Practice (Intermediate)

3026

As shown, prove $$\frac{\sin(\alpha+\beta)}{PC}=\frac{\sin{\alpha}}{PB}+\frac{\sin{\beta}}{PA}$$

3035

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.

3038

Let $P$ be a point inside parallelogram $ABCD$. If $\angle{PAB}=\angle{PCB}$, show $\angle{PBA} = \angle{PDA}$.

3039

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.

3042

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.

3043

Let $P$ be a point inside a unit square $ABCD$. Find the minimal value of $AP+BP+CP$

3044

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.

3067

Initially, a chip is placed in the upper-left corner square of a $n\times m$ grid of squares as shown. The chip can move in an $L$-shaped pattern, moving two squares in one direction (up, right, down or left) and then moving one square in a corresponding perpendicular direction. What is the minimum number of $L$-shaped moves needed to move the chip from its initial location to the square marked “$X$”?

3094

As shown, $ABCD$ is a square with side length equaling 10 cm, $CE\perp BE$, and $CE=8$. Find the area of the shaded triangle.

3102

Find all positive integer $n$ such that $n^2 + 2^n$ is a perfect square.

3104

Let $ABCD$ be a trapezoid. Points $M$ and $N$ are the mid points of its diagonal $AC$ and $BD$, respectively. Show that $MN \parallel AB$ and $MN = \frac{1}{2}\mid AB - CD\mid$.

3105

A plane passing through the vertex $A$ and the center of its inscribed sphere of a tetrahedron $ABCD$ intersects its edge $BC$ and $CD$ at point $E$ and $F$, as shown. If $AEF$ divides this tetrahedron into two equal volume parts: $A-BDEF$ and $A-CEF$, what is the relationship between these two parts' surface areas $S_1$ and $S_2$ where $S_1 = S_{A-BDEF}$ and $S_1=S_{A-CEF}$? $(A) S_1 < S_2\quad(B) S_1 > S_2\quad (C) S_1 = S_2 \quad(D) $ cannot determine

3116

Compute the least possible area of a non-degenerate right triangle with sides of lengths $\sin{x}$, $\cos{x}$ and $\tan{x}$ where $x$ is a real number.

3117

Let $P(x)$ be the polynomial $x^3 + Ax^2 +Bx+C$ for some constants $A, B,$ and $C$. There exists constant $D$ and $E$ such that for all $x$, $P(x+1)=x^3 + Dx^2 + 54x +37$ and $P(x+2)=x^3 + 26x + Ex+115$. Compute the ordered triple $(A, B, C)$.

3119

Find the largest of three prime divisors of $13^4+16^5-172^2$.

3120

In $\triangle{ABC}$, $\angle{BAC} = 40^\circ$ and $\angle{ABC} = 60^\circ$. Points $D$ and $E$ are on sides $AC$ and $AB$, respectively, such that $\angle{DBC}=40^\circ$ and $\angle{ECB}=70^\circ$. Let $F$ be the intersection point of $BD$ and $CE$. Show that $AF\perp BC$.

3125

Let $x$ be a negative real number. Find the maximum value of $y=x+\frac{4}{x} +2007$.

3126

Let real numbers $a$ and $b$ satisfy $a^2 + ab + b^2 = 1$. Find the range of $a^2 - ab + b^2$.

3130

In $\triangle{ABC}$ show that $$\tan nA + \tan nB + \tan nC = \tan nA \tan nB \tan nC$$ where $n$ is an integer.

3132

In $\triangle{ABC}$, if $A:B:C=4:2:1$, prove $$\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$$

3135

Let $O$ be a point inside a convex pentagon, as shown, such that $\angle{1} = \angle{2}, \angle{3} = \angle{4}, \angle{5} = \angle{6},$ and $\angle{7} = \angle{8}$. Show that either $\angle{9} = \angle{10}$ or $\angle{9} + \angle{10} = 180^\circ$ holds.

3136

In $\triangle{ABC}$, $AB = 33, AC=21,$ and $BC=m$ where $m$ is an integer. There exist points $D$ and $E$ on $AB$ and $AC$, respectively, such that $AD=DE=EC=n$ where $n$ is also an integer. Find all the possible values of $m$.

3137

Let quadrilateral $ABCD$ inscribe a circle. If $BE=ED$, prove $$AB^2+BC^2 +CD^2 + DA^2 = 2AC^2$$

3138

Let $H$ be the orthocenter of acute $\triangle{ABC}$. Show that $$a\cdot BH\cdot CH + b\cdot CH\cdot AH+c\cdot AH\cdot BH=abc$$ where $a=BC, b=CA,$ and $c=AB$.

3140

As shown, $\angle{ACB} = 90^\circ$, $AD=DB$, $DE=DC$, $EM\perp AB$, and $EN\perp CD$. Prove $$MN\cdot AB = AC\cdot CB$$

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