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2018_AMC_10A_Problems/Problem_24 | Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$ . Let $D$ be the midpoint of $\overline{AB}$ , and let $E$ be the midpoint of $\overline{AC}$ . The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$ , respectively. What is the area of quadrilateral $FDBG$ | 75 | AMC_10 | 2,743 |
2018_AMC_10A_Problems/Problem_25 | For a positive integer $n$ and nonzero digits $a$ $b$ , and $c$ , let $A_n$ be the $n$ -digit integer each of whose digits is equal to $a$ ; let $B_n$ be the $n$ -digit integer each of whose digits is equal to $b$ , and let $C_n$ be the $2n$ -digit (not $n$ -digit) integer each of whose digits is equal to $c$ . What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$ | 18 | AMC_10 | 2,748 |
2018_AMC_10B_Problems/Problem_1 | Kate bakes a $20$ -inch by $18$ -inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain? | 90 | AMC_10 | 2,752 |
2018_AMC_10B_Problems/Problem_2 | Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes? | 67 | AMC_10 | 2,754 |
2018_AMC_10B_Problems/Problem_3 | In the expression $\left(\underline{\qquad}\times\underline{\qquad}\right)+\left(\underline{\qquad}\times\underline{\qquad}\right)$ each blank is to be filled in with one of the digits $1,2,3,$ or $4,$ with each digit being used once. How many different values can be obtained? | 3 | AMC_10 | 2,756 |
2018_AMC_10B_Problems/Problem_5 | How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number? | 240 | AMC_10 | 2,759 |
2018_AMC_10B_Problems/Problem_9 | The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$ . Let $p$ be the probabilities that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$ . What other sum occurs with the same probability as $p$ | 39 | AMC_10 | 2,771 |
2018_AMC_10B_Problems/Problem_12 | Line segment $\overline{AB}$ is a diameter of a circle with $AB = 24$ . Point $C$ , not equal to $A$ or $B$ , lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle ABC$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve? | 50 | AMC_10 | 2,782 |
2018_AMC_10B_Problems/Problem_13 | How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \dots$ are divisible by $101$ | 505 | AMC_10 | 2,785 |
2018_AMC_10B_Problems/Problem_14 | A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? | 225 | AMC_10 | 2,790 |
2018_AMC_10B_Problems/Problem_16 | Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that \[a_1+a_2+\cdots+a_{2018}=2018^{2018}.\] What is the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$ | 4 | AMC_10 | 2,792 |
2018_AMC_10B_Problems/Problem_17 | In rectangle $PQRS$ $PQ=8$ and $QR=6$ . Points $A$ and $B$ lie on $\overline{PQ}$ , points $C$ and $D$ lie on $\overline{QR}$ , points $E$ and $F$ lie on $\overline{RS}$ , and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$ , where $k$ $m$ , and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$ | 7 | AMC_10 | 2,797 |
2018_AMC_10B_Problems/Problem_18 | Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip? | 96 | AMC_10 | 2,801 |
2018_AMC_10B_Problems/Problem_19 | Joey and Chloe and their daughter Zoe all have the same birthday. Joey is $1$ year older than Chloe, and Zoe is exactly $1$ year old today. Today is the first of the $9$ birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age? | 11 | AMC_10 | 2,807 |
2018_AMC_10B_Problems/Problem_20 | A function $f$ is defined recursively by $f(1)=f(2)=1$ and \[f(n)=f(n-1)-f(n-2)+n\] for all integers $n \geq 3$ . What is $f(2018)$ | 2,017 | AMC_10 | 2,811 |
2018_AMC_10B_Problems/Problem_21 | Mary chose an even $4$ -digit number $n$ . She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,\ldots,\dfrac{n}{2},n$ . At some moment Mary wrote $323$ as a divisor of $n$ . What is the smallest possible value of the next divisor written to the right of $323$ | 340 | AMC_10 | 2,818 |
