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What is ?
For real numbers and , define . What is ?
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?
A rectangle and a rectangle are contained within a square without overlapping at any point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure , as shown. What is the perimeter of this region?
Which of the following is equivalent to when ?
A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle?
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
What is the tens digit in the sum
The lines and intersect at the point . What is ?
Joe and JoAnn each bought 12 ounces of coffee in a 16 ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
Let and be the roots of the equation . Suppose that and are the roots of the equation . What is ?
Rhombus is similar to rhombus . The area of rhombus is and . What is the area of rhombus ?
Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?
Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same?
Let be a sequence for which
, , and for each positive integer .
What is ?
A circle of radius is centered at . Square has side length . Sides and are extended past to meet the circle at and , respectively. What is the area of the shaded region in the figure, which is bounded by , , and the minor arc connecting and ?
In rectangle , we have , , , for some integer . What is the area of rectangle ?
For a particular peculiar pair of dice, the probabilities of rolling , , , , , and , on each die are in the ratio . What is the probability of rolling a total of on the two dice?
Elmo makes sandwiches for a fundraiser. For each sandwich he uses globs of peanut butter at per glob and blobs of jam at per blob. The cost of the peanut butter and jam to make all the sandwiches is . Assume that , , and are positive integers with . What is the cost of the jam Elmo uses to make the sandwiches?
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7 as shown. What is the area of the shaded quadrilateral?
Circles with centers and have radii and , respectively, and are externally tangent. Points and on the circle with center and points and on the circle with center are such that and are common external tangents to the circles. What is the area of the concave hexagon ?
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
Since , . So, the area of triangle is . Therefore, the shaded area is
From the pythagorean theorem, we can see that is . Then, . The area of the shaded element is the area of sector minus the areas of triangle and triangle combined. Below is an image to help.
Using the Base Altitude formula, where and are the bases and and are the altitudes, respectively, . The area of sector is of circle . The area of circle is , and therefore we have the area of sector to be
Like the first solutions, you find that the area of sector is . We also know that the triangles will not be in terms of . Looking at the answers, choices and both contain . However, based on the diagram, we observe that the answer must be less than . Only consists of a value less than .
Therefore the area of rectangle is
This solution is the same as Solution 1 up to the point where we find that .
We build right triangles so we can use the Pythagorean Theorem. The triangle with hypotenuse has legs and , while the triangle with hypotenuse has legs and . Aha! The two triangles are similar, with one triangle having side lengths times the other!
Let . Then from our reasoning above, we have . Finally, the area of the rectangle is .
Note that the prime factorization of .
Hence, we want or
Now, we have two cases to test.
Case 1:
Notice that we want or
Taking
Hence, .
Hence, the price of the Jam is .
Triangles and also share an altitude. The ratio of their areas is also equal to the ratio of bases and . The ratio is
Because the two ratios are equal, we get the equation . We add the area of triangle to get that the total area of the quadrilateral is .
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