Found problems: 85335
In a convex quadrilateral $ ABCD, $ let $ A’,B’ $ be the orthogonal projections to $ CD $ of $ A, $ respectively, $ B. $
[b]a)[/b] Assuming that $ BB’\le AA’ $ and that the perimeter of $ ABCD $ is $ (AB+CD)\cdot BB’, $ is $ ABCD $ necessarily a trapezoid?
[b]b)[/b] The same question with the addition that $ \angle BAD $ is obtuse.
There are $n$ pawns on $n$ distinct squares of a $19\times 19$ chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in two successive moves. What is largest number of pawns can a player place on the board (being able to arrange them freely) so as to be able to continue the game indefinitely?
In the United States, coins have the following thicknesses: penny, $ 1.55$ mm; nickel, $ 1.95$ mm; dime, $ 1.35$ mm; quarter, $ 1.75$ mm. If a stack of these coins is exactly $ 14$ mm high, how many coins are in the stack?
$ \textbf{(A)}\ 7\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 9\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ 11$
The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n$ fish for various values of $n$.
\[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\ \hline \text{number of contestants who caught} \ n \ \text{fish} & 9 & 5 & 7 & 23 & \dots & 5 & 2 & 1 \\ \hline \end{array} \]
In the newspaper story covering the event, it was reported that
(a) the winner caught 15 fish;
(b) those who caught 3 or more fish averaged 6 fish each;
(c) those who caught 12 or fewer fish averaged 5 fish each.
What was the total number of fish caught during the festival?
Find all real solutions of the equation \[\sqrt[3]{x-1}+\sqrt[3]{3x-1}=\sqrt[3]{x+1}.\]
Let $A_1A_2A_3...A_{72}$ be a regurar $72$-gon with center $O$. Calculate an extenral angle of that polygon and the angles $\angle A_{45} OA_{46}$, $\angle A_{44} A_{45}A_{46}$. How many diagonals does this polygon have?
Let $ABCD$ be a square of side length $1$ made of paper. Let $M$ be the midpoint of side $AB.$ Let $E$ be a point on side $BC$ and let $F$ be a point on side $AD$ such that $A$ lands on line $ME$ when $\triangle AMF$ is folded across line $MF$ and such that $B$ lands on line $MF$ when $\triangle BME$ is folded across line $ME.$ The area of the resulting shape when both folds are made can be written as $\tfrac{a-\sqrt{b}}{c}$ for positive integers $a, b, c$ with $b$ squarefree. Find $a + b + c.$
GM Bisain's IQ is so high that he can move around in $10$ dimensional space. He starts at the origin and moves in a straight line away from the origin, stopping after $3$ units. How many lattice points can he land on? A lattice point is one with all integer coordinates.
[i]2019 CCA Math Bonanza Lightning Round #4.2[/i]
Find all positive integers $n$ for which there exists an integer multiple of $2022$ such that the sum of the squares of its digits is equal to $n$.
Twelve places have been arranged at a round table for members of the Jury, with a name tag at each place . Professor K. being absent-minded instead of occupying his place, sits down at the next place (clockwise) . Each of the other Jury members in turn either occupies the place assigned to this member or, if it has been already occupied, sits down at the first free place in the clockwise order. The resulting seating arrangement depends on the order in which the Jury members come to the table. How many different seating arrangements of this kind are possible?
(A Shapovalov)
Given acute triangle $ABC$ with $AB>AC$, let $M$ be the midpoint of $BC$. $P$ is a point in triangle $AMC$ such that $\angle MAB=\angle PAC$. Let $O,O_1,O_2$ be the circumcenters of $\triangle ABC,\triangle ABP,\triangle ACP$ respectively. Prove that line $AO$ passes through the midpoint of $O_1 O_2$.