2018_AMC_10B_Problems/Problem_23 | How many ordered pairs $(a, b)$ of positive integers satisfy the equation \[a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),\] where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$ , and $\text{lcm}(a,b)$ denotes their least common multiple? | 2 | AMC_10 | 2,823 |
2018_AMC_10B_Problems/Problem_25 | Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$ . How many real numbers $x$ satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$ | 199 | AMC_10 | 2,824 |
2017_AMC_10A_Problems/Problem_1 | What is the value of $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$ | 127 | AMC_10 | 2,831 |
2017_AMC_10A_Problems/Problem_2 | Pablo buys popsicles for his friends. The store sells single popsicles for $$1$ each, $3$ -popsicle boxes for $$2$ each, and $5$ -popsicle boxes for $$3$ . What is the greatest number of popsicles that Pablo can buy with $$8$ | 13 | AMC_10 | 2,837 |
2017_AMC_10A_Problems/Problem_4 | Mia is "helping" her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time? | 14 | AMC_10 | 2,840 |
2017_AMC_10A_Problems/Problem_5 | The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers? | 4 | AMC_10 | 2,841 |
2017_AMC_10A_Problems/Problem_7 | Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip? | 30 | AMC_10 | 2,844 |
2017_AMC_10A_Problems/Problem_8 | At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur within the group? | 245 | AMC_10 | 2,845 |
2017_AMC_10A_Problems/Problem_9 | Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$ , a distance of $10$ km all uphill, then from town $B$ to town $C$ , a distance of $15$ km all downhill, and then back to town $A$ , a distance of $20$ km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the $45$ -km ride than it takes Penny? | 65 | AMC_10 | 2,849 |
2017_AMC_10A_Problems/Problem_10 | Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod? | 17 | AMC_10 | 2,850 |
2017_AMC_10A_Problems/Problem_11 | The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$ . What is the length $\textit{AB}$ | 20 | AMC_10 | 2,851 |
2017_AMC_10A_Problems/Problem_13 | Define a sequence recursively by $F_{0}=0,~F_{1}=1,$ and $F_{n}=$ the remainder when $F_{n-1}+F_{n-2}$ is divided by $3,$ for all $n\geq 2.$ Thus the sequence starts $0,1,1,2,0,2,\ldots$ What is $F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}?$ | 9 | AMC_10 | 2,853 |
2017_AMC_10A_Problems/Problem_14 | Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda? | 23 | AMC_10 | 2,854 |
2017_AMC_10A_Problems/Problem_16 | There are $10$ horses, named Horse $1$ , Horse $2$ , . . . , Horse $10$ . They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$ , in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S=2520$ . Let $T > 0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T?$ | 3 | AMC_10 | 2,857 |
2017_AMC_10A_Problems/Problem_17 | Distinct points $P$ $Q$ $R$ $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS}$ | 7 | AMC_10 | 2,861 |
2017_AMC_10A_Problems/Problem_18 | Amelia has a coin that lands heads with probability $\frac{1}{3}\,$ , and Blaine has a coin that lands on heads with probability $\frac{2}{5}$ . Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. What is $q-p$ | 4 | AMC_10 | 2,864 |
2017_AMC_10A_Problems/Problem_19 | Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? | 28 | AMC_10 | 2,867 |
2017_AMC_10A_Problems/Problem_20 | Let $S(n)$ equal the sum of the digits of positive integer $n$ . For example, $S(1507) = 13$ . For a particular positive integer $n$ $S(n) = 1274$ . Which of the following could be the value of $S(n+1)$ | 1,239 | AMC_10 | 2,871 |
2017_AMC_10A_Problems/Problem_23 | How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$ , inclusive? | 2,148 | AMC_10 | 2,875 |
