question_id stringlengths 1 6 | correct_count int64 1 4 | incorrect_count int64 4 7 | total_responses int64 8 8 | problem stringlengths 18 2.09k | answer stringlengths 1 84 | solution stringlengths 0 11.1k |
|---|---|---|---|---|---|---|
8 | 2 | 6 | 8 | During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single? | 80\% | 1. **Identify the total number of hits**: Judy had a total of 35 hits during the season.
2. **Determine the number of hits that were not singles**:
- Home runs: 1
- Triples: 1
- Doubles: 5
- Total non-single hits = 1 (home run) + 1 (triple) + 5 (doubles) = 7
3. **Calculate the number of singles**:
\[
... |
19 | 3 | 5 | 8 | Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$ | 360 | 1. **Assign Variables and Use Pythagorean Theorem in $\triangle ABC$:**
Let $AB = x$ and $BC = y$. Since $\angle ABC = 90^\circ$, by the Pythagorean theorem, we have:
\[
x^2 + y^2 = AC^2 = 20^2 = 400.
\]
2. **Calculate Area of $\triangle ACD$:**
Since $\angle ACD = 90^\circ$, the area of $\triangle ACD$... |
22 | 2 | 6 | 8 | Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$? | $a+4$ | 1. **Define the sequence and calculate the average $b$:**
The five consecutive integers starting with $a$ are $a, a+1, a+2, a+3, a+4$. The average of these integers, $b$, is calculated as follows:
\[
b = \frac{a + (a+1) + (a+2) + (a+3) + (a+4)}{5} = \frac{5a + 10}{5} = a + 2
\]
2. **Determine the new seq... |
51 | 3 | 5 | 8 | Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ... | 192 | 1. **Understanding the Problem**: We are given four circles with centers at $A$, $B$, $C$, and $D$. Points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and similarly for circles $C$ and $D$. The distances $AB$ and $CD$ are both 39, and the length of segm... |
101 | 4 | 4 | 8 | For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? | 19 | To solve this problem, we need to consider the possible number of intersection points formed by four distinct lines in a plane. Each pair of lines can intersect at most once, and the maximum number of intersection points is determined by the number of ways to choose 2 lines from 4, which is given by the binomial coeffi... |
226 | 2 | 6 | 8 | For each positive integer $n$, let
$a_n = \frac{(n+9)!}{(n-1)!}$.
Let $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is | 9 | 1. **Expression Simplification**:
Given $a_n = \frac{(n+9)!}{(n-1)!}$, we can simplify this as:
\[
a_n = n(n+1)(n+2)\cdots(n+9)
\]
This is the product of 10 consecutive integers starting from $n$.
2. **Factorization**:
We can express $a_n$ in terms of its prime factors as $2^{x_n} 5^{y_n} r_n$, where... |
247 | 1 | 7 | 8 | Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | 85 | 1. **Understanding the Problem:**
The problem asks us to find the possible values of the slope $m$ of a line such that exactly $300$ lattice points from the set $S$ (where $S$ consists of points $(x,y)$ with $1 \leq x, y \leq 30$) lie on or below the line $y = mx$. The total number of lattice points in $S$ is $30 \t... |
276 | 4 | 4 | 8 | The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? | \frac{1}{14} |
To solve this problem, we need to ensure that the sum of the numbers in each row and each column is odd. We can achieve this by having either three odd numbers or one odd and two even numbers in each row and column. Let's analyze the possible configurations and calculate the probability.
#### Step 1: Understanding th... |
282 | 1 | 7 | 8 | Two circles lie outside regular hexagon $ABCDEF$. The first is tangent to $\overline{AB}$, and the second is tangent to $\overline{DE}$. Both are tangent to lines $BC$ and $FA$. What is the ratio of the area of the second circle to that of the first circle? | 81 | 1. **Assumption and Setup**: Assume without loss of generality (WLOG) that the regular hexagon $ABCDEF$ has a side length of 1. The first circle is tangent to $\overline{AB}$ and the second circle is tangent to $\overline{DE}$. Both circles are tangent to lines $BC$ and $FA$.
2. **First Circle's Radius**: The first ci... |
290 | 1 | 7 | 8 | Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt {50}$ and $BE = 1$. What is the area of the inner square $EFGH$? | 36 | 1. **Understanding the Problem Setup:**
- We have two squares, $ABCD$ and $EFGH$, where $EFGH$ is inside $ABCD$.
- Each side of $EFGH$ can be extended to pass through a vertex of $ABCD$.
- The side length of $ABCD$ is given as $\sqrt{50}$.