Define a permutation of the set $\{1,2,3,...,n\}$ to be $\textit{sortable}$ if upon cancelling an appropriate term of such permutation, the remaining $n-1$ terms are in increasing order. If $f(n)$ is the number of sortable permutations of $\{1,2,3,...,n\}$, find the remainder when $$\sum\limits_{i=1}^{2018} (-1)^i \cdot f(i)$$ is divided by $1000$. Note that the empty set is considered sortable.
[i]Proposed by [b]FedeX333X[/b][/i]
Two players play a game involving an $n \times n$ grid of chocolate. Each turn, a player may either eat a piece of chocolate (of any size), or split an existing piece of chocolate into two rectangles along a grid-line. The player who moves last loses. For how many positive integers $n$ less than $1000$ does the second player win?
(Splitting a piece of chocolate refers to taking an $a \times b$ piece, and breaking it into an $(a-c) \times b$ and a $c \times b$ piece, or an $a \times (b-d)$ and an $a \times d$ piece.)
[i]Proposed by Lewis Chen[/i]
Ler $a$ be the number $123456789$. Compare the numbers
$$2014^{9^{9^a}}, 2015^{a^{a^9}}$$
Find the sum of the solutions to $\dfrac{1}{1+\dfrac{1}{|x-25|}}=\frac{49}{50}$.
Let $n\ge 2$ be an integer. $a_1,a_2,\dotsc,a_n$ are arbitrarily chosen positive integers with $(a_1,a_2,\dotsc,a_n)=1$. Let $A=a_1+a_2+\dotsb+a_n$ and $(A,a_i)=d_i$. Let $(a_2,a_3,\dotsc,a_n)=D_1, (a_1,a_3,\dotsc,a_n)=D_2,\dotsc, (a_1,a_2,\dotsc,a_{n-1})=D_n$.
Find the minimum of $\prod\limits_{i=1}^n\dfrac{A-a_i}{d_iD_i}$
For $a,n\in\mathbb{Z}^+$, consider the following equation: \[ a^2x+6ay+36z=n\quad (1) \] where $x,y,z\in\mathbb{N}$.
a) Find all $a$ such that for all $n\geq 250$, $(1)$ always has natural roots $(x,y,z)$.
b) Given that $a>1$ and $\gcd (a,6)=1$. Find the greatest value of $n$ in terms of $a$ such that $(1)$ doesn't have natural root $(x,y,z)$.
Two functions of $x$ are differentiable and not identically zero. Find an example of two such functions having the property that the derivative of their quotient is the quotient of their derivatives.
Does there exist a bijection $f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}$, such that there exist a positive integer $k$, and it's possible to have each positive integer colored by one of $k$ chosen colors, such that for any $x \neq y$ , $f(x)+y$ and $f(y)+x$ are not the same color?
If $a = 2b + c$, $b = 2c + d$, $2c = d + a -1$, $d = a - c$, what is $b$?
Let $A$ be a point in the plane and $3$ lines which pass through this point divide the plane in $6$ regions.
In each region there are $5$ points. We know that no three of the $30$ points existing in these regions are collinear. Prove that there exist at least $1000$ triangles whose vertices are points of those regions such that $A$ lies either in the interior or on the side of the triangle.
Let be given an arbitrary tetrahedron $ABCD$ with volume $V$. Consider all lines which pass through the barycenter $S$ of the tetrahedron and intersect the edges $AD,BD,CD$ at points $A',B',C$ respectively. It is known that among the obtained tetrahedra there exists one with the minimal volume. Express this minimal volume in terms of $V$
Given is the non-right triangle $ABC$. $D,E$ and $F$ are the feet of the respective altitudes from $A,B$ and $C$. $P,Q$ and $R$ are the respective midpoints of the line segments $EF$, $FD$ and $DE$. $p \perp BC$ passes through $P$, $q \perp CA$ passes through $Q$ and $r \perp AB$ passes through $R$. Prove that the lines $p, q$ and $r$ pass through one point.
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.
Find all the integer solutions $(x,y,z)$ of the equation $(x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)$,