Found problems: 85335
The three distinct points$ B, C, D$ are collinear with C between B and D. Another point A not on
the line BD is such that $|AB| = |AC| = |CD|.$
Prove that ∠$BAC = 36$ if and only if $1/|CD|-1/|BD|=1/(|CD| + |BD|)$
.
The expression $ \frac{2x^2-x}{(x+1)(x-2)}-\frac{4+x}{(x+1)(x-2)}$ cannot be evaluated for $ x=-1$ or $ x=2$, since division by zero is not allowed. For other values of $ x$:
$\textbf{(A)}\ \text{The expression takes on many different values.} \\
\textbf{(B)}\ \text{The expression has only the value 2.} \\
\textbf{(C)}\ \text{The expression has only the value 1.} \\
\textbf{(D)}\ \text{The expression always has a value between } -1 \text{ and } +2. \\
\textbf{(E)}\ \text{The expression has a value greater than 2 or less than } -1.$
Consider triangle $ABC$ in $xy$-plane where $ A$ is at the origin, $ B$ lies on the positive $x$-axis, $C$ is on the upper right quadrant, and $\angle A = 30^o$, $\angle B = 60^o$ ,$\angle C = 90^o$. Let the length $BC = 1$. Draw the angle bisector of angle $\angle C$, and let this intersect the $y$-axis at $D$. What is the area of quadrilateral $ADBC$?
A holey triangle is an upward equilateral triangle of side length $n$ with $n$ upward unit triangular holes cut out. A diamond is a $60^\circ-120^\circ$ unit rhombus.
Prove that a holey triangle $T$ can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length $k$ in $T$ contains at most $k$ holes, for $1\leq k\leq n$.
[i]Proposed by Federico Ardila, Colombia [/i]
Determine all sequences of strictly positive integers $a_1, a_2, a_3, \ldots$ satisfying the following two conditions:
[list]
[*]There exists an integer $M > 0$ such that, for all indices $n \geqslant 1$, $0 < a_n \leqslant M$.
[*]For any prime number $p$ and for any index $n \geqslant 1$, the number
\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]
is a multiple of $p$.
[/list]
Let natural $n \ge 2$ be given. Let Laura be a student in a class of more than $n+2$ students, all of which participated in an olympiad and solved some problems. Additionally, it is known that:
[list]
[*] for every pair of students there is exactly one problem that was solved by both students;
[*] for every pair of problems there is exactly one student who solved both of them;
[*] one specific problem was solved by Laura and exactly $n$ other students.
[/list]
Determine the number of students in Laura's class.
Triangle $ABC$ with $AB = 10$ cm ¸and $\angle C= 15^o$, is right at $B$. Point $D \in (AC)$ is the foot of the altitude taken from $B$. Find the distance from point $D$ to the line $AB$.
Let $ABC$ be a triangle with $D$ the midpoint of side $AB, E$ the midpoint of side $BC$, and $F$ the midpoint of side $AC$. Let $k_1$ be the circle passing through points $A, D$, and $F$, let $k_2$ be the circle passing through points $B, E$, and $D$, and let $k_3$ be the circle passing through $C, F$, and $E$. Prove that circles $k_1, k_2$, and $k_3$ intersect in a point.
Let $ABC$ be an acute-angled triangle. $M ,N$ are any two points on the sides $AB , AC$ respectively. The circles with the diameters $BN$ and $CM$ intersect at points $P$ and $Q$. Show that the points $P, Q$ and the orthocenter of the triangle $ABC$ lie on a straight line.
Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.
An equiangular octagon has four sides of length $ 1$ and four sides of length $ \frac{\sqrt{2}}{2}$, arranged so that no two consecutive sides have the same length. What is the area of the octagon?
$ \textbf{(A)}\ \frac{7}{2}\qquad
\textbf{(B)}\ \frac{7\sqrt{2}}{2}\qquad
\textbf{(C)}\ \frac{5 \plus{} 4\sqrt{2}}{2}\qquad
\textbf{(D)}\ \frac{4 \plus{} 5\sqrt{2}}{2}\qquad
\textbf{(E)}\ 7$
Let $p$ be a prime number. Prove that there exists $n \in Z$ such that $p | n^2 -n+3$ if and only if there exists $m \in Z$ such that $p | m^2 -m+25$.
Compute the number of ways to shade exactly $4$ distinct cells of a $4\times4$ grid such that no two shaded cells share one or more vertices.
Lyuba, Tanya, Lena and Ira ran across a flat field. At some point it turned out that among the pairwise distances between them there are distances of $1, 2, 3, 4$ and $5$ meters, and there are no other distances. Give an example of how this could be.
For $i=1,2,\ldots,7$, Zadam Heng chooses a positive integer $m_i$ at random such that each positive integer $k$ is chosen with probability $\frac{2^i-1}{2^{ik}}$. If $m_1+2m_2+\ldots+7m_7\neq35$, Zadam keeps rechoosing the $m_i$ until this equality holds. Given that he eventually stops, what is the probability that $m_4=1$ when Zadam stops?
[i]2019 CCA Math Bonanza Tiebreaker Round #3[/i]
Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\displaystyle2\leq\frac{a}{b}\leq5$.
