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# Question
"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x, y <m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\equiv x^2\pmod 5$ would consist of the ... |
# Question
"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x, y <m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\equiv x^2\pmod 5$ would consist of the ... |
# Question
"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x<m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\equiv x^2\pmod 5$ would consist of the poin... |
# Question
"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x<m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\equiv x^2\pmod 5$ would consist of the poin... |
# Question
$100_{10}$ in base $b$ has exactly $5$ digits. What is the value of $b$?# Solution
[TURN] [STEP] 0. I need to find a base $b$ such that $100_{10}$ can be expressed as a $5$-digit number in base $b$. [/STEP]
[STEP] 1. That means that $100_{10} = a_4 b^4 + a_3 b^3 + a_2 b^2 + a_1 b + a_0$, where $a_4, a_3,... |
# Question
$100_{10}$ in base $b$ has exactly $5$ digits. What is the value of $b$?# Solution
[TURN] [STEP] 0. I need to find a base $b$ such that $100_{10} = 5$ digits in base $b$. [/STEP]
[STEP] 1. I can rewrite $100_{10}$ as $1 \times b^2 + 0 \times b^1 + 0 \times b^0$ in base $b$. [/STEP]
[STEP] 2. This means ... |
# Question
$24$ lemons are required to make $32$ gallons of lemonade. How many lemons are needed to make $4$ gallons of lemonade?# Solution
[TURN] [STEP] 0. I need to find the relationship between the number of lemons and the number of gallons of lemonade. [/STEP]
[STEP] 1. I can use a proportion to compare the two... |
# Question
$24$ lemons are required to make $32$ gallons of lemonade. How many lemons are needed to make $4$ gallons of lemonade?# Solution
[TURN] [STEP] 0. To solve this problem, I need to find the rate of lemons per gallon of lemonade. [/STEP]
[STEP] 1. I can use the given information that $24$ lemons make $32$ g... |
# Question
$361+2(19)(6)+36=x$. Solve for $x$.# Solution
[TURN] [STEP] 0. I notice that the left-hand side of the equation is very close to a perfect square: $361=19^2$ and $36=6^2$. [/STEP]
[STEP] 1. So I wonder if I can rewrite the equation as a perfect square plus some other term. [/STEP]
[STEP] 2. To do that, ... |
# Question
$361+2(19)(6)+36=x$. Solve for $x$.# Solution
[TURN] [STEP] 0. I notice that $x$ is a sum of three terms, and the last two terms have a common factor of $6$. [/STEP]
[STEP] 1. I can use the distributive property to rewrite the expression as $(361+36)+2(19)(6)$. [/STEP]
[STEP] 2. Then I can simplify the ... |
# Question
$3^n = 3 \cdot 9^3 \cdot 81^2$. What is the value of $n$?# Solution
[TURN] [STEP] 0. So we can rewrite $9^3$ as $3^6$ and $81^2$ as $3^8$. [/STEP]
[STEP] 1. That's right. So we're trying to solve $3^n = 3^6 \cdot 3^8$. [/STEP]
[STEP] 2. Right. So we can just add $6+8$ to get $n$. [/STEP]
[STEP] 3. That... |
# Question
$3^n = 3 \cdot 9^3 \cdot 81^2$. What is the value of $n$?# Solution
[TURN] [STEP] 0. To solve for $n$, I need to express both sides of the equation in terms of the same base. [/STEP]
[STEP] 1. I notice that all the numbers on the right side are powers of 3, so I can rewrite them using exponent rules. [/S... |
# Question
$A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP < DP.$
[asy]
unitsize(0.6 inch);
draw(circle((0,0),1));
draw((-0.3,0.94)--(0.3,-0.94));
draw((-0.7,-0.7)--(0.7,-0.7));
label("... |
# Question
$A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP < DP.$
[asy]
unitsize(0.6 inch);
draw(circle((0,0),1));
draw((-0.3,0.94)--(0.3,-0.94));
draw((-0.7,-0.7)--(0.7,-0.7));
label("... |
# Question
$A, B, C$ and $D$ are distinct positive integers such that the product $AB = 60$, the product $CD = 60$ and $A - B = C + D$ . What is the value of $A$?# Solution
[TURN] [STEP] 0. I know that $60 = 2^2 \times 3 \times 5$, so the possible values for $A$ and $B$ are $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30$ a... |
# Question
$A, B, C$ and $D$ are distinct positive integers such that the product $AB = 60$, the product $CD = 60$ and $A - B = C + D$ . What is the value of $A$?# Solution
[TURN] [STEP] 0. Let's write out the prime factorization of 60. [/STEP]
[STEP] 1. Ok. [/STEP]
[STEP] 2. So $60=2^2\cdot3\cdot5$. [/STEP]
[S... |
# Question
$ABC$ is an equilateral triangle with sides equal to 2cm. $\overline{BC}$ is extended its own length to $D$, and $E$ is the midpoint of $\overline{AB}$. Suppose $\overline{ED}$ meets $\overline{AC}$ at $F$. Find the area of the quadrilateral $BEFC$ in square centimeters.
