spaced points on a circle $O$. $13$ circles of equal radius lie inside $O$ in the configuration below, where all centers lie on $\overline{AC}$ or $\overline{BD}$, adjacent circles are externally tangent, and the outer circles are internally tangent to $O$. Find the ratio of the area of the region inside $O$ but outside the smaller circles to the total area of the smaller circles.
[img]https://cdn.artofproblemsolving.com/attachments/9/7/7ff192baf58f40df0e4cfae4009836eab57094.png[/img]
[b]p8.[/b] Find the greatest divisor of $40!$ that has exactly three divisors.
[b]p9.[/b] Suppose we have positive integers $a, b, c$ such that $a = 30$, lcm $(a, b) = 210$, lcm $(b, c) = 126$. What is the minimum value of lcm $(a, c)$?
PS. You should use hide for answers.
2022 CMWMC, R1
[u]Set 1[/u]
[b]p1.[/b] Assume the speed of sound is $343$ m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters?
[b]p2.[/b] Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get?
[b]p3.[/b] Twain is trying to crack a $4$-digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain $10$ seconds to enter a combination, how many hours would it take them to try every possible combination that satisfies these rules?
PS. You should use hide for answers.
2023 CMWMC, R1
[b]p1.[/b] Sherry starts with a three-digit positive integer. She subtracts $7$ from it, then multiplies the result by $7$, and then adds $7$ to that. If she ends up with $2023$, what number did she start with?
[b]p2.[/b] Square $ABCD$ has side length $1$. Point $X$ lies on $\overline{AB}$ such that $\frac{AX}{XB} = 2$, and point $Y$ lies on $\overline{DX}$ such that $\frac{DY}{YX} = 3$. Compute the area of triangle $DAY$ .
[b]p3.[/b] A fair six-sided die is labeled $1-6$ such that opposite faces sum to $7$. The die is rolled, but before you can look at the outcome, the die gets tipped over to an adjacent face. If the new face shows a $4$, what is the probability the original roll was a $1$?
PS. You should use hide for answers.
2023 CMWMC, R1
[u]Set 1[/u]
[b]1.1[/b] How many positive integer divisors are there of $2^2 \cdot 3^3 \cdot 5^4$?
[b]1.2[/b] Let $T$ be the answer from the previous problem. For how many integers $n$ between $1$ and $T$ (inclusive) is $\frac{(n)(n - 1)(n - 2)}{12}$ an integer?
[b]1.3[/b] Let $T$ be the answer from the previous problem. Find $\frac{lcm(T, 36)}{gcd(T, 36)}$.
PS. You should use hide for answers.
2022 CMWMC, R7
[u]Set 7[/u]
[b]p19.[/b] The polynomial $x^4 + ax^3 + bx^2 - 32x$, where$ a$ and $b$ are real numbers, has roots that form a square in the complex plane. Compute the area of this square.
[b]p20.[/b] Tetrahedron $ABCD$ has equilateral triangle base $ABC$ and apex $D$ such that the altitude from $D$ to $ABC$ intersects the midpoint of $\overline{BC}$. Let $M$ be the midpoint of $\overline{AC}$. If the measure of $\angle DBA$ is $67^o$, find the measure of $\angle MDC$ in degrees.
[b]p21.[/b] Last year’s high school graduates started high school in year $n- 4 = 2017$, a prime year. They graduated high school and started college in year $n = 2021$, a product of two consecutive primes. They will graduate college in year $n + 4 = 2025$, a square number. Find the sum of all $n < 2021$ for which these three properties hold. That is, find the sum of those $n < 2021$ such that $n -4$ is prime, n is a product of two consecutive primes, and $n + 4$ is a square.
PS. You should use hide for answers.
2023 CMWMC, R2
[b]p4.[/b] What is gcd $(2^6 - 1, 2^9 - 1)$?
[b]p5.[/b] Sarah is walking along a sidewalk at a leisurely speed of $\frac12$ m/s. Annie is some distance behind her, walking in the same direction at a faster speed of $s$ m/s. What is the minimum value of $s$ such that Sarah and Annie spend no more than one second within one meter of each other?
