Practice (EndingDigits,TheDivideByNineMethod,MODBasic)
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We are given a triangle with the following property: one of its angles is quadrisected (divided into four equal angles) by the height, the angle bisector, and the median from that vertex. This property uniquely determines the triangle (up to scaling). Find the measure of the quadrisected angle.
Fagnoan's Problem
Fermat's Point
Let $P$ be a point inside parallelogram $ABCD$. If $\angle{PAB}=\angle{PCB}$, show $\angle{PBA} = \angle{PDA}$.
Two sides of a triangle are 4 and 9; the median drawn to the third side has length 6. Find the length of the third side.
A right triangle has legs $a$ and $b$ and hypotenuse $c$. Two segments from the right angle to the hypotenuse are drawn,
dividing it into three equal parts of length $x=\frac{c}{3}$. If the segments have length $p$ and $q$, prove that $p^2 +q^2 =5x^2$.
A circle inscribed in $\triangle{ABC}$ (the incircle) is tangent to $BC$ at $X$, to $AC$ at $Y$ , to $AB$ at $Z$. Show that $AX$, $BY$, and $CZ$ are concurrent.
Three squares are drawn on the sides of $\triangle{ABC}$ (i.e. the square on $AB$ has $AB$ as one of its sides and lies outside $\triangle{ABC}$). Show that the lines drawn from the vertices $A, B, C$ to the centers of the opposite squares are concurrent.
Let $P$ be a point inside a unit square $ABCD$. Find the minimal value of $AP+BP+CP$
The diagonals $AC$ and $CD$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ such that $AM:AC = CN:CE=r$. Determine $r$ if $B, M,$ and $N$ are collinear.
Let $a @ b = a$. What is the value of $5 @ 3$? Express your answer as a common fraction.
How many rectangles of any size are in the grid shown here?
Given $7x + 13 = 328$, what is the value of $14x + 13$?
What is the median of the positive perfect squares less than 250?
If $\frac{x + 5}{x-2} = \frac{2}{3}$, what is the value of $x$?
In rectangle $TUVW$, shown here, $WX = 4$ units, $XY = 2$ units, $YV = 1$ unit and $UV = 6$ units. What is the absolute difference between the areas of triangles $TXZ$ and $UYZ$.
A bag contains 4 blue, 5 green and 3 red marbles. How many green marbles must be added to the bag so that 75 percent of the marbles are green?
MD rides a three wheeled motorcycle called a trike. MD has a spare tire for his trike and wants to occasionally swap out his tires so that all four will
have been used for the same distance as he drives 25,000 miles. How many miles will each tire drive?
Lucy and her father share the same birthday. When Lucy turned 15 her father turned 3 times her age. On their birthday this year, Lucy’s father turned exactly twice as old as she turned. How old did Lucy turn this year?
The sum of three distinct 2-digit primes is 53. Two of the primes have a units digit of 3, and the other prime has a units digit of 7. What is the greatest of the three primes?
Ross and Max have a combined weight of 184 pounds. Ross and Seth have a combined weight of 197 pounds. Max and Seth have a combined weight of 189 pounds. How many pounds does Ross weigh?
What is the least possible denominator of a positive rational number whose repeating decimal representation is $0.\overline{AB}$ , where $A$ and $B$ are distinct digits?
A taxi charges \$3.25 for the first mile and \$0.45 for each additional 14 mile thereafter. At most, how many miles can a passenger travel using \$13.60? Express your answer as a mixed number.
Kali is mixing soil for a container garden. If she mixes $2 m^3$ of soil containing 35% sand with $6 m^3$ of soil containing 15% sand, what percent of the new mixture is sand?
Alex can run a complete lap around the school track in 1 minute, 28 seconds, and Becky can run a complete lap in 1 minute, 16 seconds. If they begin running at the same time and location, how many complete laps will Alex have run when Becky passes him for the first time?