2017_AMC_10A_Problems/Problem_24 | For certain real numbers $a$ $b$ , and $c$ , the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$ | 7,007 | AMC_10 | 2,877 |
2017_AMC_10A_Problems/Problem_25 | How many integers between $100$ and $999$ , inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $211$ have this property. | 226 | AMC_10 | 2,886 |
2017_AMC_10B_Problems/Problem_1 | Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$ . Then she switched the digits of the result, obtaining a number between $71$ and $75$ , inclusive. What was Mary's number? | 12 | AMC_10 | 2,890 |
2017_AMC_10B_Problems/Problem_4 | Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3x+y}{x-3y}=-2$ . What is the value of $\frac{x+3y}{3x-y}$ | 2 | AMC_10 | 2,895 |
2017_AMC_10B_Problems/Problem_5 | Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have? | 40 | AMC_10 | 2,899 |
2017_AMC_10B_Problems/Problem_6 | What is the largest number of solid $2\text{-in} \times 2\text{-in} \times 1\text{-in}$ blocks that can fit in a $3\text{-in} \times 2\text{-in}\times3\text{-in}$ box? | 4 | AMC_10 | 2,902 |
2017_AMC_10B_Problems/Problem_10 | The lines with equations $ax-2y=c$ and $2x+by=-c$ are perpendicular and intersect at $(1, -5)$ . What is $c$ | 13 | AMC_10 | 2,911 |
2017_AMC_10B_Problems/Problem_11 | At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it? | 25 | AMC_10 | 2,912 |
2017_AMC_10B_Problems/Problem_12 | Elmer's new car gives $50\%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is $20\%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip? | 20 | AMC_10 | 2,915 |
2017_AMC_10B_Problems/Problem_13 | There are $20$ students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are $10$ students taking yoga, $13$ taking bridge, and $9$ taking painting. There are $9$ students taking at least two classes. How many students are taking all three classes? | 3 | AMC_10 | 2,919 |
2017_AMC_10B_Problems/Problem_14 | An integer $N$ is selected at random in the range $1\leq N \leq 2020$ . What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$ | 45 | AMC_10 | 2,924 |
2017_AMC_10B_Problems/Problem_16 | How many of the base-ten numerals for the positive integers less than or equal to $2017$ contain the digit $0$ | 469 | AMC_10 | 2,926 |
2017_AMC_10B_Problems/Problem_17 | Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$ $23578$ , and $987620$ are monotonous, but $88$ $7434$ , and $23557$ are not. How many monotonous positive integers are there? | 1,524 | AMC_10 | 2,928 |
2017_AMC_10B_Problems/Problem_23 | Let $N=123456789101112\dots4344$ be the $79$ -digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$ | 9 | AMC_10 | 2,937 |
2017_AMC_10B_Problems/Problem_24 | The vertices of an equilateral triangle lie on the hyperbola $xy=1$ , and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? | 108 | AMC_10 | 2,941 |
2017_AMC_10B_Problems/Problem_25 | Last year Isabella took $7$ math tests and received $7$ different scores, each an integer between $91$ and $100$ , inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95$ . What was her score on the sixth test? | 100 | AMC_10 | 2,945 |
2016_AMC_10A_Problems/Problem_1 | What is the value of $\dfrac{11!-10!}{9!}$ | 100 | AMC_10 | 2,950 |
2016_AMC_10A_Problems/Problem_2 | For what value of $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$ | 3 | AMC_10 | 2,953 |
2016_AMC_10A_Problems/Problem_7 | The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$ . What is the value of $x$ | 90 | AMC_10 | 2,958 |
2016_AMC_10A_Problems/Problem_8 | Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning? | 35 | AMC_10 | 2,960 |
2016_AMC_10A_Problems/Problem_9 | A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$ th row. What is the sum of the digits of $N$ | 9 | AMC_10 | 2,962 |
