Found problems: 85335
Let $A$ be a set of $n$ points in the space. From the family of all segments with endpoints in $A$, $q$ segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of $m$ yellow segments, where $m \geq \frac{2q}{n}$, arranged in order of increasing length.
Solve the equation $|x^2 -1|+ |x^2 - 4| = mx$ as a function of the parameter $m$. Which pairs $(x,m)$ of integers satisfy this equation?
Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$
Find, with proof, all functions $f$ mapping integers to integers with the property that for all integers $m,n$, $$f(m)+f(n)= \max\left(f(m+n),f(m-n)\right).$$
Let $T_1$ be an isosceles triangle with sides of length 8, 11, and 11. Let $T_2$ be an isosceles triangle with sides of length $b$, 1, and 1. Suppose that the radius of the incircle of $T_1$ divided by the radius of the circumcircle of $T_1$ is equal to the radius of the incircle of $T_2$ divided by the radius of the circumcircle of $T_2$. Determine the largest possible value of $b$.
Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$.
a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$.
b) Give an example of a non-constant function $f$ with property $(P)$.
c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.
A circle $\omega$ has center $O$ and radius $r$. A chord $BC$ of $\omega$ also has length $r$, and the tangents to $\omega$ at $B$ and $C$ meet at $A$. Ray $AO$ meets $\omega$ at $D$ past $O$, and ray $OA$ meets the circle centered at $A$ with radius $AB$ at $E$ past $A$. Compute the degree measure of $\angle DBE$.
[i]Author: Ray Li[/i]
Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area of the original index card?
[asy]
unitsize(0.6 cm);
pair A, B, C, D, E, F, G, H;
real x, y;
x = 9;
y = 5;
A = (0,y);
B = (x - 1,y);
C = (x - 1,y - 1);
D = (x,y - 1);
E = (x,0);
F = (1,0);
G = (1,1);
H = (0,1);
draw(A--B--C--D--E--F--G--H--cycle);
draw(interp(C,G,0.03)--interp(C,G,0.97), dashed, Arrows(6));
draw(interp(A,E,0.03)--interp(A,E,0.97), dashed, Arrows(6));
label("$1$", (B + C)/2, W);
label("$1$", (C + D)/2, S);
label("$8$", interp(A,E,0.3), NE);
label("$4 \sqrt{2}$", interp(G,C,0.2), SE);
[/asy]
$\textbf{(A) }14\qquad\textbf{(B) }10\sqrt{2}\qquad\textbf{(C) }16\qquad\textbf{(D) }12\sqrt{2}\qquad\textbf{(E) }18$
The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that
\[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\]
for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that
\[\varphi(x,y,z) = h(x+y+z)\]
for all real numbers $x,y$ and $z.$
Statement $P$: solution set to inequalities $a_1x^2+b_1x+c_1>0$ and $a_2x^2+b_2x+c_2>0$ are the same; statement $Q$: $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.
$\text{(A)}$ $Q$ is sufficient and necessary condition of $P$.
$\text{(B)}$ $Q$ is sufficient but unnecessary condition of $P$.
$\text{(C)}$ $Q$ is insufficient but necessary condition of $P$.
$\text{(D)}$ $Q$ is insufficient and unnecessary condition of $P$.
Problem Shortlist BMO 2017
Let $ a $,$ b$,$ c$, be positive real numbers such that $abc= 1 $. Prove that
$$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+b^{5}+b^{2}}\leq 1 . $$
$xxxx$
$xx$
$x$
$x$
In how many ways can you fill in the $x$s with the numbers $1-8$ so that for each $x$, the numbers below and to the right are higher.
Two $n$-sided polygons are said to be of the same type if we can label their vertices in clockwise order as $A_1$, $A_2$, ..., $A_n$ and $B_1$, $B_2$, ..., $B_n$ respectively such that each pair of interior angles $A_i$ and $B_i$ are either both reflex angles or both non-reflex angles. How many different types of $11$-sided polygons are there?
Let $(a_n), (b_n)$, $n = 1,2,...$ be two sequences of integers defined by $a_1 = 1, b_1 = 0$ and for $n \geq 1$
$a_{n+1} = 7a_n + 12b_n + 6$
$b_{n+1} = 4a_n + 7b_n + 3$
Prove that $a_n^2$ is the difference of two consecutive cubes.
[b]2.[/b] Omit the vertices of a closed rectangle; the configuration obtained in such a way will be called a reduced rectangle. Prove tha the set-union of any system of reduced rectangles with parallel sides is equal to the union of countably many elements of the system. [b](St. 3)[/b]
Given that nonzero reals $a,b,c,d$ satisfy $a^b=c^d$ and $\frac{a}{2c}=\frac{b}{d}=2$, compute $\frac{1}{c}$.
[i]2021 CCA Math Bonanza Lightning Round #2.2[/i]
$ABCD$ is a trapezoid with $AB \parallel CD$. The diagonals intersect at $P$. Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\omega _3$ is the circumcircle of triangle $BPC$.
Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$.
[i]Proposed by Kasra Ahmadi[/i]
Let $f$ be a function on non-negative integers defined as follows $$f(2n)=f(f(n))~~~\text{and}~~~f(2n+1)=f(2n)+1$$
[b](a)[/b] If $f(0)=0$ , find $f(n)$ for every $n$.
[b](b)[/b] Show that $f(0)$ cannot equal $1$.
[b](c)[/b] For what non-negative integers $k$ (if any) can $f(0)$ equal $2^k$ ?
Let $A=\{1,2,\ldots, 2006\}$. Find the maximal number of subsets of $A$ that can be chosen such that the intersection of any 2 such distinct subsets has 2004 elements.
On the island of knights and liars, a tennis tournament was held, in which $100$ people participated in. Each two of them played exactly $1$ time with the other one. After the tournament, each of the participants declared: “I have beaten as many knights as liars,” while all the knights told the truth, and all the liars lied. What is the largest number of knights that could participate in the tournament?
Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.
In each square of the $100\times 100$ square table, type $1, 2$, or $3$. Consider all subtables $m \times n$, where $m = 2$ and $n = 2$. A subtable will be called [i]balanced [/i] if it has in its corner boxes of four identical numbers boxes . For as large a number $k$ prove, that we can always find $k$ balanced subtables, of which no two overlap, i.e. do not have a common box.
The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to $ 100$. How many cells are there that are on the distance $ 50$ from each of the three cells?
A plane intersects a sphere of radius $10$ such that the distance from the center of the sphere to the plane is $9$. The plane moves toward the center of the bubble at such a rate that the increase in the area of the intersection of the plane and sphere is constant, and it stops once it reaches the center of the circle. Determine the distance from the center of the sphere to the plane after two-thirds of the time has passed.
Find all real $x$ such that $0 < x < \pi $ and $\frac{8}{3 sin x - sin 3x} + 3 sin^2x \le 5$.