[asy]
size(200);
draw( (0,0) -- (1/... |
# Question
$ABCD$ is a rectangle whose area is 12 square units. How many square units are contained in the area of trapezoid $EFBA$?
[asy]
size(4cm,4cm);
for(int i=0; i < 4; ++i){
for(int k=0; k < 5; ++k){
draw((0,i)--(4,i));
draw((k,0)--(k,3));
} }
draw((0,0)--(1,3));
draw((3,3)--(4,0));
label("$A$",(0,0),SW);
l... |
# Question
$ABCD$ is a rectangle whose area is 12 square units. How many square units are contained in the area of trapezoid $EFBA$?
[asy]
size(4cm,4cm);
for(int i=0; i < 4; ++i){
for(int k=0; k < 5; ++k){
draw((0,i)--(4,i));
draw((k,0)--(k,3));
} }
draw((0,0)--(1,3));
draw((3,3)--(4,0));
label("$A$",(0,0),SW);
l... |
# Question
$ABCD$ is a regular tetrahedron (right triangular pyramid). If $M$ is the midpoint of $\overline{CD}$, then what is $\cos \angle AMB$?# Solution
[TURN] [STEP] 0. I need to find the cosine of the angle between two vertices of the tetrahedron, which involves the lengths of the edges and the height of the t... |
# Question
$ABCD$ is a square and $M$ and $N$ are the midpoints of $\overline{BC}$ and $\overline{CD},$ respectively. Find $\sin \theta.$
[asy]
unitsize(1.5 cm);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((0,0)--(2,1));
draw((0,0)--(1,2));
label("$A$", (0,0), SW);
label("$B$", (0,2), NW);
label("$C$", (2,2), NE)... |
# Question
$ABCD$ is a trapezoid with the measure of base $\overline{AB}$ twice the measure of the base $\overline{CD}$. Point $E$ is the point of intersection of the diagonals. The measure of diagonal $\overline{AC}$ is 11. Find the length of segment $\overline{EC}$. Express your answer as a common fraction.
[asy]
... |
# Question
$ABCDE$ is a regular pentagon. $AP$, $AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD$, $CB$ extended and $DE$ extended, respectively. Let $O$ be the center of the pentagon. If $OP = 1$, then find $AO + AQ + AR$.
[asy]
unitsize(2 cm);
pair A, B, C, D, E, O, P, Q, R;
A = dir(90);
B = di... |
# Question
$ABCDE$ is a regular pentagon. $AP$, $AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD$, $CB$ extended and $DE$ extended, respectively. Let $O$ be the center of the pentagon. If $OP = 1$, then find $AO + AQ + AR$.
[asy]
unitsize(2 cm);
pair A, B, C, D, E, O, P, Q, R;
A = dir(90);
B = di... |
# Question
$ABCDE$ is inscribed in a circle with $AB = BC = CD = DE = 4$ and $AE = 1.$ Compute $(1 - \cos \angle B)(1 - \cos \angle ACE).$# Solution
[TURN] [STEP] 0. I notice that the problem involves inscribed angles and cosine, so I wonder if there is a trigonometric identity or theorem that can help me simplify ... |
# Question
$ABCDEFGH$ shown below is a cube with volume 1. Find the volume of pyramid $ABCH$.