2016_AMC_10A_Problems/Problem_13 | Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up? | 2 | AMC_10 | 2,965 |
2016_AMC_10A_Problems/Problem_14 | How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008\cdot 2 + 0\cdot 3$ and $402\cdot 2 + 404\cdot 3$ are two such ways.) | 337 | AMC_10 | 2,968 |
2016_AMC_10A_Problems/Problem_17 | Let $N$ be a positive multiple of $5$ . One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$ | 12 | AMC_10 | 2,973 |
2016_AMC_10A_Problems/Problem_19 | In rectangle $ABCD,$ $AB=6$ and $BC=3$ . Point $E$ between $B$ and $C$ , and point $F$ between $E$ and $C$ are such that $BE=EF=FC$ . Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$ , respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s,$ and $t$ is $1.$ What is $r+s+t$ | 20 | AMC_10 | 2,980 |
2016_AMC_10A_Problems/Problem_22 | For some positive integer $n$ , the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$ . How many positive integer divisors does the number $81n^4$ have? | 325 | AMC_10 | 2,988 |
2016_AMC_10A_Problems/Problem_23 | A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$ . (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. What is $p+q?$ | 109 | AMC_10 | 2,990 |
2016_AMC_10A_Problems/Problem_24 | A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$ . Three of the sides of this quadrilateral have length $200$ . What is the length of the fourth side? | 500 | AMC_10 | 2,996 |
2016_AMC_10A_Problems/Problem_25 | How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600 \text{ and lcm}(y,z)=900$ | 15 | AMC_10 | 3,008 |
2016_AMC_10B_Problems/Problem_1 | What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \tfrac{1}{2}$ | 10 | AMC_10 | 3,011 |
2016_AMC_10B_Problems/Problem_3 | Let $x=-2016$ . What is the value of $\Bigg\vert\Big\vert |x|-x\Big\vert-|x|\Bigg\vert-x$ | 4,032 | AMC_10 | 3,013 |
2016_AMC_10B_Problems/Problem_5 | The mean age of Amanda's $4$ cousins is $8$ , and their median age is $5$ . What is the sum of the ages of Amanda's youngest and oldest cousins? | 22 | AMC_10 | 3,015 |
2016_AMC_10B_Problems/Problem_6 | Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$ . What is the smallest possible value for the sum of the digits of $S$ | 4 | AMC_10 | 3,016 |
2016_AMC_10B_Problems/Problem_7 | The ratio of the measures of two acute angles is $5:4$ , and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles? | 135 | AMC_10 | 3,018 |
2016_AMC_10B_Problems/Problem_8 | What is the tens digit of $2015^{2016}-2017?$ | 0 | AMC_10 | 3,020 |
2016_AMC_10B_Problems/Problem_9 | All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$ , with $A$ at the origin and $\overline{BC}$ parallel to the $x$ -axis. The area of the triangle is $64$ . What is the length of $BC$ | 8 | AMC_10 | 3,022 |
2016_AMC_10B_Problems/Problem_11 | Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden? | 336 | AMC_10 | 3,026 |
2016_AMC_10B_Problems/Problem_13 | At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets? | 100 | AMC_10 | 3,031 |
2016_AMC_10B_Problems/Problem_14 | How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$ , the line $y=-0.1$ and the line $x=5.1?$ | 50 | AMC_10 | 3,033 |
2016_AMC_10B_Problems/Problem_15 | All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$ . What is the number in the center? | 7 | AMC_10 | 3,036 |
2016_AMC_10B_Problems/Problem_16 | The sum of an infinite geometric series is a positive number $S$ , and the second term in the series is $1$ . What is the smallest possible value of $S?$ | 4 | AMC_10 | 3,040 |
2016_AMC_10B_Problems/Problem_17 | All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products? | 729 | AMC_10 | 3,046 |
2016_AMC_10B_Problems/Problem_18 | In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? | 7 | AMC_10 | 3,050 |