- The distance from vertex $B$ of square $ABCD$ to the nearest point... |
310 | 1 | 7 | 8 | Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd. | 2016 | 1. **Observation of $f(n)$**: We start by observing the function $f(n)$, which counts the number of ways to write $n$ as a sum of powers of 2, considering the order of terms. For example, $f(4) = 6$ because we can write $4$ as:
- $4$
- $2 + 2$
- $2 + 1 + 1$
- $1 + 2 + 1$
- $1 + 1 + 2$
- $1 + 1 + 1 + 1... |
311 | 3 | 5 | 8 | There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$. What is $N$? | 78 | 1. **Identify the integer solution condition**:
The quadratic equation given is:
\[
5x^2 + kx + 12 = 0
\]
For this equation to have an integer solution, let's denote that integer by $-n$. Thus, the equation can be factored as:
\[
(5x + \frac{12}{n})(x + n) = 0
\]
This factorization is valid... |
325 | 2 | 6 | 8 | The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
some rotation ar... | 2 | To determine which transformations will map the figure onto itself, we analyze each type of transformation given in the problem:
1. **Some rotation around a point on line $\ell$:**
- Consider a rotation of $180^\circ$ around a point exactly halfway between an up-facing square and a down-facing square on line $\ell$... |
328 | 4 | 4 | 8 | Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have? | $20\%$ | Let's denote the amount of soda Jacqueline has as $J$. According to the problem, Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. We can express the amounts of soda Liliane and Alice have in terms of $J$.
1. **Calculate the amount of soda Liliane has:**
Liliane has $50\%... |
332 | 2 | 6 | 8 | The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$? | 279 | To solve this problem, we need to understand the relationship between the powers of $5$ and $2$ and how they are distributed between $5^n$ and $5^{n+1}$.
1. **Understanding the relationship between $5^n$ and $2^m$:**
We know that $5^{867}$ is between $2^{2013}$ and $2^{2014}$. This gives us a way to compare the gro... |
340 | 3 | 5 | 8 | A triangle with vertices $(6, 5)$, $(8, -3)$, and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles? | \frac{32}{3} | 1. **Identify the vertices and their reflections**:
- Let $A = (6, 5)$, $B = (8, -3)$, and $C = (9, 1)$.
- Reflecting over the line $x=8$, we find:
- $A' = (10, 5)$ since reflecting $(6, 5)$ over $x=8$ moves it 2 units to the left of $x=8$, so it moves 2 units to the right of $x=8$ to $(10, 5)$.
- $B'... |
358 | 4 | 4 | 8 | How many ways are there to paint each of the integers $2, 3, \cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? | 432 | To solve this problem, we need to consider the constraints imposed by the requirement that each number must have a different color from each of its proper divisors. We start by identifying the divisors of each number from $2$ to $9$:
- $2$: No proper divisors in the list.
- $3$: No proper divisors in the list.
- $4$: ... |
370 | 4 | 4 | 8 | John scores 93 on this year's AHSME. Had the old scoring system still been in effect, he would score only 84 for the same answers.
How many questions does he leave unanswered? | 9 | Let $c$, $w$, and $u$ be the number of correct, wrong, and unanswered questions respectively. We are given three pieces of information:
1. Under the old scoring system, John's score is $84$. The old scoring system awards $30$ points initially, $4$ points for each correct answer, subtracts $1$ point for each wrong answ... |
374 | 4 | 4 | 8 | The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
[asy]
unitsize(3mm); defaultpen(linewidth(0.8pt));
path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0);
path p2=(0,1)--(1,1)--(1,0);
path p3=(2,0)--(2,1)--(3,1);
path p4=(3,2)--(2... | 56 | 1. **Identify the basic unit of tiling**: The tiling pattern consists of a large square divided into 9 smaller squares, each with side length $a$. The large square thus has a side length of $3a$.
2. **Calculate the area of the large square**: The area of the large square is $(3a)^2 = 9a^2$.
3. **Identify the areas co... |
387 | 1 | 7 | 8 | If $|x-\log y|=x+\log y$ where $x$ and $\log y$ are real, then | x(y-1)=0 | Given the equation $|x-\log y|=x+\log y$, we need to consider the properties of the absolute value function and the possible values of $x$ and $\log y$.
1. **Understanding the absolute value equation**:
The absolute value equation $|a| = b$ holds if and only if $a = b$ or $a = -b$, and $b \geq 0$. Applying this to ... |
391 | 1 | 7 | 8 | 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 | 1. **Define the problem**: We need to find the smallest possible value for the sum of the digits of $S$, where $S$ is the sum of two three-digit numbers $a$ and $b$. Each digit among $a$ and $b$ is unique.
