Found problems: 85335
Prove that if
$$ (a + b + c)^2 = 3 (ab + bc + ac - x^2 - y^2 - z^2),$$
where $ a $, $ b $, $ c $, $ x $, $ y $, $ z $ denote real numbers, then $ a = b = c $ and $ x = y = z = 0 $.
Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds
$$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$
A set of ten two-digit numbers is given. Prove that one can always choose two disjoint subsets of this set such that the sum of their elements is the same.
Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )
For every positive integer $n{}$ denote $a_n$ as the last digit of the sum of the number from $1$ to $n{}$. For example $a_5=5, a_6=1.$
a) Find $a_{21}.$
b) Compute the sum $a_1+a_2+\ldots+a_{2015}.$
Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, ...$ Find $n$ such that the first $n$ terms sum up to $2010.$
Solve the equation $2z^3-(5+6i)z^2+9iz+1-3i=0$ over $\mathbb C$ given that one of the solutions is real.
$ABC$ is a right triangle with $\angle C$ the right angle. $X$ is some point inside $ABC$ satisfying $CA=AX$. Let $D$ be the feet of altitude from $C$ to $AB$, and $Y(\neq X)$ the point of intersection of $DX$ and the circumcircle of $ABX$. Prove that $AX=AY$.
Prove that the squares with sides $\frac{1}{1}, \frac{1}{2}, \frac{1}{3},\ldots$ may be put into the square with side $\frac{3}{2} $ in such a way that no two of them have any interior point in common.
From a point $P$ outside of a circle with centre $O$, tangent segments $\overline{PA}$ and $\overline{PB}$ are drawn. If $\frac1{\left|\overline{OA}\right|^2}+\frac1{\left|\overline{PA}\right|^2}=\frac1{16}$, then $\left|\overline{AB}\right|=$?
$\textbf{(A)}~4$
$\textbf{(B)}~6$
$\textbf{(C)}~8$
$\textbf{(D)}~10$
Which operation makes the following expression true: $(4 \underline{~~~~} 1) \times (3 \underline{~~~~} 2 - 1) = 2$?
$\textbf{(A) } +\qquad\textbf{(B) } -\qquad\textbf{(C) } \times\qquad\textbf{(D) } \div\qquad\textbf{(E) } \text{There is no such operation}$
Let $f(x)=x^5-3x^4+2x^3+6x^2+x-14=a(x-1)^5+b(x-1)^4+c(x-1)^3+d(x-1)^2+e(x-1)+f,$ for some real constants $a,b,c,d,e,f.$ Determine the value of $ab+bc+cd+de+ad+be.$
Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Construct the square $BDEC$ such as $A$ and the square are at opposite sides of $BC$. Let the angle bisector of $\angle BAC$ cut the sides $[BC]$ and $[DE]$ at $F$ and $G$, respectively. If $|AB|=24$ and $|AC|=10$, calculate the area of quadrilateral $BDGF$.
Show that any positive integer has at least as many positive divisors of the form $3k+1$ as of the form $3k-1$. [b](N. 7)[/b]
A pack of $2008$ cards, numbered from $1$ to $2008$, is shuffled in order to play a game in which each move has two steps:
(i) the top card is placed at the bottom;
(ii) the new top card is removed.
It turns out that the cards are removed in the order $1,2,\dots,2008$. Which card was at the top before the game started?
Let $ABC$ be a triangle and let $M$ be the midpoint of the side $AB$. Let $BD$ be the interior angle bisector of $\angle ABC$, $D\in AC$. Prove that if $MD \perp BD$ then $AB=3BC$.
Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, "At least two of us have the same score." Marty thinks, "I didn't get the lowest score." Shana thinks, "I didn't get the highest score." List the scores from lowest to highest for Marty (M), Quay (Q) and Shana (S).
$ \text{(A)}\ \text{S,Q,M}\qquad\text{(B)}\ \text{Q,M,S}\qquad\text{(C)}\ \text{Q,S,M}\qquad\text{(D)}\ \text{M,S,Q}\qquad\text{(E)}\ \text{S,M,Q} $
Bhairav the Bat lives next to a town where $12.5$% of the inhabitants have Type AB blood. When Bhairav the Bat leaves his cave at night to suck of the inhabitants blood, chooses individuals at random until he bites one with type AB blood, after which he stops. What is the expected value of the number of individuals Bhairav the Bat will bite in any given night?
[i]2015 CCA Math Bonanza Lightning Round #3.1[/i]
How many real numbers $x$ are solutions to the following equation? \[ 2003^x + 2004^x = 2005^x \]
There are $n$ holes in a circle. The holes are numbered $1,2,3$ and so on to $n$. In the beginning, there is a peg in every hole except for hole $1$. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if $n=4$ the puzzle can be solved in two jumps: peg $3$ jumps peg $4$ to hole $1$, then peg $2$ jumps the peg in $1$ to hole $4$. (See illustration below, in which black circles indicate pegs and white circles are holes.)
[center][img]http://i.imgur.com/4ggOa8m.png[/img][/center]
[list=a]
[*]Can the puzzle be solved for $n=5$?
[*]Can the puzzle be solved for $n=2014$?
[/list]
In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.
Let $ABC$ be an scalene triangle and $I$ and $H$ its incenter, ortocenter respectively.
The incircle touchs $BC$, $CA$ and $AB$ at $D,E$ an $F$. $DF$ and $AC$ intersects at $K$ while $EF$ and $BC$ intersets at $M$.
Shows that $KM$ cannot be paralel to $IH$.
PS1: The original problem without the adaptation apeared at the Brazilian Olympic Revenge 2011 but it was incorrect.
PS2:The Brazilian Olympic Revenge is a competition for teachers, and the problems are created by the students.
Sorry if I had some English mistakes here.
Show that there are at least $3$ and at most $4$ powers of $2$ with $m$ digits. For which $m$ are there $4$?
Consider $13$ weights of integer mass (in grams). It is known that any $6$ of them may be placed onto two pans of a balance achieving equilibrium. Prove that all the weights are of equal mass.
Let $f_1(x)=x^2-1$, and for each positive integer $n \geq 2$ define $f_n(x) = f_{n-1}(f_1(x))$. How many distinct real roots does the polynomial $f_{2004}$ have?
Let $AD$ be the altitude $ABC$ of an acute triangle. On the line $AD$ are chosen different points $E$ and $F$ so that $|DE |= |DF|$ and point $E$ is in the interior of triangle $ABC$. The circumcircle of triangle $BEF$ intersects $BC$ and $BA$ for second time at points $K$ and $M$ respectively. The circumcircle of the triangle $CEF$ intersects the $CB$ and $CA$ for the second time at points $L$ and $N$ respectively. Prove that the lines $AD, KM$ and $LN$ intersect at one point.
Let the notation $\underline{ABC}$ denote the number compromised of the digits $A$, $B$, and $C$ with $0\leq A,B,C\leq9$. That is, $\underline{ABC}=100A+10B+C$ and $\underline{CCAAC}=10000C+1000C+100A+10A+C$. Now, if $(\underline{ABC})^2=\underline{CCAAC}$, where $A$, $B$, and $C$ are distinct nonzero digits, find the $3$ digit number $\underline{ABC}$.