[asy]
import three;
triple A,B,C,D,EE,F,G,H;
A = (0,0,0);
B = (1,0,0);
C = (1,1,0);
D= (0,1,0);
EE = (0,0,1);
F = B+EE;
G = C + EE;
H = D + EE;
draw(B--C--D);
draw(B--A--D,dashed);
draw(EE--F--G--H--EE);
draw(A--EE,dashed);... |
# Question
$ABCDEFGH$ shown below is a cube with volume 1. Find the volume of pyramid $ABCH$.
[asy]
import three;
triple A,B,C,D,EE,F,G,H;
A = (0,0,0);
B = (1,0,0);
C = (1,1,0);
D= (0,1,0);
EE = (0,0,1);
F = B+EE;
G = C + EE;
H = D + EE;
draw(B--C--D);
draw(B--A--D,dashed);
draw(EE--F--G--H--EE);
draw(A--EE,dashed);... |
# Question
$ABCDEFGH$ shown below is a cube. Find $\sin \angle GAC$.
[asy]
import three;
triple A,B,C,D,EE,F,G,H;
A = (0,0,0);
B = (1,0,0);
C = (1,1,0);
D= (0,1,0);
EE = (0,0,1);
F = B+EE;
G = C + EE;
H = D + EE;
draw(B--C--D);
draw(B--A--D,dashed);
draw(EE--F--G--H--EE);
draw(A--EE,dashed);
draw(B--F)... |
# Question
$ABCDEFGH$ shown below is a cube. Find $\sin \angle HAC$.
[asy]
import three;
triple A,B,C,D,EE,F,G,H;
A = (0,0,0);
B = (1,0,0);
C = (1,1,0);
D= (0,1,0);
EE = (0,0,1);
F = B+EE;
G = C + EE;
H = D + EE;
draw(B--C--D);
draw(B--A--D,dashed);
draw(EE--F--G--H--EE);
draw(A--EE,dashed);
draw(B--F)... |
# Question
$B$ and $C$ trisect $\overline{AD}$ and $M$ is the midpoint of $\overline{AD}$. $MC = 8$. How many units are in the length of $\overline{AD}$?# Solution
[TURN] [STEP] 0. I know that trisection means dividing into three equal parts, so I can use that to find the lengths of the segments $\overline{AB}$, $\o... |
# Question
$B$ and $C$ trisect $\overline{AD}$ and $M$ is the midpoint of $\overline{AD}$. $MC = 8$. How many units are in the length of $\overline{AD}$?# Solution
[TURN] [STEP] 0. I know that trisection means dividing into three equal parts, so $B$ and $C$ must be equidistant from $A$ and $D$. [/STEP]
[STEP] 1. Th... |
# Question
$\Delta ABC$ is isosceles with $AC = BC$. If $m\angle C = 40^{\circ}$, what is the number of degrees in $m\angle CBD$? [asy] pair A,B,C,D,E;
C = dir(65); B = C + dir(-65); D = (1.5,0); E = (2,0);
draw(B--C--A--E); dot(D);
label("$A$",A,S); label("$B$",B,S); label("$D$",D,S); label("$C$",C,N);
[/asy]# Soluti... |
# Question
$\Delta ABC$ is isosceles with $AC = BC$. If $m\angle C = 40^{\circ}$, what is the number of degrees in $m\angle CBD$? [asy] pair A,B,C,D,E;
C = dir(65); B = C + dir(-65); D = (1.5,0); E = (2,0);
draw(B--C--A--E); dot(D);
label("$A$",A,S); label("$B$",B,S); label("$D$",D,S); label("$C$",C,N);
[/asy]# Soluti... |
# Question
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, $6$ cents each, Peru $4$ cents each, and Spain $5$ cents each. (Brazil and Peru are South American ... |
# Question
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, $6$ cents each, Peru $4$ cents each, and Spain $5$ cents each. (Brazil and Peru are South American ... |
# Question
$\triangle ABC$ and $\triangle DBC$ share $BC$. $AB = 5\ \text{cm}$, $AC = 12\ \text{cm}$, $DC = 8\ \text{cm}$, and $BD = 20\ \text{cm}$. What is the least possible integral number of centimeters in $BC$?