2. **Set constraints on $a$ and $b$**: Since $a$ and $b$ are three-digit numbers and all digits are different, we... |
409 | 1 | 7 | 8 | In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D, E$, and $F$ lie on sides $BC, AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals | 50^\circ | 1. **Identify the properties of triangle $ABC$**: Given that $AB = AC$, triangle $ABC$ is isosceles. Also, $\angle A = 80^\circ$. Since the sum of angles in a triangle is $180^\circ$, and $AB = AC$, the base angles $\angle B$ and $\angle C$ are equal. Thus, we calculate:
\[
\angle B = \angle C = \frac{180^\circ -... |
413 | 4 | 4 | 8 | Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.
Suppose $PQ=6$ and... | 81 | 1. **Identify the given information and setup:**
- $ABCD$ is a parallelogram with area $15$.
- $PQ = 6$ and $RS = 8$ are the lengths of the projections of $A$ and $C$ onto $BD$, and $B$ and $D$ onto $AC$, respectively.
- We need to find $d^2$, where $d$ is the length of diagonal $BD$.
2. **Use the properties ... |
426 | 3 | 5 | 8 | Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle{ABC}$? | $3\sqrt{3}$ | 1. **Understanding the Configuration**: We are given a central regular hexagon surrounded by six regular hexagons, all of the same side length $1$. We need to find the area of $\triangle{ABC}$, which is formed by connecting the centers of three adjacent outer hexagons.
2. **Hexagon and Its Properties**: Each hexagon i... |
434 | 3 | 5 | 8 | Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 = m + n\sqrt p$, where $m$, $n$, and $p$ ar... | 752 | 1. **Understanding the relationship between $f$ and $g$:**
Given that $g(x) = -f(100 - x)$, we can infer that the graph of $g$ is a $180^\circ$ rotation of the graph of $f$ around the point $(50, 0)$. This is because replacing $x$ with $100 - x$ reflects the graph across the line $x = 50$, and the negative sign refl... |
487 | 2 | 6 | 8 | For how many integers $n$ between $1$ and $50$, inclusive, is $\frac{(n^2-1)!}{(n!)^n}$ an integer? | 34 | 1. **Understanding the Expression**: We start by analyzing the expression \[\frac{(n^2-1)!}{(n!)^n}.\] We need to determine for how many integers $n$ between $1$ and $50$, inclusive, this expression is an integer.
2. **Rewriting the Expression**: We can rewrite the expression as:
\[\frac{(n^2-1)!}{(n!)^n} = \frac{(... |
506 | 1 | 7 | 8 | Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $1$ through $46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property— the sum of the ba... | \frac{1}{4} | 1. **Understanding the Problem**: Professor Gamble needs to pick six different integers from 1 to 46 such that the sum of the base-ten logarithms of these numbers is an integer. This implies that the product of these numbers must be a power of 10, as the logarithm of a power of 10 is an integer.
2. **Identifying Eligi... |
534 | 2 | 6 | 8 | Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\overline{AB}$ and $\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$? | 12 | 1. **Identify the Triangle and Hypotenuse**:
Given a right triangle $ABC$ with legs $AB = 20$ and $BC = 21$, we first calculate the length of the hypotenuse $AC$ using the Pythagorean theorem:
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{20^2 + 21^2} = \sqrt{400 + 441} = \sqrt{841} = 29.
\]
2. **Determine the Altit... |
541 | 2 | 6 | 8 | Regular polygons with $5, 6, 7,$ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect? | 68 | To solve this problem, we need to calculate the number of intersection points inside the circle where two sides from different polygons intersect. We consider pairs of polygons and count the intersections for each pair.
1. **Understanding the Intersection Rule**:
- When two polygons with $m$ and $n$ sides ($m > n$)... |
550 | 1 | 7 | 8 | Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice? | 40 | 1. **Determine the amount of juice from pears and oranges:**
- Miki extracts 8 ounces of pear juice from 3 pears. Therefore, the amount of pear juice per pear is:
\[
\frac{8 \text{ ounces}}{3 \text{ pears}} = \frac{8}{3} \text{ ounces per pear}
\]
- Miki extracts 8 ounces of orange juice from 2 ora... |
582 | 3 | 5 | 8 | Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide? | 5 | 1. **Identify the total number of triangles in each half**: Each half of the figure contains 3 red triangles, 5 blue triangles, and 8 white triangles.
2. **Analyze the red triangles**: When the upper half is folded down, 2 pairs of red triangles coincide. This means 4 red triangles (2 from each half) are involved in c... |
598 | 1 | 7 | 8 | Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$? | \frac{1}{5} | 1. **Identify Key Points and Setup**: Let's denote the vertices of $\triangle ABC$ as follows: $A$ and $B$ are the endpoints of one diagonal of a unit square, and $C$ is a point on an adjacent side of another unit square. We need to find the area of this triangle.