Let $ p>5$ be a prime number. Prove that every algebraic integer of the $ p$th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the field.
[i]K. Gyory[/i]
Inside a square with sides of length $1$ unit several non-overlapping smaller squares with sides parallel to the sides of the large square are placed (the small squares may differ in size). Draw a diagonal of the large square and consider all of the small squares intersecting it. Can the sum of their perimeters be greater than $1993$?
(AN Vblmogorov)
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
$ \textbf{(A)}\ 5\sqrt{2}-7 \qquad
\textbf{(B)}\ 7-4\sqrt{3} \qquad
\textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad
\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad
\textbf{(E)}\ \frac{\sqrt{3}}{9} $
Initially five variables are defined: $a_1=1, a_2=0, a_3=0, a_4=0, a_5=0.$ On a turn, Evan can choose an integer $2 \le i \le 5.$ Then, the integer $a_{i-1}$ will be added to $a_i$. For example, if Evan initially chooses $i = 2,$ then now $a_1=1, a_2=0+1=1, a_3=0, a_4=0, a_5=0.$ Find the minimum number of turns Evan needs to make $a_5$ exceed $1,000,000.$
[b]p1[/b]. Points $P_1,P_2,P_3,... ,P_n$ lie on a plane such that $P_aP_b = 1$,$P_cP_d = 2$, and $P_eP_f = 2018$ for not necessarily distinct indices $a,b,c,d,e, f \in \{1, 2,... ,n\}$. Find the minimum possible value of $n$.
[b]p2.[/b] Find the coefficient of the $x^2y^4$ term in the expansion of $(3x +2y)^6$.
[b]p3.[/b] Find the number of positive integers $n < 1000$ such that $n$ is a multiple of $27$ and the digit sum of $n$ is a multiple of $11$.
[b]p4.[/b] How many times do the minute hand and hour hand of a $ 12$-hour analog clock overlap in a $366$-day leap year?
[b]p5.[/b] Find the number of ordered triples of integers $(a,b,c)$ such that $(a +b)(b +c)(c + a) = 2018$.
[b]p6.[/b] Let $S$ denote the set of the first $2018$ positive integers. Call the score of a subset the sum of its maximal element and its minimal element. Find the sum of score $(x)$ over all subsets $s \in S$
[b]p7.[/b] How many ordered pairs of integers $(a,b)$ exist such that $1 \le a,b \le 20$ and $a^a$ divides $b^b$?
[b]p8.[/b] Let $f$ be a function such that for every non-negative integer $p$, $f (p)$ equals the number of ordered pairs of positive integers $(a,n)$ such that $a^n = a^p \cdot n$. Find $\sum^{2018}_{p=0}f (p)$.
[b]p9.[/b] A point $P$ is randomly chosen inside a regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$. What is the probability that the projections of $P$ onto the lines $\overleftrightarrow{A_i A_{i+1}}$ for $i = 1,2,... ,8$ lie on the segments $\overline{A_iA_{i+1}}$ for $i = 1,2,... ,8$ (where indices are taken $mod \,\, 8$)?
[b]p10. [/b]A person keeps flipping an unfair coin until it flips $3$ tails in a row. The probability of it landing on heads is $\frac23$ and the probability it lands on tails is $\frac13$ . What is the expected value of the number of the times the coin flips?
PS. You had better use hide for answers.
For any system $ x_1,x_2,...,x_n$ of positive real numbers, let
$ f_1(x_1,x_2,...,x_n) \equal{} x_1$,
and
$ f_{\nu} \equal{} \frac {x_1 \plus{} 2x_2 \plus{} \cdots \plus{} \nu x_{\nu}}{\nu \plus{} (\nu \minus{} 1)x_1 \plus{} (\nu \minus{} 2)x_2 \plus{} \cdots \plus{} 1\cdot x_{\nu \minus{} 1}}$
for $ \nu \equal{} 2,3,...,n$. Show that for any $ \epsilon > 0$, a positive integer $ n_0 < n_0(\epsilon)$ can be found such that for every $ n > n_0$ there exists a system $ x_1',x_2',...,x_n'$ of positive real numbers
with $ x_1' \plus{} x_2' \plus{} \cdots \plus{} x_n' \equal{} 1$ and $ f_{\nu}(x_1',x_2',...,x_n')\le \epsilon$ for $ \nu \equal{} 1,2,...,n$ .
Let $a,b,c\in \mathbb{R}^+$, prove that: $$\frac{2a}{\sqrt{3a+b}}+\frac{2b}{\sqrt{3b+c}}+\frac{2c}{\sqrt{3c+a}}\leq \sqrt{3(a+b+c)}$$
A positive integer is [b]superb[/b] if it is the least common multiple of $1,2,\ldots, n$ for some positive integer $n$.
Find all superb $x,y,z$ such that $x+y=z$.
[i]
Proposed by usjl[/i]
Let $\vartriangle ABC$ be such that $\angle BAC$ is acute. The line perpendicular on side $AB$ from $C$ and the line perpendicular on $AC$ from $B$ intersect the circumscribed circle of $\vartriangle ABC$ at $D$ and $E$ respectively. If $DE = BC$ , calculate $\angle BAC$.