[asy]
size(100); import graph; currentpen = fontsize(10pt);
pair B = (0,0), C = (13,0), A = (-5,7), D ... |
# Question
$\triangle ABC$ and $\triangle DBC$ share $BC$. $AB = 5\ \text{cm}$, $AC = 12\ \text{cm}$, $DC = 8\ \text{cm}$, and $BD = 20\ \text{cm}$. What is the least possible integral number of centimeters in $BC$?
[asy]
size(100); import graph; currentpen = fontsize(10pt);
pair B = (0,0), C = (13,0), A = (-5,7), D ... |
# Question
$\triangle ABC$ is similar to $\triangle DEF$ . What is the number of centimeters in the length of $\overline{EF}$ ? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(8,-2)--(5,4)--cycle);
label("8cm",(2.5,2),NW);
label("5cm",(6.1,1),NE);
draw((12,0)--(18,-1.5)--(15.7,2.5)--cycle);
... |
# Question
$\triangle ABC$ is similar to $\triangle DEF$ . What is the number of centimeters in the length of $\overline{EF}$ ? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(8,-2)--(5,4)--cycle);
label("8cm",(2.5,2),NW);
label("5cm",(6.1,1),NE);
draw((12,0)--(18,-1.5)--(15.7,2.5)--cycle);
... |
# Question
$\triangle ABC\sim\triangle DBE$, $BC=20\text{ cm}.$ How many centimeters long is $DE$? Express your answer as a decimal to the nearest tenth. [asy]
draw((0,0)--(20,0)--(20,12)--cycle);
draw((13,0)--(13,7.8));
label("$B$",(0,0),SW);
label("$E$",(13,0),S);
label("$D$",(13,7.8),NW);
label("$A$",(20,12),NE);
l... |
# Question
$\triangle ABC\sim\triangle DBE$, $BC=20\text{ cm}.$ How many centimeters long is $DE$? Express your answer as a decimal to the nearest tenth. [asy]
draw((0,0)--(20,0)--(20,12)--cycle);
draw((13,0)--(13,7.8));
label("$B$",(0,0),SW);
label("$E$",(13,0),S);
label("$D$",(13,7.8),NW);
label("$A$",(20,12),NE);
l... |
# Question
$f (x) = x + 3$ and $g(x) = x^2 -6$, what is the value of $f (g(2))$?# Solution
[TURN] [STEP] 0. To evaluate $f (g(2))$, I need to first find $g(2)$ and then plug that into $f(x)$. [/STEP]
[STEP] 1. To find $g(2)$, I just replace $x$ with $2$ in the definition of $g(x)$ and simplify. [/STEP]
[STEP] 2. $... |
# Question
$f (x) = x + 3$ and $g(x) = x^2 -6$, what is the value of $f (g(2))$?# Solution
[TURN] [STEP] 0. To find the value of $f (g(2))$, I need to first evaluate $g(2)$ and then plug that into $f$. [/STEP]
[STEP] 1. So, $g(2) = 2^2 - 6 = 4 - 6 = -2$. [/STEP]
[STEP] 2. Now, $f (g(2)) = f (-2) = -2 + 3 = 1$. [/S... |
# Question
$r(x)$ has domain $\{-1,0,1,2\}$ and range $\{0,2,4,6\}$. $s(x)$ has domain $\{1,2,3,4\}$ and is defined by $s(x)=x+1$. What is the sum of all possible values of $s(r(x))$?# Solution
[TURN] [STEP] 0. I need to find the possible values of $s(r(x))$, which means I need to plug in the values of $x$ from the ... |
# Question
$r(x)$ has domain $\{-1,0,1,2\}$ and range $\{0,2,4,6\}$. $s(x)$ has domain $\{1,2,3,4\}$ and is defined by $s(x)=x+1$. What is the sum of all possible values of $s(r(x))$?# Solution
[TURN] [STEP] 0. To find the sum of all possible values of $s(r(x))$, I need to know what values of $r(x)$ are in the domai... |
# Question
$x = {1+\frac{\sqrt{2}}{1+\frac{\sqrt{2}}{1+...}}}$. Find $\frac{1}{(x+1)(x-2)}$. When your answer is in the form $\frac{A+\sqrt{B}}{C}$, where $A$, $B$, and $C$ are integers, and $B$ is not divisible by the square of a prime, what is $|A|+|B|+|C|$?# Solution
[TURN] [STEP] 0. I notice that the expression ... |
# Question
$x = {1+\frac{\sqrt{2}}{1+\frac{\sqrt{2}}{1+...}}}$. Find $\frac{1}{(x+1)(x-2)}$. When your answer is in the form $\frac{A+\sqrt{B}}{C}$, where $A$, $B$, and $C$ are integers, and $B$ is not divisible by the square of a prime, what is $|A|+|B|+|C|$?# Solution
[TURN] [STEP] 0. This problem involves an infi... |
# Question
100 people who were surveyed and asked the question: ``Is fuchsia kinda pink, or purply?'' Of them, 60 believe that fuchsia is ``kinda pink'' and 27 believe it is both ``kinda pink,'' and also ``purply.'' Another 17 think that fuchsia is neither ``kinda pink'' nor ``purply.''
How many of those 100 people ... |
# Question
100 people who were surveyed and asked the question: ``Is fuchsia kinda pink, or purply?'' Of them, 60 believe that fuchsia is ``kinda pink'' and 27 believe it is both ``kinda pink,'' and also ``purply.'' Another 17 think that fuchsia is neither ``kinda pink'' nor ``purply.''
How many of those 100 people ... |
