MathFun4Kids

Stengel’s Theorem – Extended Cross Ladders Theorem

Posted on November 7, 2024 by kevin

If you remember, the Classic Cross Ladders Theorem was used in one of the solutions to this MathCounts problem. The Extended Cross Ladders Theorem, also known as Stengel’s Theorem, applies to cross ladders in a triangle. Specifically, in $\triangle{ABC}$ as shown below, if $BG\parallel DH\parallel FJ\parallel EI$, then $$\dfrac{1}{EI}+\dfrac{1}{DH}=\dfrac{1}{BG}+\dfrac{1}{FJ}$$

It also implies that $$\dfrac{1}{\triangle{AEC}}+\dfrac{1}{\triangle{ADC}}=\dfrac{1}{\triangle{ABC}}+\dfrac{1}{\triangle{AFC}}$$

Question: Let $\triangle{AEF}=3$, $\triangle{ACF}=4$, $\triangle{CDF}=1$. Find the area of quadrilateral $BEFD$.

Solution: Let $x$ be the area of quadrilateral $BEFD$. By Stengel’s Theorem, we have $$\dfrac{1}{3+4}+\dfrac{1}{1+4}=\dfrac{1}{x+4}+\dfrac{1}{4}$$

Solving the above equation, we have $x=\boxed{\dfrac{88}{13}}$.

Challenge: Can you apply Stengel’s Theorem to find the Area of Irregular Pentagon?

Posted in Daily Problems, Geometry | Comments Off

AMC 10 Exercise – 2

Posted on October 20, 2024 by kevin

What is the tens digit of $2019^{2020}-2021$?
All the numbers $2$,$3$,$4$,$5$ are assigned to the four faces of a tetrahedron. For each of the four vertices of the tetrahedron, a product of three numbers are computed, where the three numbers are the numbers assigned to the three faces that include that vertex. Then the sum of these four products is calculated. What is the difference between the greatest and the smallest possible values of the sum?
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?

Posted in Math Classes, MATHCOUNTS | Comments Off

Geometry Challenge – Area of Irregular Pentagon

Posted on October 15, 2024 by kevin

As shown in the diagram below, a star sign consists of five straight lines. It produces five triangles and a pentagon. If areas of five triangles are 3, 10, 7, 15, and 8 square unit respectively. Find the area of the pentagon.

Click here for the solution.

Solution: Draw line AB, BC, CD, DE, EA, and label the area of $5$ newly formed triangles as $a$, $b$, $c$, $d$, and $e$ respectively. Additionally, label the area of the pentagon $FGHIJ$ as $f$, we have the following diagram:

Based on the common ratios of areas for triangles sharing the same bases, we have the following:

$$\dfrac{Area\ \triangle{CDG}}{Area\ \triangle{DEG}}=\dfrac{Area\ \triangle{CBG}}{Area\ \triangle{BEG}}$$

i.e. $$\dfrac{a}{b+15}=\dfrac{e+7}{f+18}\tag{1}$$

Similarly, we have the following additional equations:

$$\dfrac{b}{a+15}=\dfrac{c+8}{f+10}\tag{2}$$

$$\dfrac{c}{b+8}=\dfrac{d+3}{f+25}\tag{3}$$

$$\dfrac{d}{c+3}=\dfrac{e+10}{f+15}\tag{4}$$

$$\dfrac{e}{d+10}=\dfrac{a+7}{f+18}\tag{5}$$

Since there are $6$ variables in the above 5 equations, we can add co-linear constraint $C$, $F$, and $G$.

By using Barycentric Coordinate System for triangles, we have the following non-normalized coordinates for $C$, $F$, and $G$, relative to $\triangle{ABD}$:

$$C=(d+e+10,-(a+e+7),a+f+25)\ \ \ \ \ \ F=(c+8,b,0)\ \ \ \ \ \ G=(e+7,0,a)$$

Because $C$, $F$, and $G$ are co-linear, with Barycentric Coordinate System, we have:

$$\begin{vmatrix} d+e+10 & -(a+e+7) & a+f+25\\ c+8 & b & 0 \\ e+7 & 0 & a \end{vmatrix}=0\tag{6}$$

Expanding the above 6 equations, we have:

$$\begin{cases} \ \ a(f+18)=(e+7)(b+15)\\ \ \ b(f+10)=(a+15)(c+8)\\ \ \ c(f+25)=(b+8)(d+3)\\ \ \ d(f+15)=(c+3)(e+10)\\ \ \ e(f+18)=(d+10)(a+7)\\ \ \ ab(d+e+10)+a(c+8)(a+e+7)-b(e+7)(a+f+25)=0 \end{cases}$$

By using a symbolic Computer Algebra System, such as Wolfram Alpha or Wolfree Alpha, with an input such as:

a>0,b>0,c>0,d>0,e>0,f>0,a(f+18)=(b+15)(e+7), b(f+10)=(c+8)(a+15), 
c(f+25)=(d+3)(b+8), d(f+15)=(e+10)(c+3), e(f+18)=(a+7)(d+10),
ab(d+e+10)+a(c+8)(a+e+7)-b(e+7)(a+f+25)=0

We have the following unique solution:

$$a=21,\ \ \ \ \ b=20,\ \ \ \ \ c=7,\ \ \ \ \ d=\dfrac{15}{2},\ \ \ \ \ e=14,\ \ \ \ \ f=17$$

Therefore the area of the pentagon is $\boxed{17}$ square unit.

