Found problems: 85335
Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that
$$k\le2\lfloor\sqrt n\rfloor.$$
Compute the greatest constant $K$ such that for all positive real numbers $a,b,c,d$ measuring the sides of a cyclic quadrilateral, we have
\[
\left(\frac{1}{a+b+c-d}+\frac{1}{a+b-c+d}+\frac{1}{a-b+c+d}+\frac{1}{-a+b+c+d}\right)(a+b+c+d)\geq K.
\]
[i]2015 CCA Math Bonanza Lightning Round #3.4[/i]
A row of $1000$ numbers is written on the blackboard. We write a new row, below the first according to the rule:
We write under every number $a$ the natural number, indicating how many times the number $a$ is encountered in the first line. Then we write down the third line: under every number $b$ -- the natural number, indicating how many times the number $b$ is encountered in the second line, and so on.
a) Prove that there is a line that coincides with the preceding one.
b) Prove that the eleventh line coincides with the twelfth.
c) Give an example of the initial line such, that the tenth row differs from the eleventh.
Let $a, b, c$ be positive reals such that $abc=1$. Show that \[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle.
[asy]
unitsize(0.25cm);
path p=polygon(3);
for(int m=0; m<=11;++m){
for(int n=0 ; n<= 11-m; ++n){
draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p);
}
}
[/asy]
An integer $a$ is a quadratic nonresidue modulo a prime $p$ if there does not exist $x \in Z$ such that $x^2 \equiv a$ (mod $p$). How many ordered pairs $(a, b)$ modulo $29$ exist such that
$$a + b\equiv 1 \,\,\, (mod \,\,\, 29)$$
where both $a$ and $b$ are quadratic nonresidues modulo $29$?
a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$.
b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$.
Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$
Calculate number of the hamiltonian cycles of the graph below: (15 points)
Let $ABC$ be an isosceles triangle with $CA = CB$. Let $D$ be the foot of the alttiude from $C$, and $\ell$ be the external angle bisector at $C$. Take a point $N$ on $\ell$ so that $AN> AC$ , on the same side as $A$ wrt $CD$. The bisector of the angle $NAC$ cuts $\ell$'at $F$. Show that $\angle NCD + \angle BAF> 180^o.$
Suppose that $f: \mathbb{R}\rightarrow\mathbb{R}$ fulfils $\left|\sum^n_{k=1}3^k\left(f(x+ky)-f(x-ky)\right)\right|\le1$ for all $n\in\mathbb{N},x,y\in\mathbb{R}$. Prove that $f$ is a constant function.
Let $A=(0,0), B=(1,0), C = (-1,0)$, and $D = (-1,1)$. Let $\mathcal C$ be the closed curve given by the segment $AB$, the minor arc of the circle $x^2 + (y-1)^2 = 2$ connecting $B$ to $C$, the segment $CD$, and the minor arc of the circle $x^2 + (y-1)^2=1$ connecting $D$ to $A$. Let $\mathcal D$ be a piece of paper whose boundary is $\mathcal C$. Compute the sum of all integers $2\le n\le 2019$ such that it is possible to cut $\mathcal D$ into $n$ congruent pieces of paper.
[i]Proposed by Vincent Huang[/i]
In the set of integers solve the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{p}$, where $p$ is a prime number.
Let be a natural number $ n. $ Calculate
$$ \sum_{k=1}^{n^2}\#\left\{ d\in\mathbb{N}| 1\le d\le k\le d^2\le n^2\wedge k\equiv 0\pmod d \right\} . $$
Here, $ \# $ means cardinal.
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$
The diagram shows an octagon consisting of $10$ unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base $5$. If $\overline{PQ}$ bisects the area of the octagon, what is the ratio $\frac{XQ}{QY}$?
[asy] import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((0,0)--(6,0),linewidth(1.2pt)); draw((0,0)--(0,1),linewidth(1.2pt)); draw((0,1)--(1,1),linewidth(1.2pt)); draw((1,1)--(1,2),linewidth(1.2pt)); draw((1,2)--(5,2),linewidth(1.2pt)); draw((5,2)--(5,1),linewidth(1.2pt)); draw((5,1)--(6,1),linewidth(1.2pt)); draw((6,1)--(6,0),linewidth(1.2pt));draw((1,1)--(5,1),linewidth(1.2pt)); draw((1,1)--(1,0),linewidth(1.2pt));draw((2,2)--(2,0),linewidth(1.2pt)); draw((3,2)--(3,0),linewidth(1.2pt)); draw((4,2)--(4,0),linewidth(1.2pt)); draw((5,1)--(5,0),linewidth(1.2pt)); draw((0,0)--(5,1.5),linewidth(1.2pt)); dot((0,0),ds); label("$P$", (-0.23,-0.26),NE*lsf); dot((0,1),ds); dot((1,1),ds); dot((1,2),ds); dot((5,2),ds); label("$X$", (5.14,2.02),NE*lsf); dot((5,1),ds); label("$Y$", (5.12,1.14),NE*lsf); dot((6,1),ds); dot((6,0),ds); dot((1,0),ds); dot((2,0),ds); dot((3,0),ds); dot((4,0),ds); dot((5,0),ds); dot((2,2),ds); dot((3,2),ds); dot((4,2),ds); dot((5,1.5),ds); label("$Q$", (5.14,1.51),NE*lsf); clip((-4.19,-5.52)--(-4.19,6.5)--(10.08,6.5)--(10.08,-5.52)--cycle); [/asy]
$\textbf{(A)}\ \frac 25 \qquad
\textbf{(B)}\ \frac 12 \qquad
\textbf{(C)}\ \frac 35 \qquad
\textbf{(D)}\ \frac 23 \qquad
\textbf{(E)}\ \frac 34$
There is an $n*n$ table with some unit cells colored black and the others are white.
In each step , Amin takes a $row$ with exactly one black cell in it , and color all cells in that black cell's $column$ red.
While Ali , takes a $column$ with exactly one black cell in it , and color all cells in that black cell's $row$ red.
Prove that Amin can color all the cells red , iff Ali can do so.
In equiangular octagon $CAROLINE$, $CA = RO = LI = NE = \sqrt{2}$ and $AR = OL = IN = EC = 1$. The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$, that is, that total area of the six triangular regions. Then $K=\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
Let $n>2$ be a positive integer. Find the largest value $h$ and the smallest value $H$ for which
\[h<{a_1\over a_1+a_2}+{a_2\over a_2+a_3}+\cdots+{a_n\over a_n+a_1}<H\]
holds for any positive reals $a_1,\dots,a_n$.
Let $ABC$ be an acute triangle with $AC\neq BC$, $M$ the midpoint of side $AB$, $H$ is the orthocenter of $\triangle ABC$, $D$ on $BC$ is the foot of the altitude from $A$ and $E$ on $AC$ is the foot of the perpendicular from $B$. Prove that the lines $AB, DE$ and the perpendicular to $MH$ through $C$ are concurrent.
If $A, B$ and $C$ are real angles such that
$$\cos (B-C)+\cos (C-A)+\cos (A-B)=-3/2,$$
find
$$\cos (A)+\cos (B)+\cos (C)$$
Point $ D$ is chosen on the side $ AB$ of triangle $ ABC$. Point $ L$ inside the triangle $ ABC$ is such that $ BD=LD$ and $ \angle LAB=\angle LCA=\angle DCB$. It is known that $ \angle ALD+\angle ABC=180^\circ$. Prove that $ \angle BLC=90^\circ$.