Found problems: 85335
Given a triangle $ABC$, a circle $\Omega$ is tangent to $AB,AC$ at $B,C,$ respectively. Point $D$ is the midpoint of $AC$, $O$ is the circumcenter of triangle $ABC$. A circle $\Gamma$ passing through $A,C$ intersects the minor arc $BC$ on $\Omega$ at $P$, and intersects $AB$ at $Q$. It is known that the midpoint $R$ of minor arc $PQ$ satisfies that $CR \perp AB$. Ray $PQ$ intersects line $AC$ at $L$, $M$ is the midpoint of $AL$, $N$ is the midpoint of $DR$, and $X$ is the projection of $M$ onto $ON$. Prove that the circumcircle of triangle $DNX$ passes through the center of $\Gamma$.
Thee tiles are marked $ X$ and two other tiles are marked $ O$. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads $ XOXOX$?
$ \textbf{(A)}\ \frac{1}{12}\qquad
\textbf{(B)}\ \frac{1}{10}\qquad
\textbf{(C)}\ \frac{1}{6}\qquad
\textbf{(D)}\ \frac{1}{4}\qquad
\textbf{(E)}\ \frac{1}{3}$
Let $A_{1}$, $B_{1}$, $C_{1}$ be the feet of the altitudes of an acute-angled triangle $ABC$ issuing from the vertices $A$, $B$, $C$, respectively. Let $K$ and $M$ be points on the segments $A_{1}C_{1}$ and $B_{1}C_{1}$, respectively, such that $\measuredangle KAM = \measuredangle A_{1}AC$. Prove that the line $AK$ is the angle bisector of the angle $C_{1}KM$.
Let $S_{n}$ be the sum of the digits of $2^n$. Prove or disprove that $S_{n+1}=S_{n}$ for some positive integer $n$.
There are $n \ge 3$ distinct positive real numbers. Show that there are at most $n-2$ different integer power of three that can be written as the sum of three distinct elements from these $n$ numbers.
The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.
a) Each of the numbers $x_1,x_2,...,x_n$ can be $1, 0$, or $-1$. What is the minimal possible value of the sum of all products of couples of those numbers.
b) Each absolute value of the numbers $x_1,x_2,...,x_n$ doesn't exceed $1$. What is the minimal possible value of the sum of all products of couples of those numbers.
Prove that the number
$$2^{2^k-1}-2^k-1$$is composite (not prime) for all positive integers $k>2$.
Prove that in every triangle the diameter of the incircle is not greater than the radius of the circumcircle.
Batman, who has a mass of $\text{M = 100 kg}$, climbs to the roof of a $\text{30 m}$ building and then lowers one end of a massless rope to his sidekick Robin. Batman then pulls Robin, who has a mass of $\text{m = 75 kg}$, up the roof of the building. Approximately how much total work has Batman done after Robin is on the roof?
(A) $\text{60 J}$
(B) $\text{7} \times \text{10}^3 \text{J}$
(C) $\text{5} \times \text{10}^4 \text{J}$
(D) $\text{600 J}$
(E) $\text{3} \times \text{10}^4 \text{J}$
Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$. Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$.
[i]2017 CCA Math Bonanza Lightning Round #2.4[/i]
For a triangle $\triangle ABC$ with an obtuse angle $\angle A$ , let $E , F$ be feet of altitudes from $B , C$ on sides $AC , AB$ respectively. The tangents from $B , C$ to circumcircle of triangle $\triangle ABC$ intersect line $EF$ at points $K , L$ respectively and we know that $\angle CLB=135$. Point $R$ lies on segment $BK$ in such a way that $KR=KL$ and let $S$ be a point on line $BK$ such that $K$ is between $B , S$ and $\angle BLS=135$. Prove that the circle with diameter $RS$ is tangent to circumcircle of triangle $\triangle ABC$.
[i]Proposed by Mehran Talaei[/i]
Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly nonconsecutive) days total, and he cannot wear the same outfit (which consists of one T-shirt and one pair of pants) on two different (not necessarily consecutive) days. How many ways can he choose the outfits for these seven days?
[i]Proposed by Yannick Yao[/i]
Find all pairs of real numbers $a, b$ for which the equation in the domain of the real numbers
\[\frac{ax^2-24x+b}{x^2-1}=x\]
has two solutions and the sum of them equals $12$.
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .
[i]Proposed by Serbia[/i]
Let $f_n=\left(1+\frac{1}{n}\right)^n\left((2n-1)!F_n\right)^{\frac{1}{n}}$. Find $\lim \limits_{n \to \infty}(f_{n+1} - f_n)$ where $F_n$ denotes the $n$th Fibonacci number (given by $F_0 = 0$, $F_1 = 1$, and by $F_{n+1} = F_n + F_{n-1}$ for all $n \geq 1$
Given real numbers $b \geq a>0$, find all solutions of the system
\begin{align*}
&x_1^2+2ax_1+b^2=x_2,\\
&x_2^2+2ax_2+b^2=x_3,\\
&\qquad\cdots\cdots\cdots\\
&x_n^2+2ax_n+b^2=x_1.
\end{align*}
Let $a, b, c$ be side lengths of a right triangle and $c$ be the length of the hypotenuse .Find the minimum value of $\frac{a^3+b^3+c^3}{abc}$.
Prove that the product of 4 consecutive natural numbers cannot be a perfect cube.
Suppose that $X$ is a compact metric space and $T: X\rightarrow X$ is a continous function. Prove that $T$ has a returning point. It means there is a strictly increasing sequence $n_i$ such that $\lim_{k\rightarrow \infty} T^{n_k}(x_0)=x_0$ for some $x_0$.
Let $n \geq 2$ be a positive integer and $\{x_i\}_{i=0}^n$ a sequence such that not all of its elements are zero and there is a positive constant $C_n$ for which:
(i) $x_1+ \dots +x_n=0$, and
(ii) for each $i$ either $x_i\leq x_{i+1}$ or $x_i\leq x_{i+1} + C_n x_{i+2}$ (all indexes are assumed modulo $n$).
Prove that
a) $C_n\geq 2$, and
b) $C_n=2$ if and only $2 \mid n$.
[b]p1.[/b] Trung has $2$ bells. One bell rings $6$ times per hour and the other bell rings $10$ times per hour. At the start of the hour both bells ring. After how much time will the bells ring again at the same time? Express your answer in hours.
[b]p2.[/b] In a soccer tournament there are $n$ teams participating. Each team plays every other team once. The matches can end in a win for one team or in a draw. If the match ends with a win, the winner gets $3$ points and the loser gets $0$. If the match ends in a draw, each team gets $1$ point. At the end of the tournament the total number of points of all the teams is $21$. Let $p$ be the number of points of the team in the first place. Find $n + p$.
[b]p3.[/b] What is the largest $3$ digit number $\overline{abc}$ such that $b \cdot \overline{ac} = c \cdot \overline{ab} + 50$?
[b]p4.[/b] Let s(n) be the number of quadruplets $(x, y, z, t)$ of positive integers with the property that $n = x + y + z + t$. Find the smallest $n$ such that $s(n) > 2014$.
[b]p5.[/b] Consider a decomposition of a $10 \times 10$ chessboard into p disjoint rectangles such that each rectangle contains an integral number of squares and each rectangle contains an equal number of white squares as black squares. Furthermore, each rectangle has different number of squares inside. What is the maximum of $p$?
[b]p6.[/b] If two points are selected at random from a straight line segment of length $\pi$, what is the probability that the distance between them is at least $\pi- 1$?
[b]p7.[/b] Find the length $n$ of the longest possible geometric progression $a_1, a_2,..,, a_n$ such that the $a_i$ are distinct positive integers between $100$ and $2014$ inclusive.
[b]p8.[/b] Feng is standing in front of a $100$ story building with two identical crystal balls. A crystal ball will break if dropped from a certain floor $m$ of the building or higher, but it will not break if it is dropped from a floor lower than $m$. What is the minimum number of times Feng needs to drop a ball in order to guarantee he determined $m$ by the time all the crystal balls break?
[b]p9.[/b] Let $A$ and $B$ be disjoint subsets of $\{1, 2,..., 10\}$ such that the product of the elements of $A$ is equal to the sum of the elements in $B$. Find how many such $A$ and $B$ exist.
[b]p10.[/b] During the semester, the students in a math class are divided into groups of four such that every two groups have exactly $2$ students in common and no two students are in all the groups together. Find the maximum number of such groups.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Prove that $\binom{n+k}{n}$ can be written as product of $n$ pairwise coprime numbers $a_1,a_2,\dots,a_n$ such that $k+i$ is divisible by $a_i$ for all indices $i$.
A positive integer $n$ is called special if it can be represented in the form $n=\frac{x^2+y^2}{u^2+v^2}$, for some positive integers $x,y,u,v$. Prove that
(a) $25$ is special;
(b) $2014$ is not special;
(c) $2015$ is not special.
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.
Proposed by United Kingdom