# Question
12 balls numbered 1 through 12 are placed in a bin. In how many ways can 3 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn?# Solution
[TURN] [STEP] 0. I need to count the number of permutations of 3 balls out of 12, since the order of drawing matters. [/STEP... |
# Question
12 balls numbered 1 through 12 are placed in a bin. In how many ways can 3 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn?# Solution
[TURN] [STEP] 0. This problem involves counting the number of possible outcomes of a sequence of events, so I will use the m... |
# Question
12 balls numbered 1 through 12 are placed in a bin. Joe produces a list of three numbers by performing the following sequence three times: he chooses a ball, records the number, and places the ball back in the bin. How many different lists are possible?# Solution
[TURN] [STEP] 0. I want to count the n... |
# Question
12 balls numbered 1 through 12 are placed in a bin. Joe produces a list of three numbers by performing the following sequence three times: he chooses a ball, records the number, and places the ball back in the bin. How many different lists are possible?# Solution
[TURN] [STEP] 0. This is a problem abo... |
# Question
2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?# Solution
[TURN] [STEP] 0. A regular nonagon has 9 vertices, and any two vertices determine a diagonal. [/STEP]
[STEP] 1. So there are $\binom{9}{2} = 36$ possible pa... |
# Question
2 distinct vertices of an octagon are chosen at random. What is the probability that they are adjacent?# Solution
[TURN] [STEP] 0. To find the probability of an event, I need to divide the number of favorable outcomes by the number of possible outcomes. [/STEP]
[STEP] 1. In this case, the event is choos... |
# Question
23 people attend a party. Each person shakes hands with at least two other people. What is the minimum possible number of handshakes?# Solution
[TURN] [STEP] 0. Let's think about the problem. [/STEP]
[STEP] 1. Ok. So we have 23 people and each one shakes hands with at least two other people. [/STEP]
[ST... |
# Question
23 people attend a party. Each person shakes hands with at least two other people. What is the minimum possible number of handshakes?# Solution
[TURN] [STEP] 0. I want to minimize the number of handshakes, so I want to avoid having any person shake hands with more than two other people. [/STEP]
[STEP] 1.... |
# Question
23 people attend a party. Each person shakes hands with at most 22 other people. What is the maximum possible number of handshakes, assuming that any two people can shake hands at most once?# Solution
[TURN] [STEP] 0. I need to find the maximum number of handshakes, not the average. [/STEP]
[STEP] 1. If ... |
# Question
23 people attend a party. Each person shakes hands with at most 22 other people. What is the maximum possible number of handshakes, assuming that any two people can shake hands at most once?# Solution
[TURN] [STEP] 0. I need to find the maximum possible number of handshakes, so I want to make sure that ea... |
# Question
8 coins are simultaneously flipped. What is the probability that heads are showing on at most 2 of them?# Solution
[TURN] [STEP] 0. To solve this problem, I need to find the total number of possible outcomes and the number of outcomes that satisfy the given condition. [/STEP]
[STEP] 1. I can use the bin... |
# Question
8 people attend a party. During the party everyone shakes hands with everyone else. How many handshakes take place at the party?# Solution
[TURN] [STEP] 0. I need to count the total number of handshakes that occur when 8 people shake hands with each other. [/STEP]
[STEP] 1. One way to approach this probl... |
# Question
8 people attend a party. During the party everyone shakes hands with everyone else. How many handshakes take place at the party?# Solution
[TURN] [STEP] 0. To solve this problem, I need to figure out how to count the number of handshakes without repeating or missing any. [/STEP]
[STEP] 1. One way to do t... |
# Question
A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $100$ vertices equals 2009. The midpoints of the sides of $P_1$ form a second $100$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $100$-gon, $P_3$. Find the sum of the $x$-coordinates of the ... |