[b]p6.[/b] You have a choice to play one of two games. In both games, a coin is flipped four times. In game $1$, if (at least) two flips land heads, you win. In game $2$, if (at least) two consecutive flips land heads, you win. Let $N$ be the number of the game that gives you a better chance of winning, and let $p$ be the absolute difference in the probabilities of winning each game. Find $N + p$.
PS. You should use hide for answers.
2023 CMWMC, R3
[u]Set 3[/u]
[b]3.1[/b] Find the number of distinct values that can be made by inserting parentheses into the expression
$$1\,\,\,\,\, - \,\,\,\,\, 1 \,\,\,\,\, -\,\,\,\,\, 1 \,\,\,\,\, - \,\,\,\,\, 1 \,\,\,\,\, - \,\,\,\,\, 1\,\,\,\,\, - \,\,\,\,\, 1$$
such that you don’t introduce any multiplication. For example, $(1-1)-((1-1)-1-1)$ is a valid way to insert parentheses, but $1 - 1(-1 - 1) - 1 - 1$ is not.
[b]3.2[/b] Let $T$ be the answer from the previous problem. Katie rolls T fair 4-sided dice with faces labeled $0-3$. Considering all possible sums of these rolls, there are two sums that have the highest probability of occurring. Find the smaller of these two sums.
[b]3.3[/b] Let $T$ be the answer from the previous problem. Amy has a fair coin that she will repeatedly flip until her total number of heads is strictly greater than her total number of tails. Find the probability she will flip the coin exactly T times. (Hint: Finding a general formula in terms of T is hard, try solving some small cases while you wait for $T$.)
PS. You should use hide for answers.
2022 CMWMC, R6
[u]Set 6[/u]
[b]p16.[/b] Let $x$ and $y$ be non-negative integers. We say point $(x, y)$ is square if $x^2 + y$ is a perfect square. Find the sum of the coordinates of all distinct square points which also satisfy $x^2 + y \le 64$.
[b]p17.[/b] Two integers $a$ and $b$ are randomly chosen from the set $\{1, 2, 13, 17, 19, 87, 115, 121\}$, with $a > b$. What is the expected value of the number of factors of $ab$?
[b]p18.[/b] Marnie the Magical Cello is jumping on nonnegative integers on number line. She starts at $0$ and jumps following two specific rules. For each jump she can either jump forward by $1$ or jump to the next multiple of $4$ (the next multiple must be strictly greater than the number she is currently on). How many ways are there for her to jump to $2022$? (Two ways are considered distinct only if the sequence of numbers she lands on is different.)
PS. You should use hide for answers.
2022 CMWMC, R4
[u]Set 4[/u]
[b]4.1[/b] Quadrilateral $ABCD$ (with $A, B, C$ not collinear and $A, D, C$ not collinear) has $AB = 4$, $BC = 7$, $CD = 10$, and $DA = 5$. Compute the number of possible integer lengths $AC$.
[img]https://cdn.artofproblemsolving.com/attachments/1/6/4f43873a64bc00a0e6173002ccd80e8f1529a9.png[/img]
[b]4.2[/b] Let $T$ be the answer from the previous part. $2T$ congruent isosceles triangles with base length $b$ and leg length $\ell$ are arranged to form a parallelogram as shown below (not necessarily the correct number of triangles). If the total length of all drawn line segments (not double counting overlapping sides) is exactly three times the perimeter of the parallelogram, find $\frac{\ell}{b}$.
[img]https://cdn.artofproblemsolving.com/attachments/5/c/744f503ed822bc43acafe2633e6108022f2c88.png[/img]
[b]4.3[/b] Let $T$ be the answer from the previous part. Rectangle $R$ has length $T$ times its width. $R$ is inscribed in a square $S$ such that the diagonals of $ S$ are parallel to the sides of $R$. What proportion of the area of $S$ is contained within $R$?
[img]https://cdn.artofproblemsolving.com/attachments/a/1/0928dd1ffbeb4d7dee9b697fdb7696cc70c03d.png[/img]
PS. You should use hide for answers.
2022 CMWMC, R3
[u]Set 3[/u]
[b]3.1[/b] Annie has $24$ letter tiles in a bag; $8$ C’s, $8$ M’s, and $8$ W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw?
