答案解析请参考文末
What is the difference between the sum of the first even counting numbers and the sum of the first odd counting numbers?
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?
A solid box is cm by cm by cm. A new solid is formed by removing a cube cm on a side from each corner of this box. What percent of the original volume is removed?
It takes Mary minutes to walk uphill km from her home to school, but it takes her only minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?
Let and denote the solutions of . What is the value of ?
Define to be for all real numbers and . Which of the following statements is not true?
for all and
for all and
for all
for all
if
How many non-congruent triangles with perimeter have integer side lengths?
What is the probability that a randomly drawn positive factor of is less than ?
Simplify
.
The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?
The sum of the two 5-digit numbers and is . What is ?
A point is randomly picked from inside the rectangle with vertices , , , and . What is the probability that ?
The sum of three numbers is . The first is four times the sum of the other two. The second is seven times the third. What is the product of all three?
Let be the largest integer that is the product of exactly 3 distinct prime numbers , , and , where and are single digits. What is the sum of the digits of ?
What is the probability that an integer in the set is divisible by and not divisible by ?
What is the units digit of ?
The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?
What is the sum of the reciprocals of the roots of the equation
?
A semicircle of diameter sits at the top of a semicircle of diameter , as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
A base-10 three digit number is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of are both three-digit numerals?
Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?
In rectangle , we have , , is on with , is on with , line intersects line at , and is on line with . Find the length of .
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have rows of small congruent equilateral triangles, with small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of small equilateral triangles?
Sally has five red cards numbered through and four blue cards numbered through . She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
Let be a -digit number, and let and be the quotient and the remainder, respectively, when is divided by . For how many values of is divisible by ?
The formula for the sum of the first even numbers, is , (E standing for even).
Sum of first odd numbers, is , (O standing for odd).
Knowing this, plug for ,
.
We can use the sum and product of a quadratic:
Thus the answer is .
Another way to think of it is that a cube missing one face has of its faces. Since the shape has faces already, we need another face. The only way to add another face is if the added square does not overlap any of the others. ,, and overlap, while squares to do not. The answer is
Therefore, .
To solve this matrix equation, we can rearrange it thus:
Solving this matrix equation by using inverse matrices and matrix multiplication yields
Which means that , , and . Therefore,
The largest possible value of is .
So, the sum of the digits of is
So the answer is .
Dividing both sides by ,
,
we see by Vieta's formulas that the sum of the roots is .
The area of the triangle is .
So the shaded area is .
Pat needs to select more cookies that are either oatmeal or peanut butter.
The assortments are: assortments.
Pat selects chocolate chip cookies:
Pat needs to select more cookies that are either oatmeal or peanut butter.
The assortments are: assortments.
Pat selects chocolate chip cookies:
Pat needs to select more cookies that are either oatmeal or peanut butter.
The assortments are: assortments.
Pat selects chocolate chip cookies:
Pat needs to select more cookies that are either oatmeal or peanut butter.
The assortments are: assortments.
Pat selects chocolate chip cookies:
Pat needs to select more cookies that are either oatmeal or peanut butter.
The assortments are: assortments.
Pat selects chocolate chip cookies:
Pat needs to select more cookies that are either oatmeal or peanut butter.
The only assortment is: assortment.
The total number of assortments of cookies that can be collected is
There is a much faster way to do casework.
Case 1: 1 type of cookie - 3 ways
Case 2: 2 types of cookies - 3 ways to choose the 2 types of cookies, and 5 ways to choose how of each there are 1 and 5 2 and 4 3 and 3 4 and 2 5 and 1
cases for case 2
Case 3: 3 types of cookies - quick examination shows us that the only ways to use all three cookies are the following: 4, 1, 1: this gives us 3!/2!*1! = 3 ways 3, 2, 1: this gives us 3! = 6 ways 2, 2, 2: this gives us 1 way total of 10 cases for this case 10+15+3= total
It is given that it is possible to select at least 6 of each. Therefore, we can make a bijection to the number of ways to divide the six choices into three categories, since it is assumed that their order is unimportant. Using the ball and urns/sticks and stones/stars and bars formula, the number of ways to do this is
Suppose the six cookies to be chosen are the stars, as we attempt to implement stars and bars. We take two dividers, and place them between the cookies, such that the six cookies are split into 3 groups, where the groups are the number of chocolate chip, oatmeal, and peanut butter cookies, and each group can have 0. First, assume that the two dividers cannot go in between the same two cookies. By stars and bars, there are ways to make the groups.
Finally, since the two dividers can be together, we must add those cases where the two dividers are in the same space between cookies. There are 7 spaces, and hence 7 cases.
Our final answer is
.
So .
.
Therefore .
Since is a rectangle, .
Since is a rectangle and , .
Since is a rectangle, .
So, is a transversal, and .
This is sufficient to prove that and .
Using ratios:
Since can't have 2 different lengths, both expressions for must be equal.
Since is a rectangle, , , and . From the Pythagorean Theorem, .
Statement:
Proof: , obviously.
Since two angles of the triangles are equal, the third angles must equal each other. Therefore, the triangles are similar.
Let .
Also, , therefore
We can multiply both sides by to get that is twice of 10, or
We extend such that it intersects at . Since is a rectangle, it follows that , therefore, . Let . From the similarity of triangles and , we have the ratio (as , and ). and are the altitudes of and , respectively. Thus, , from which we have , thus
Since and we have Thus, Suppose and Thus, we have Additionally, now note that which is pretty obvious from insight, but can be proven by AA with extending to meet From this new pair of similar triangles, we have Therefore, we have by combining those two equations,Solving, we have and therefore
Experimenting a bit we find that the number of toothpicks needs a triangle with , and rows is , and respectively. Since , and are triangular numbers we know that depending on how many rows there are in the triangle, the number we multiply by to find total no.toothpicks is the corresponding triangular number. Since the triangle in question has rows, we can use to find the triangular number for that row and multiply by , hence finding the total no.toothpicks; this is just .
This yields the following arrangement from top to bottom:
Therefore, the sum of the numbers on the middle three cards is .
Let equal , where through are digits. Therefore,
We now take :
The divisor trick for 11 is as follows:
"Let be an digit integer. If is divisible by , then is also divisible by ."
Therefore, the five digit number is divisible by 11. The 5-digit multiples of 11 range from to . There are divisors of 11 between those inclusive.
Since is a quotient and is a remainder when is divided by . So we have . Since we are counting choices where is divisible by , we have for some . This means that is the sum of two multiples of and would thus itself be a multiple of . Then we can count all the five digit multiples of as in Solution 2. (This solution is essentially the same as Solution 2, but it does not necessarily involve mods and so could potentially be faster.)
以上解析方式仅供参考
学术活动报名扫码了解!免费领取历年真题!
翰林课程体验,退费流程快速投诉邮箱: yuxi@linstitute.net 沪ICP备2023009024号-1