2. **Understanding the Geometry**: The problem states ... |
601 | 2 | 6 | 8 | For how many positive integers $n \le 1000$ is$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$not divisible by $3$? | 22 | We are tasked with finding how many positive integers \( n \leq 1000 \) make the expression
\[
\left\lfloor \frac{998}{n} \right\rfloor + \left\lfloor \frac{999}{n} \right\rfloor + \left\lfloor \frac{1000}{n} \right\rfloor
\]
not divisible by 3. We start by analyzing the behavior of the floor function in this context.
... |
615 | 3 | 5 | 8 | Samuel's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exact... | 20 | To solve this problem, we need to determine how many of the smaller $1 \times 1 \times 1$ inch cubes have icing on exactly two sides. We will analyze the positions of these cubes on the cake.
1. **Understanding the Cake Structure**:
- The cake is a $4 \times 4 \times 4$ cube.
- Icing is on the top and all four v... |
619 | 2 | 6 | 8 | The bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory? | \frac{3}{4} | 1. **Identify Satisfactory Grades**: According to the problem, grades A, B, C, and D are considered satisfactory.
2. **Count the Number of Satisfactory Grades**:
- Number of students with grade A = 5
- Number of students with grade B = 4
- Number of students with grade C = 3
- Number of students with grade... |
626 | 2 | 6 | 8 | Part of an \(n\)-pointed regular star is shown. It is a simple closed polygon in which all \(2n\) edges are congruent, angles \(A_1,A_2,\cdots,A_n\) are congruent, and angles \(B_1,B_2,\cdots,B_n\) are congruent. If the acute angle at \(A_1\) is \(10^\circ\) less than the acute angle at \(B_1\), then \(n=\) | 36 | 1. **Understanding the Star Polygon**: In the given problem, we have a regular star polygon with $n$ points. Each point of the star has two angles associated with it: one at $A_i$ and one at $B_i$ for $i = 1, 2, \ldots, n$. All $A_i$ angles are congruent, and all $B_i$ angles are congruent. The acute angle at each $A_i... |
655 | 3 | 5 | 8 | If the pattern in the diagram continues, what fraction of eighth triangle would be shaded?
[asy] unitsize(10); draw((0,0)--(12,0)--(6,6sqrt(3))--cycle); draw((15,0)--(27,0)--(21,6sqrt(3))--cycle); fill((21,0)--(18,3sqrt(3))--(24,3sqrt(3))--cycle,black); draw((30,0)--(42,0)--(36,6sqrt(3))--cycle); fill((34,0)--(32,2sq... | \frac{7}{16} | 1. **Identify the pattern of shaded triangles**: Observing the given sequence of diagrams, we notice that the number of shaded triangles in each subsequent diagram follows the sequence of triangular numbers. The sequence of triangular numbers is defined by the formula $T_n = \frac{n(n+1)}{2}$, where $n$ is the term num... |
676 | 4 | 4 | 8 | 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 | 1. **Identify the region and lattice points**:
The region is bounded by the lines $y = \pi x$, $y = -0.1$, and $x = 5.1$. We consider only the lattice points (points with integer coordinates) that lie within this region. The relevant lattice points along the x-axis from $x = 0$ to $x = 5$ are:
- $(0,0)$
- $(1,... |
719 | 1 | 7 | 8 | A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\frac{n}{11}$ is also a palindrome? | \frac{11}{30} | To solve this problem, we first need to understand the structure of a 6-digit palindrome and then determine how many of these palindromes are divisible by 11 and also form a palindrome when divided by 11.
1. **Structure of a 6-digit palindrome**:
A 6-digit palindrome can be represented as $n = \overline{abcba}$, w... |
745 | 4 | 4 | 8 | Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ and $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares? | \frac{1}{3} | 1. **Assign Side Lengths**: Let the side length of each square $ABCD$, $EFGH$, and $GHIJ$ be $s = 1$ for simplicity.
2. **Identify Key Points and Relationships**:
- Points $C$ and $D$ are midpoints of sides $IH$ and $HE$ respectively, so $HC = CI = \frac{1}{2}$ and $HE = ED = \frac{1}{2}$.
- Let $X$ be the inter... |
746 | 3 | 5 | 8 | The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$? | 12 | 1. **Count the number of factors of 10 in $90!$:**
The number of factors of 10 in $90!$ is determined by the number of factors of 5, as there are more factors of 2 than 5. We calculate this using the formula for the number of factors of a prime $p$ in $n!$:
\[
\left\lfloor \frac{90}{5} \right\rfloor + \left\lf... |
762 | 4 | 4 | 8 | A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is: | 540 | Let's consider a quadrilateral $ABCD$ inscribed in a circle. We need to find the sum of the angles inscribed in the four segments outside the quadrilateral.
1. **Identify the Segments and Angles**:
- The four segments outside the quadrilateral are the regions outside $ABCD$ but inside the circle.