2016_AMC_10B_Problems/Problem_20 | A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$ . What distance does the origin $O(0,0)$ , move under this transformation? | 13 | AMC_10 | 3,054 |
2016_AMC_10B_Problems/Problem_21 | What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$ | 2 | AMC_10 | 3,059 |
2016_AMC_10B_Problems/Problem_22 | A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$ $B$ beat $C$ , and $C$ beat $A?$ | 385 | AMC_10 | 3,061 |
2016_AMC_10B_Problems/Problem_24 | How many four-digit integers $abcd$ , with $a \neq 0$ , have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$ , where $a=4$ $b=6$ $c=9$ , and $d=2$ | 17 | AMC_10 | 3,065 |
2015_AMC_10A_Problems/Problem_2 | A box contains a collection of triangular and square tiles. There are $25$ tiles in the box, containing $84$ edges total. How many square tiles are there in the box? | 9 | AMC_10 | 3,067 |
2015_AMC_10A_Problems/Problem_5 | Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$ . After he graded Payton's test, the test average became $81$ . What was Payton's score on the test? | 95 | AMC_10 | 3,071 |
2015_AMC_10A_Problems/Problem_6 | The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number? | 32 | AMC_10 | 3,073 |
2015_AMC_10A_Problems/Problem_7 | How many terms are in the arithmetic sequence $13$ $16$ $19$ $\dotsc$ $70$ $73$ | 21 | AMC_10 | 3,075 |
2015_AMC_10A_Problems/Problem_8 | Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ $1$ | 4 | AMC_10 | 3,079 |
2015_AMC_10A_Problems/Problem_9 | Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders? | 21 | AMC_10 | 3,080 |
2015_AMC_10A_Problems/Problem_10 | How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $ba$ | 2 | AMC_10 | 3,081 |
2015_AMC_10A_Problems/Problem_12 | Points $( \sqrt{\pi} , a)$ and $( \sqrt{\pi} , b)$ are distinct points on the graph of $y^2 + x^4 = 2x^2 y + 1$ . What is $|a-b|$ | 2 | AMC_10 | 3,083 |
2015_AMC_10A_Problems/Problem_13 | Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have? | 5 | AMC_10 | 3,086 |
2015_AMC_10A_Problems/Problem_15 | Consider the set of all fractions $\frac{x}{y}$ , where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$ , the value of the fraction is increased by $10\%$ | 1 | AMC_10 | 3,097 |
2015_AMC_10A_Problems/Problem_18 | Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$ . Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$ | 21 | AMC_10 | 3,099 |
2015_AMC_10A_Problems/Problem_20 | A rectangle with positive integer side lengths in $\text{cm}$ has area $A$ $\text{cm}^2$ and perimeter $P$ $\text{cm}$ . Which of the following numbers cannot equal $A+P$ | 102 | AMC_10 | 3,102 |
2015_AMC_10A_Problems/Problem_23 | The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of the possible values of $a?$ | 16 | AMC_10 | 3,103 |
2015_AMC_10A_Problems/Problem_24 | For some positive integers $p$ , there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$ , right angles at $B$ and $C$ $AB=2$ , and $CD=AD$ . How many different values of $p<2015$ are possible? | 31 | AMC_10 | 3,105 |
2015_AMC_10A_Problems/Problem_25 | Let $S$ be a square of side length $1$ . Two points are chosen independently at random on the sides of $S$ . The probability that the straight-line distance between the points is at least $\dfrac{1}{2}$ is $\dfrac{a-b\pi}{c}$ , where $a$ $b$ , and $c$ are positive integers with $\gcd(a,b,c)=1$ . What is $a+b+c$ | 59 | AMC_10 | 3,107 |
2015_AMC_10B_Problems/Problem_3 | Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$ , and one of the numbers is $28.$ What is the other number? | 8 | AMC_10 | 3,109 |
2015_AMC_10B_Problems/Problem_15 | The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet? | 47 | AMC_10 | 3,113 |
2015_AMC_10B_Problems/Problem_18 | Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads? | 56 | AMC_10 | 3,117 |