Note: Please refer to Barycentric Coordinates in Olympiad Geometry or Barycentric Coordinates for the Impatient for additional information related to Barycentric Coordinates.

Posted in Algebra, Geometry | Comments Off

AMC 10 Exercise – 1

Posted on September 20, 2024 by kevin

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 $2:1$?
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 ratio between the height of the first cylinder and that of the second one?
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 $cd$.
The diagram below shows the circular face of a clock with radius $20$cm, and a circular disk with radius $12$cm externally tangent to the clock face at $12$ o’clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?
Consider the set of all fractions $\dfrac{x}{y}$, 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 $10%$?
If $y+4=(x-2)^2$, $x+4=(y-2)^2$,and $x\ne y$, what is the value of $x^2+y^2$?
A line that passes through the origin intersects both the line $x=1$ and the line $y=1+\dfrac{\sqrt{3}}{3}x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?
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 $n$?
A rectangle with positive integer side lengths in $\text{cm}$ has area $A$ ${cm}^2$ and perimeter $P$ $cm$. Which of the following numbers cannot equal to $A+P$? (A) 100 (B) 102 (C) 104 (D) 106 (E) 108
Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\dfrac{12}{5}\sqrt{2}$. What is the volume of the tetrahedron?
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
The zeroes of the function $f(x)=x^2-ax+2a$are integers. What is the sum of the possible values of $a$?
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 $p\lt 2015$ are possible?
Let $S$ be a square of side length $1$. Two points are chosen 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 $a+b+c$?
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? $(A) 41 (B) 47 (C) 59 (D) 61 (E) 66$
When the centers of the faces of the right rectangular prism, with three side length as $3$, $4$, and $5$, are joined to create an octahedron, what is the volume of the octahedron?
In $\triangle{ABC}$, $\angle{C}=90^\circ$, and $AB=12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$ and $W$lie on a circle. What is the perimeter of the triangle?
Erin the ant starts at a given corner of a cube and crawls along exactly $7$ edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose Dash takes $19$ fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?
In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG+JH+CD$?
A rectangular box measures $a\times b\times c$, where $a$,$b$, and $c$ are integers and $1\le a \le b \le c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a, b, c)$ are possible?

Posted in Math Classes, MATHCOUNTS | Comments Off

Geometry Probability – 3

Posted on September 18, 2024 by kevin

Two points are randomly and uniformly selected from the interior of a circle. The center of the circle and the two points joined together form a triangle. What is the probability that the triangle is acute? Click here for the solution.

Solution: First pick up the first point A randomly and uniformly from the interior of the unit circle centered at $O$ and orientate $OA$ be on positive side of the $x$-axis, with the distance between $O$ and $A$ as $t$:

Draw another circle with $OA$ as its diameter. Draw a chord $DE \perp OA$ and link $OD$ and $OE$.

If point $B$ is selected from the left side of the circle, $\triangle{OAB}$ would be obtuse. Therefore, in order for the another point $B$ randomly and uniformly selected from the interior of the circle so that $\triangle{OAB}$ is acute, point B must be on the right half of the unit circle.

Additionally, point $B$ must be not inside the smaller circle with its diameter as $OA$, and not on the right side of line $DE$.

Therefore, the probability $\triangle{OAB}$ is obtuse when $B$ is on the right half of the unit circle for $t$ would be:

$$f(t)=\dfrac{area\ of\ the\ smaller\ circle\ + \ area\ of\ unit\ circle\ on\ the\ rigit\ side\ of \ line\ DE}{area\ of\ the\ semi\ unit\ circle}$$ $$=\dfrac{\dfrac{1}{4}\pi t^2 + cos^{-1}(t)-t\sqrt{1-t^2}}{\dfrac{1}{2}\pi} \tag{1}$$

Therefore the probability of $\triangle{OAB}$ is obtuse on the right half of the unit circle would be:

$$P=\int_{0}^{1}f(\sqrt{x})dx \tag{2}$$

$\sqrt{x}$ is used to ensure that point $A$ is uniformly distributed across whole unit circle area. The square root function compensates for the quadratic increase in area, spreading the points evenly.

Simplifying $(2)$ we have:

$$P=\dfrac{1}{2}\int_{0}^{1} x dx + \dfrac{2}{\pi}\int_{0}^{1}cos^{-1}(\sqrt{x}) dx -\dfrac{2}{\pi}\int_{0}^{1}\sqrt{x(1-x)} dx$$

$$=\dfrac{1}{2}\cdot \dfrac{1}{2}+\dfrac{2}{\pi}\cdot \dfrac{\pi}{4}-\dfrac{2}{\pi}\cdot \dfrac{\pi}{8}=\dfrac{1}{2}$$

Therefore the probability of $\triangle{OAB}$ is acute on the right half of the unit circle is $1-P=\dfrac{1}{2}$.

Since the probability for $B$ to be in the right half of the unit circle is $\dfrac{1}{2}$, the overall probability is $\dfrac{1}{2}\cdot \dfrac{1}{2}=\boxed{\dfrac{1}{4}}$

Posted in Algebra, Calculus, Geometry, Probability | Comments Off

Stengel’s Theorem – Extended Cross Ladders Theorem

AMC 10 Exercise – 2

Geometry Challenge – Area of Irregular Pentagon

AMC 10 Exercise – 1

Geometry Probability – 3

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