Practice (EndingDigits,TheDivideByNineMethod,MODBasic)
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Find the number of sets $\{a,b,c\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$.
The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$.
Triangle $ABC$ is inscribed in circle $\omega$. Points $P$ and $Q$ are on side $\overline{AB}$ with $AP < AQ$. Rays $CP$ and $CQ$ meet $\omega$ again at $S$ and $T$ (other than $C$), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.
The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color
Beatrix is going to place six rooks on a $6 \times 6$ chessboard where both the rows and columns are labeled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The $value$ of a square is the sum of its row number and column number. The $score$ of an arrangement of rooks is the least value of any occupied square. Find the average score over all valid configurations.
Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\triangle ABC$ and are on opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (the angle between the two planes). There is a point $O$ whose distance from each of $A,B,C,P,$ and $Q$ is $d$. Find $d$.
For $1 \leq i \leq 215$ let $a_i = \dfrac{1}{2^{i}}$ and $a_{216} = \dfrac{1}{2^{215}}$. Let $x_1, x_2, ..., x_{215}$ be positive real numbers such that $\sum_{i=1}^{215} x_i=1$ and $\sum_{i \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}$. The maximum possible value of $x_2=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
As shown, in quadrilateral $ABCD$, $AB=AD$, $\angle{BAD} = \angle{DCB} = 90^\circ$. Draw altitude from $A$ towards $BC$ and let the foot be $E$. If $AE=1$, find the area of $ABCD$.
In pentagon $ABCDE$, if $AB=AE$, $BC+DE=CD$, and $\angle{ABC} + \angle{AED} = 180^\circ$, show that $\angle{ADE}=\angle{ADC}$.
Let point $P$ be inside equilateral $\triangle{ABC}$ such that $PA=2$, $PB=2\sqrt{3}$, and $PC=4$. Find the side length of this equilateral triangle.
Let point $P$ be inside of equilateral $\triangle{ABC}$ such that $PA=3, PB=4,$ and $PC=5$. Find the measurement of $\angle{APB}$.
Let point $P$ be inside equilateral $\triangle{ABC}$ such that $\angle{APB}=115^\circ$ and $\angle{BPC}=125^\circ$. Find the measurements of the three internal angles if we construct a triangle using $PA$, $PB$, and $PC$.
Let $\triangle{ABC}$ be an isosceles right triangle where $\angle{C}=90^\circ$. If points $M$ and $N$ are on $AB$ such that $\angle{MCN}=45^\circ$, $AM=4$, and $BN=3$, find the length of $MN$.
Given an acute $\triangle$, let $D$ be the middle point of $AB$, and $DE\perp DF$ where points $E$ and $F$ are on the other two sides respectively. Show that $S_{\triangle{DEF}} < S_{\triangle{ADF}} + S_{\triangle{BDE}}$
In $\triangle{ABC}$, let $AB=c$, $AC=b$, and $\angle{BAC}=\alpha$. If $AD$ bisects $\angle{BAC}$ and intersects $BC$ at $D$, find the length of $AD$.
As shown, prove $$\frac{\sin(\alpha+\beta)}{PC}=\frac{\sin{\alpha}}{PB}+\frac{\sin{\beta}}{PA}$$
(Weitzenbock's Inequality) Let $a, b, c$, and $S$ be a triangle's three sides' lengths and its area, respectively. Show that $$a^2 + b^2 + c^2 \ge 4\sqrt{3}\cdot S$$
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?