# Question
A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $100$ vertices equals 2009. The midpoints of the sides of $P_1$ form a second $100$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $100$-gon, $P_3$. Find the sum of the $x$-coordinates of the ... |
# Question
A $180^\circ$ rotation around the origin in the counter-clockwise direction is applied to $-6 - 3i.$ What is the resulting complex number?# Solution
[TURN] [STEP] 0. A 180-degree rotation around the origin in the counter-clockwise direction is equivalent to multiplying the complex number by -1. [/STEP]
... |
# Question
A $180^\circ$ rotation around the origin in the counter-clockwise direction is applied to $-6 - 3i.$ What is the resulting complex number?# Solution
[TURN] [STEP] 0. To apply a rotation to a complex number, I can use the formula $z' = z \cdot e^{i\theta},$ where $z$ is the original complex number, $z'$ i... |
# Question
A $2\times 3$ rectangle and a $3\times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?# Solution
[TURN] [STEP] 0. I need to find the sm... |
# Question
A $2\times 3$ rectangle and a $3\times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?# Solution
[TURN] [STEP] 0. I need to find the sm... |
# Question
A $33$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $33$ vertices equals $99$. The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$. Find the sum of the $x$-coordinates of the vert... |
# Question
A $33$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $33$ vertices equals $99$. The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$. Find the sum of the $x$-coordinates of the vert... |
# Question
A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece?# Solu... |
# Question
A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece?# Solu... |
# Question
A $5 \times 8$ rectangle can be rolled to form two different cylinders with different maximum volumes. What is the ratio of the larger volume to the smaller volume? Express your answer as a common fraction.# Solution
[TURN] [STEP] 0. To roll a rectangle into a cylinder, I need to choose the height and the... |
# Question
A $\emph{convex}$ polygon is a polygon in which every interior angle is less than 180 degrees. A $\emph{diagonal}$ of a convex polygon is a line segment that connects two non-adjacent vertices. How many diagonals does a convex polygon with 20 sides have?# Solution
[TURN] [STEP] 0. I know that a diagonal... |
# Question
A $\emph{convex}$ polygon is a polygon in which every interior angle is less than 180 degrees. A $\emph{diagonal}$ of a convex polygon is a line segment that connects two non-adjacent vertices. How many diagonals does a convex polygon with 20 sides have?# Solution
[TURN] [STEP] 0. I know that the number... |
# Question
A $\textit{composite number}$ is a number that has two or more prime factors. The number 87 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers?# Solution
[TURN] [STEP] 0. I need to find two composite numbers that add up to 87... |
# Question
A $\textit{composite number}$ is a number that has two or more prime factors. The number 87 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers?# Solution
[TURN] [STEP] 0. I want to find two composite numbers, x and y, such th... |
# Question
A $\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit hour v... |
# Question
A $\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit hour v... |
# Question
A $\textit{palindrome}$ is a number which reads the same forward as backward. For example, 343 and 1221 are palindromes. What is the least natural number that can be added to 40,305 to create a palindrome?# Solution
[TURN] [STEP] 0. I know that 40305 is not a palindrome, because the first and last digits ... |
# Question
A $\textit{palindrome}$ is a number which reads the same forward as backward. For example, 343 and 1221 are palindromes. What is the least natural number that can be added to 40,305 to create a palindrome?# Solution
[TURN] [STEP] 0. I want to find the smallest palindrome that is greater than 40,305. [/STE... |
# Question
A $\textit{palindrome}$ is a positive integer which reads the same forward and backward, like $12321$ or $4884$.
