Practice (90/1000)
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As show, three squares are arranged side-by-side such that their bases are collinear. The sides of two squares are known and marked. Find the area of shaded triangle.
Let $ABCD$ be a rectangle where $AB=4$ and $BC=6$. If $AE=CG=3$, $BF=DH=4$, and $S_{AEPH}=5$. Find the area of $PFCG$.
Let $ABCD$ be a rectangle where $AB=3$ and $AD=4$. Point $P$ is on the side $AD$. If points $E$ and $F$ are on $AC$ and $BD$ respectively such that $PE \perp AC$ and $PF \perp BD$. Compute $PE+PF$.
As shown in the diagram, both $ABCD$ and $BEFG$ are squares, where point $E$ is on $AB$. If $AD=2$, compute the area of $\triangle{AFC}$.
Let $P$ be a point inside $\triangle{ABC}$. If $AP$, $BP$, and $CP$ intersect the opposite sides at $D$, $E$, and $F$, respectively. Show that $$\frac{PD}{AD}+\frac{PE}{BE}+\frac{PF}{CF}=1$$
Let real numbers $x_1$ and $x_2$ satisfy $ \frac{\pi}{2} > x_1 > x_2 > 0$, show $$\frac{\tan x_1}{x_1} > \frac{\tan x_2}{x_2}$$
The numbers from 1 to 7 are separated into two non-empty sets A and B. The numbers in A are multiplied together to get a. The numbers in B are multiplied together to get b. The larger of the two numbers a and b is written down. What is the smallest number that can be written down using this procedure?
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.
How many rectangles of any size are in the grid shown here?
Given $7x + 13 = 328$, what is the value of $14x + 13$?
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 $UYZ$.
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?
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.
The Beavers, Ducks, Platypuses and Narwhals are the only four basketball teams remaining in a single-elimination tournament. Each round consists of the teams playing in pairs with the winner of each game continuing to the next round. If the teams are randomly paired and each has an equal probability of winning any game, what is the probability that the Ducks and the Beavers will play each other in one of the two rounds? Express your answer as a common fraction.