[b]3.2[/b] Let $T$ be the answer from the previous problem. Charlotte is initially standing at $(0, 0)$ in the coordinate plane. She takes $T$ steps, each of which moves her by $1$ unit in either the $+x$, $-x$, $+y$, or $-y$ direction (e.g. her first step takes her to $(1, 0)$, $(1, 0)$, $(0, 1)$ or $(0, -1)$). After the T steps, how many possibilities are there for Charlotte’s location?
[b]3.3[/b] Let $T$ be the answer from the previous problem, and let $S$ be the sum of the digits of $T$. Francesca has an unfair coin with an unknown probability $p$ of landing heads on a given flip. If she flips the coin $S$ times, the probability she gets exactly one head is equal to the probability she gets exactly two heads. Compute the probability $p$.
PS. You should use hide for answers.
2022 CMWMC, R4
[u]Set 4[/u]
[b]p10.[/b] Eve has nine letter tiles: three $C$’s, three $M$’s, and three $W$’s. If she arranges them in a random order, what is the probability that the string “$CMWMC$” appears somewhere in the arrangement?
[b]p11.[/b] Bethany’s Batteries sells two kinds of batteries: $C$ batteries for $\$4$ per package, and $D$ batteries for $\$7$ per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly $\$2021$, and no two of them purchased the same quantities of both types of battery. Bethany also notes that if any other customer had come, at least one of these two conditions would’ve had to fail. How many packages of batteries did Bethany sell?
[b]p12.[/b] A deck of cards consists of $30$ cards labeled with the integers $1$ to $30$, inclusive. The cards numbered $1$ through $15$ are purple, and the cards numbered $16$ through $30$ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $32$. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck.
PS. You should use hide for answers.
2023 CMWMC, R8
[b]p22.[/b] Find the unique ordered pair $(m, n)$ of positive integers such that $x = \sqrt[3]{m} -\sqrt[3]{n}$ satisfies $x^6 + 4x^3 - 36x^2 + 4 = 0$.
[b]p23.[/b] Jenny plays with a die by placing it flat on the ground and rolling it along any edge for each step. Initially the face with $1$ pip is face up. How many ways are there to roll the dice for $6$ steps and end with the $1$ face up again?
[b]p24.[/b] There exists a unique positive five-digit integer with all odd digits that is divisible by $5^5$. Find this integer.
PS. You should use hide for answers.
2023 CMWMC, R4
[b]p10.[/b] Square $ABCD$ has side length $n > 1$. Points $E$ and $F$ lie on $\overline{AB}$ and $\overline{BC}$ such that $AE = BF = 1$. Suppose $\overline{DE}$ and $\overline{AF}$ intersect at $X$ and $\frac{AX}{XF} = \frac{11}{111}$ . What is $n$?
[b]p11.[/b] Let $x$ be the positive root of $x^2 - 10x - 10 = 0$. Compute $\frac{1}{20}x^4 - 6x^2 - 45$.
[b]p12.[/b] Francesca has $7$ identical marbles and $5$ distinctly labeled pots. How many ways are there for her to distribute at least one (but not necessarily all) of the marbles into the pots such that at most two pots are nonempty?
PS. You should use hide for answers.
2022 CMWMC, R2
[u]Set 2[/u]
[b]p4.[/b] $\vartriangle ABC$ is an isosceles triangle with $AB = BC$. Additionally, there is $D$ on $BC$ with $AC = DA = BD = 1$. Find the perimeter of $\vartriangle ABC$.
[b]p5[/b]. Let $r$ be the positive solution to the equation $100r^2 + 2r - 1 = 0$. For an appropriate $A$, the infinite series $Ar + Ar^2 + Ar^3 + Ar^4 +...$ has sum $1$. Find $A$.
[b]p6.[/b] Let $N(k)$ denote the number of real solutions to the equation $x^4 -x^2 = k$. As $k$ ranges from $-\infty$ to $\infty$, the value of $N(k)$ changes only a finite number of times. Write the sequence of values of $N(k)$ as an ordered tuple (i.e. if $N(k)$ went from $1$ to $3$ to $2$, you would write $(1, 3, 2)$).
PS. You should use hide for answers.