How many $4$-digit palindromes are there?# Solution
[TURN] [STEP] 0. I know that a $4$-digit palindrome must have the same first and last digit, and the same second and third digit. [/STEP]
[... |
# Question
A $\textit{palindrome}$ is a positive integer which reads the same forward and backward, like $12321$ or $4884$.
How many $4$-digit palindromes are there?# Solution
[TURN] [STEP] 0. To form a $4$-digit palindrome, I need to choose the first two digits, and then the last two digits will be determined by r... |
# Question
A ''super ball'' is dropped from a window 16 meters above the ground. On each bounce it rises $\frac34$ the distance of the preceding high point. The ball is caught when it reached the high point after hitting the ground for the third time. To the nearest meter, how far has it travelled?# Solution
[TURN] ... |
# Question
A ''super ball'' is dropped from a window 16 meters above the ground. On each bounce it rises $\frac34$ the distance of the preceding high point. The ball is caught when it reached the high point after hitting the ground for the third time. To the nearest meter, how far has it travelled?# Solution
[TURN] ... |
# Question
A 1$\%$ late charge was added to Jenna's bill on the $30^{\text{th}}$ day past its due date. The resulting total was then increased by 1$\%$ because she did not pay the bill in the next 30 days either. Her original bill was $\$400$. Exactly how much is the bill now?# Solution
[TURN] [STEP] 0. I need to ca... |
# Question
A 1$\%$ late charge was added to Jenna's bill on the $30^{\text{th}}$ day past its due date. The resulting total was then increased by 1$\%$ because she did not pay the bill in the next 30 days either. Her original bill was $\$400$. Exactly how much is the bill now?# Solution
[TURN] [STEP] 0. So the origi... |
# Question
A 12-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Only six slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms?# Solution
[TURN] [STEP] 0. This is a problem about the intersection of tw... |
# Question
A 12-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Only six slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms?# Solution
[TURN] [STEP] 0. I want to find the number of slices that have ... |
# Question
A 4-inch by 6-inch picture is enlarged for framing by tripling its dimensions. A 2-inch-wide border is then placed around each side of the enlarged picture, as shown. Thin metal framing is sold only in increments of one foot. What is the minimum number of linear feet of framing that must be purchas... |
# Question
A 4-inch by 6-inch picture is enlarged for framing by tripling its dimensions. A 2-inch-wide border is then placed around each side of the enlarged picture, as shown. Thin metal framing is sold only in increments of one foot. What is the minimum number of linear feet of framing that must be purchas... |
# Question
A 40-foot by 10-foot rectangular garden is enclosed by a fence. To make the garden larger, while using the same amount of fencing, its shape is changed to a square. How many square feet larger than the old garden is the new garden?# Solution
[TURN] [STEP] 0. To make the garden larger, I need to maximize t... |
# Question
A 40-foot by 10-foot rectangular garden is enclosed by a fence. To make the garden larger, while using the same amount of fencing, its shape is changed to a square. How many square feet larger than the old garden is the new garden?# Solution
[TURN] [STEP] 0. To make the garden larger, I need to use the sa... |
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