Practice (69)

2963

Three darts are thrown at at $3\times 3$ target, each landing in a different square. What is the probability that the squares they land in form a row, either horizontally, vertically or diagonally?

2967

A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.

2972

For a permutation $p = (a_1,a_2,\ldots,a_9)$ of the digits $1,2,\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$. Let $n$ denote the number of permutations $p$ with $s(p) = m$. Find $|m - n|$.

2981

Camy made a list of every possible distinct ﬁve-digit positive even integer that can be formed using each of the digits 1,3, 4, 5 and 9 exactly once in each integer. What is the sum of the integers on Camy’s list?

3046

How many rectangles of any size are in the grid shown here?

3369

The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?

3695

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?

3764

The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?

3769

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?

3828

Find the number of possible arrangements in Fisher Random Chess. The diagram below is one possible arrangement.

In a legal arrangement, the White's position must satisfy the following criteria:

Eight pawns must be in the $2^{nd}$ row. (The same as regular chess)
Two bishops must be in opposite colored squares (e.g. $b1$ and $e1$ in the above diagram)
King must locate between two rooks (e.g. in the diagram above, King is at $c1$ and two rooks are at $a1$ and $g1$)

The Black's position will be mirroring to the White's.

4103

Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders $RRGGG$, $GGGGR$, or $RRRRR$ will make Kathy happy, but $RRRGR$ will not. Find the probability that Kathy will be happy.

4109

Find the number of four-element subsets of $\{1,2,3,4,\dots, 20\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\{3,5,13,19\}$ and $\{6,10,20,18\}$ are two such subsets.

4112

For every subset $T$ of $U = \{ 1,2,3,\ldots,18 \}$, let $s(T)$ be the sum of the elements of $T$, with $s(\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.

4126

Find the number of permutations of $1, 2, 3, 4, 5, 6$ such that for each $k$ with $1$ $\leq$ $k$ $\leq$ $5$, at least one of the first $k$ terms of the permutation is greater than $k$.

4351

Seven students took a math exam. Every problem was solved at at most $3$ students. For every pair of students, there is at least one problem which was solved by both of these two students. What is the minimal number of problem this exam contains?

4414

$\textbf{Key Set}$

A sensitive location is protected by a door with multiple locks. This place has $11$ workers. The regulation requires that any combination of six workers can open all the locks, but any combination of five cannot. What is the minimal number of locks and how to distribute the keys?

4420

How many quadratic equations are there whose coefficients are distinct and are selected from $\{0,\ 1,\ 3,\ 5,\ 7\}$? Among these equations, how many have real roots?

4729

How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

4745

Let $T$ be the triangle in the coordinate plane with vertices $\left(0,0\right)$, $\left(4,0\right)$, and $\left(0,3\right)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)

4746

How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$?

4770

Randomly draw a card twice with replacement from $1$ to $10$, inclusive. What is the probability that the product of these two cards is a multiple of $7$?

4771

How many even $4$- digit integers are there whose digits are distinct?

Combinatorics BasicCountingPrinciple AdditionPrinciple MultiplicationPrinciple CountTheOpposite Probability BasicProbability NumberTheoryBasic IndeterminateEquation BrainTeaser

AMC8 AMC10/12 AIME

With Solutions

back to index | new