- Let $\alpha$,... |
848 | 4 | 4 | 8 | A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? | 70 | 1. **Identify the fraction of pink and red flowers**:
Given that six tenths (or $\frac{6}{10}$) of the flowers are pink, we can simplify this fraction to $\frac{3}{5}$. Consequently, the remaining flowers must be red, which is $\frac{2}{5}$ of the total flowers (since the total must sum to 1, or $\frac{5}{5}$).
2.... |
854 | 1 | 7 | 8 | Let $\{a_k\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general,
\[a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k... | 1341 | To solve this problem, we need to understand the behavior of the sequence $\{a_k\}_{k=1}^{2011}$ and determine when $a_k = b_k$, where $\{b_k\}_{k=1}^{2011}$ is the sequence $\{a_k\}_{k=1}^{2011}$ rearranged in decreasing order.
#### Step 1: Analyze the sequence $\{a_k\}_{k=1}^{2011}$
The sequence is defined recursive... |
859 | 1 | 7 | 8 | The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.
$\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0 \\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}$ | 8 | 1. **Identify the Incorrect Sum**: First, we add the given numbers without changing any digits:
- $742586 + 829430 = 1572016$
- The provided sum is $1212016$.
2. **Analyze the Incorrectness**: The provided sum $1212016$ differs significantly from the actual sum $1572016$. We need to change one digit $d$ to anoth... |
860 | 3 | 5 | 8 | The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$? | 12 | 1. **Count the number of factors of 10 in $90!$:**
The number of factors of 10 in $90!$ is determined by the number of factors of 5, as there are more factors of 2 than 5. We calculate this using the formula for the number of factors of a prime $p$ in $n!$:
\[
\left\lfloor \frac{90}{5} \right\rfloor + \left\lf... |
864 | 2 | 6 | 8 | Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together? | 120 | 1. **Understanding the problem**: We are given that Brenda received $36$ votes, which represents $\frac{3}{10}$ of the total votes in the election.
2. **Calculating the total number of votes**:
- Since $36$ votes is $\frac{3}{10}$ of the total votes, we can find the total number of votes by setting up the equation:... |
872 | 4 | 4 | 8 | For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal to $97$ does $D(n) = 2$? | 26 | To solve for the number of positive integers less than or equal to $97$ for which $D(n) = 2$, we analyze the binary representations of numbers and count those with exactly two transitions between adjacent digits (from 0 to 1 or from 1 to 0).
#### Case Analysis:
For $D(n) = 2$, the binary representation of $n$ must hav... |
886 | 2 | 6 | 8 | Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. 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? | 12 | 1. **Identify the possible placements for $R_5$ and $B_5$:**
- Since $R_5$ divides $B_5$ and $B_5 = 5$ is the only blue card divisible by $R_5 = 5$, $R_5$ must be adjacent to $B_5$.
- $B_5$ cannot be adjacent to any other red card except $R_1$ and $R_5$ because $5$ is not divisible by $2, 3,$ or $4$.
- Thus, $... |
913 | 1 | 7 | 8 | Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (not the original number the person picked.)
The number pick... | 1 | Let's denote the number picked by person $i$ as $a_i$. According to the problem, each person announces the average of the numbers picked by their two immediate neighbors. Therefore, if person $i$ announces $i$, the equation relating the numbers picked by their neighbors is:
\[
\frac{a_{i-1} + a_{i+1}}{2} = i
\]
which s... |
920 | 4 | 4 | 8 | On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to | 2.1 | 1. **Identify the vertices and their relationships**: In a $4 \times 4 \times 3$ rectangular parallelepiped, let's assume $D$ is at the origin $(0,0,0)$, and $A$, $B$, $C$ are at $(4,0,0)$, $(0,4,0)$, and $(0,0,3)$ respectively. These vertices are adjacent to $D$.
2. **Volume of pyramid $ABCD$**: The volume $V$ of a p... |
925 | 3 | 5 | 8 | In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of... | 5 | 1. **Calculate the total number of games**: In a round-robin tournament with 6 teams, each team plays against every other team exactly once. The total number of games played is given by the combination formula $\binom{n}{2}$, where $n$ is the number of teams. Thus, the total number of games is:
\[
\binom{6}{2} = ... |
931 | 1 | 7 | 8 | The number $121_b$, written in the integral base $b$, is the square of an integer, for | $b > 2$ | 1. **Convert the base-b number to base-10**: The number $121_b$ in base $b$ can be expressed in base 10 as:
\[
1 \cdot b^2 + 2 \cdot b^1 + 1 \cdot b^0 = b^2 + 2b + 1
\]
2. **Factorize the expression**: The expression $b^2 + 2b + 1$ can be rewritten by recognizing it as a perfect square:
\[
b^2 + 2b + 1 ... |
935 | 3 | 5 | 8 | For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ | 10 | To solve this problem, we need to understand the function $f_j(n)$ and its behavior as $j$ increases. We start by analyzing the function $f_1(n)$, which is defined as twice the number of positive integer divisors of $n$. We then recursively apply $f_1$ to its own outputs to determine $f_j(n)$ for $j \geq 2$.
#### Step... |
946 | 2 | 6 | 8 | For what value of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$? | 5 | 1. **Identify the domain**: The equation given is $\frac{x-1}{x-2} = \frac{x-k}{x-6}$. We must exclude values of $x$ that make the denominators zero, hence the domain is $\mathbb{R} \setminus \{2,6\}$.
2. **Cross-multiply to eliminate fractions**:
\[
(x-1)(x-6) = (x-k)(x-2)
\]
Expanding both sides, we get... |
948 | 1 | 7 | 8 | Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon... | 147 | 1. **Understanding the Problem:**
We have three identical square sheets of paper, each with side length $6$. The middle sheet is rotated $30^\circ$ clockwise, and the top sheet is rotated $60^\circ$ clockwise. We need to find the area of the resulting $24$-sided polygon.
2. **Breaking Down the Polygon:**
The pol... |
950 | 4 | 4 | 8 | What is the maximum value of $\frac{(2^t-3t)t}{4^t}$ for real values of $t?$ | \frac{1}{12} | 1. **Substitute $2^t = x$**: We start by letting $2^t = x$, which implies that $\log_2{x} = t$. This substitution simplifies the expression:
\[
\frac{(2^t-3t)t}{4^t} = \frac{(x - 3\log_2{x})\log_2{x}}{x^2}.
\]
2. **Rewrite the expression**: Using the properties of logarithms, we can rewrite the expression as:... |
963 | 2 | 6 | 8 | Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What is the probability that after the bell ha... | \frac{1}{4} | 1. **Initial Setup and State Description:**
Each player starts with $1. The possible states of money distribution after each round are $(1-1-1)$ and $(2-1-0)$ in some permutation. The state $(3-0-0)$ is not possible because:
- A player cannot give money to themselves.
- A maximum of $2 is being distributed, an... |
975 | 2 | 6 | 8 | The 16 squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:
(1) fold the top half over the bottom half
(2) fold the bottom half over the top half
(3) fold the right half over the left half
(4) fold the left half over... | 9 | To solve this problem, we need to track the position of the top square through each fold. We start by visualizing the initial configuration of the squares and then follow each fold step-by-step.
#### Initial Configuration:
The squares are arranged in a $4 \times 4$ grid, numbered from 1 to 16. The numbering is assumed... |
982 | 1 | 7 | 8 | Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\overline{AC}$ intersects $\overline{BP}$ at point $D$. If $PB = 3$ and $PD= 2$, then $AD\cdot CD =$ | 5 | 1. **Identify the Circle and Key Points**: Since $P$ is equidistant from $A$ and $B$, $P$ lies on the perpendicular bisector of $\overline{AB}$. Given that $\angle APB = 2\angle ACB$, and $P$ is equidistant from $A$ and $B$, we can infer that $A$, $B$, and $C$ lie on a circle centered at $P$ with radius $PA = PB$.
2. ... |
983 | 4 | 4 | 8 | When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.
For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$ $7 \t... | 83 | We start by understanding the function used by the machine. If $N$ is the input, the output $O$ is given by:
- If $N$ is odd, $O = 3N + 1$.
- If $N$ is even, $O = \frac{N}{2}$.
We need to find the inverse of this function to trace back from $O = 1$ to the original input $N$ after six steps. The inverse function can be... |
1016 | 4 | 4 | 8 | The roots of the equation $x^{2}-2x = 0$ can be obtained graphically by finding the abscissas of the points of intersection of each of the following pairs of equations except the pair:
[Note: Abscissas means x-coordinate.] | $y = x$, $y = x-2$ | To solve this problem, we need to find the roots of the equation $x^2 - 2x = 0$ and check which pair of equations does not yield these roots when their graphs intersect.
1. **Finding the roots of the equation $x^2 - 2x = 0$:**
\[
x^2 - 2x = 0 \implies x(x - 2) = 0
\]
Setting each factor equal to zero gives... |
1022 | 2 | 6 | 8 | How many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally? | 36 | We are tasked with placing $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in a $3 \times 3$ grid such that no two chips of the same color are directly adjacent either vertically or horizontally.
#### Step 1: Fixing the position of one chip
To simplify the probl... |
1037 | 3 | 5 | 8 | A circle with center $O$ has area $156\pi$. Triangle $ABC$ is equilateral, $\overline{BC}$ is a chord on the circle, $OA = 4\sqrt{3}$, and point $O$ is outside $\triangle ABC$. What is the side length of $\triangle ABC$? | $6$ | 1. **Calculate the radius of the circle**:
The area of the circle is given by the formula $A = \pi r^2$. Given $A = 156\pi$, we solve for $r$:
\[
\pi r^2 = 156\pi \implies r^2 = 156 \implies r = \sqrt{156} = 12\sqrt{3}
\]
2. **Analyze the position of point $A$**:
Since $OA = 4\sqrt{3}$ and $r = 12\sqrt{... |
1043 | 3 | 5 | 8 | Marvin had a birthday on Tuesday, May 27 in the leap year $2008$. In what year will his birthday next fall on a Saturday? | 2017 | To determine the next year when Marvin's birthday, May 27, falls on a Saturday after 2008, we need to consider the day of the week progression from 2008 onwards, taking into account whether each year is a leap year or not.
1. **Day Increment Calculation**:
- In a non-leap year, there are 365 days, which is equivale... |
1052 | 2 | 6 | 8 | Jim starts with a positive integer $n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $n = 55$, then his sequence contains $5$ numbers:
$\begin{array... | 3 | To solve this problem, we need to construct a sequence of numbers starting from $N$ such that the sequence has exactly 8 numbers, including $N$ and $0$. Each number in the sequence is obtained by subtracting the largest perfect square less than or equal to the current number.
We start from the last step and work our w... |
1067 | 4 | 4 | 8 | A list of integers has mode 32 and mean 22. The smallest number in the list is 10. The median m of the list is a member of the list. If the list member m were replaced by m+10, the mean and median of the new list would be 24 and m+10, respectively. If m were instead replaced by m-8, the median of the new list wou... | 20 | 1. **Understanding the problem and setting up equations:**
Let the number of integers in the list be $n$. The list has a mean of $22$, so the sum of all integers in the list is $22n$.
2. **Analyzing the effect of replacing $m$ with $m+10$:**
When $m$ is replaced by $m+10$, the sum of the integers becomes $22n + ... |
1086 | 4 | 4 | 8 | Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt{2}.$ Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt{5}$ and $\angle AEC = 45^{\circ}.$ What is $CE^2+DE^2?$ | 100 | 1. **Identify the Geometry and Given Information:**
- Let $O$ be the center of the circle.
- $\overline{AB}$ is a diameter, so $A$, $O$, and $B$ are collinear, and $OA = OB = 5\sqrt{2}$.
- $\overline{CD}$ intersects $\overline{AB}$ at $E$ such that $BE = 2\sqrt{5}$ and $\angle AEC = 45^\circ$.
2. **Reflect Po... |
1088 | 1 | 7 | 8 | Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is $CF$? | 30 | 1. **Identify the triangle and given lengths**: We are given a triangle $ABC$ with side lengths $AB = 11$, $BC = 24$, and $CA = 20$.
2. **Angle bisector and point definitions**: The bisector of $\angle BAC$ intersects $BC$ at point $D$ and the circumcircle of $\triangle ABC$ at point $E \ne A$. The circumcircle of $\... |
1123 | 4 | 4 | 8 | A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating th... | 24.2 | 1. **Identify the relationship between the trees' heights:** Each tree is either twice as tall or half as tall as the one to its right. This means that for any tree $i$ and tree $i+1$, the height of tree $i$ is either $2 \times \text{height of tree } i+1$ or $\frac{1}{2} \times \text{height of tree } i+1$.
2. **Use th... |
1127 | 3 | 5 | 8 | [asy]
draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle, black+linewidth(.75));
draw((0,-1)--(0,1), black+linewidth(.75));
draw((-1,0)--(1,0), black+linewidth(.75));
draw((-1,-1/sqrt(3))--(1,1/sqrt(3)), black+linewidth(.75));
draw((-1,1/sqrt(3))--(1,-1/sqrt(3)), black+linewidth(.75));
draw((-1/sqrt(3),-1)--(1/sqrt(3),1), blac... | 2\sqrt{3}-2 | 1. **Assume the side length of the square**: Let's assume the side length of the square is 2 units for simplicity. This assumption does not affect the generality of the solution because we are interested in the ratio of areas, which is dimensionless and independent of the actual size of the square.
2. **Divide the squ... |
1128 | 1 | 7 | 8 | The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn? | 15 | 1. **Identify the distance traveled by each biker after 4 hours**: According to the problem, Bjorn biked 45 miles and Alberto biked 60 miles in the same time period.
2. **Calculate the difference in miles traveled**: To find out how many more miles Alberto biked than Bjorn, subtract the distance biked by Bjorn from th... |
1145 | 3 | 5 | 8 | A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?
[asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw(circle((2,2),1)); draw((4,0)--(6,1... | 4 | To solve this problem, we need to determine the smallest number of cubes that can be snapped together such that all the protruding snaps are hidden and only the receptacle holes are visible.
1. **Understanding the Cube Configuration**: Each cube has one protruding snap and five receptacle holes. The protruding snap pr... |
1148 | 4 | 4 | 8 | A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? | 729 | 1. **Identify the constraints and setup the problem:**
- The chef has to prepare desserts for 7 days starting from Sunday.
- The desserts options are cake, pie, ice cream, or pudding.
- The same dessert cannot be served on consecutive days.
- Cake must be served on Friday due to a birthday.
2. **Determine ... |
1159 | 3 | 5 | 8 | Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac{5}{2} \angle B$. What is the degree measure of $\angle D$? | 52.5 | 1. **Identify the properties of the triangles**: Given that $AB = BC = CD = CE$, we can conclude that $\triangle ABC$ and $\triangle CDE$ are both isosceles. Additionally, it is given that $\angle A = \frac{5}{2} \angle B$.
2. **Analyze $\triangle ABC$**:
- Since $\triangle ABC$ is isosceles with $AB = BC$, we have... |
1185 | 1 | 7 | 8 | Lines in the $xy$-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1)$. One of these lines has the equation | x-4y+13=0 | 1. **Finding the Trisection Points:**
The trisection points of the line segment joining $(-4, 5)$ and $(5, -1)$ are calculated by dividing the segment into three equal parts. We start by finding the differences in the x-coordinates and y-coordinates:
- Difference in x-coordinates: $5 - (-4) = 9$
- Difference i... |
1227 | 3 | 5 | 8 | Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid? | 750 | 1. **Identify the Components of the Trapezoid**:
Given that $ABCD$ is a trapezoid with $AB$ and $CD$ as the parallel sides, and the altitude (height) from $AB$ to $CD$ is $12$. The lengths of the sides are $AD = 15$, $AB = 50$, $BC = 20$.
2. **Draw Altitudes and Form Right Triangles**:
By drawing altitudes from ... |
1234 | 2 | 6 | 8 | How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$? | 12 | 1. **Symmetry and Reduction of Cases**:
Without loss of generality (WLOG), assume $a \geq 0$ and $a \geq b$. This assumption is valid because if $(a, b, c)$ is a solution, then $(-a, -b, c)$, $(b, a, c)$, and $(-b, -a, c)$ are also solutions due to the symmetry in the equations. If $a = b$, then $|a+b| = |2a| = 2a$,... |
1243 | 2 | 6 | 8 | The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two... | 240 | 1. **Assigning angles in arithmetic progression**: Since the internal angles of quadrilateral $ABCD$ form an arithmetic progression, we denote them as $a$, $a+d$, $a+2d$, and $a+3d$. The sum of the internal angles of any quadrilateral is $360^\circ$, so:
\[
a + (a+d) + (a+2d) + (a+3d) = 360^\circ
\]
Simplif... |
1266 | 3 | 5 | 8 | An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if th... | 191 | To solve this problem, we need to determine the maximum number of cables that can be used such that every employee can communicate with each other, under the constraint that cables can only connect a brand A computer to a brand B computer.
#### Step 1: Understand the problem constraints
- There are 30 employees: 20 wi... |
1283 | 4 | 4 | 8 | Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer stan... | \frac{3}{16} | 1. **Calculate Rachel's running details:**
- Rachel completes a lap every 90 seconds.
- In 10 minutes (600 seconds), Rachel completes $\frac{600}{90} = 6\frac{2}{3}$ laps. This means she completes 6 full laps and is $\frac{2}{3}$ of a lap into her seventh lap.
- $\frac{2}{3}$ of a lap corresponds to $\frac{2}{... |
1295 | 2 | 6 | 8 | Let $T_1$ be a triangle with side lengths $2011$, $2012$, and $2013$. For $n \geq 1$, if $T_n = \Delta ABC$ and $D, E$, and $F$ are the points of tangency of the incircle of $\Delta ABC$ to the sides $AB$, $BC$, and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE$, and $CF$, if it exists. Wha... | \frac{1509}{128} | 1. **Identify the side lengths of the initial triangle $T_1$:**
Given $T_1$ has side lengths $2011$, $2012$, and $2013$.
2. **Define the sequence of triangles $T_n$:**
For each triangle $T_n = \Delta ABC$, the incircle touches $AB$, $BC$, and $AC$ at points $D$, $E$, and $F$ respectively. The next triangle $T_{n... |
1353 | 2 | 6 | 8 | The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y| \leq 1000$? | 40 | 1. **Identify the axis of symmetry**: Given the focus of the parabola $P$ at $(0,0)$ and points $(4,3)$ and $(-4,-3)$ on $P$, we observe that the line connecting these points has a slope of $\frac{3 - (-3)}{4 - (-4)} = \frac{6}{8} = \frac{3}{4}$. This suggests that the axis of symmetry of the parabola makes an angle $\... |
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