Found problems: 85335
$200$ sticks are given whose lengths are $1, 2, 4, \ldots , 2^{199}$. What is the smallest number of sticks needed to be broken so that out of all the resulting sticks, several triangles could be created, if each stick could be broken only once, and each triangle can be created out of only three sticks?
Let $D_n$ be the set of divisors of $n$. Find all natural $n$ such that it is possible to split $D_n$ into two disjoint sets $A$ and $G$, both containing at least three elements each, such that the elements in $A$ form an arithmetic progression while the elements in $G$ form a geometric progression.
Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge.
(a) Describe one polyhedron with the above property.
(b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.
We're given the functions $f(x)=|x-1|-|x-2|$ and $g(x)=|x-3|$.
a) Draw the graph of the function $f(x)$.
b) Determine the area of the section enclosed by the functions $f(x)$ and $g(x)$.
A set S consisting of $2019$ (different) positive integers has the following property:
[i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i].
What is the maximum number of prime numbers that $S$ can contain?
Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m.$
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
In $\vartriangle ABC$ on the sides $BC, CA, AB$ mark feet of altitudes $H_1, H_2, H_3$ and the midpoint of sides $M_1, M_3, M_3$. Let $H$ be orthocenter $\vartriangle ABC$. Suppose that $X_2, X_3$ are points symmetric to $H_1$ wrt $BH_2$ and $CH_3$. Lines $M_3X_2$ and $M_2X_3$ intersect at point $X$. Similarly, $Y_3,Y_1$ are points symmetric to $H_2$ wrt $C_3H$ and $AH_1$.Lines $M_1Y_3$ and $M_3Y_1$ intersect at point $Y.$ Finally, $Z_1,Z_2$ are points symmetric to $H_3$ wrt $AH_1$ and $BH_2$. Lines $M_1Z_2$ and $M_2Z_1$ intersect at the point $Z$ Prove that $H$ is the incenter $\vartriangle XYZ$ .
Let non-integer real numbers $a, b,c,d$ are given, such that the sum of each $3$ of them is integer. May it happen that $ab + cd$ is an integer.
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?
$\textbf{(A)}\hspace{.05in}3127 \qquad \textbf{(B)}\hspace{.05in}3133 \qquad \textbf{(C)}\hspace{.05in}3137 \qquad \textbf{(D)}\hspace{.05in}3139 \qquad \textbf{(E)}\hspace{.05in}3149 $
Prove that the points $ A_1, A_2, \ldots, A_n $ ($ n \geq 7 $) located on the surface of the sphere lie on a circle if and only if the planes tangent to the surface of the sphere at these points have a common point or are parallel to one straight line.
A regular hexagon with side length $ 1 $ is given. There are $ m $ points in its interior such that no $ 3 $ are collinear. The hexagon is divided into triangles (triangulated), such that every point of the $ m $ given and every vertex of the hexagon is a vertex of such a triangle. The triangles don't have common interior points. Prove that there exists a triangle with area not greater than $ \frac{3 \sqrt{3}}{4(m+2)}$.
Let $ABC$ be an acute triangle with $AB\ne AC$ and circumcircle $\omega$. The angle bisector of $BAC$ intersects $BC$ and $\omega$ at $D$ and $E$ respectively. Circle with diameter $DE$ intersects $\omega$ again at $F \ne E$. Point $P$ is on $AF$ such that $PB = PC$ and $X$ and $Y$ are feet of perpendiculars from $P$ to $AB$ and $AC$ respectively. Let $H$ and $H'$ be the orthocenters of $ABC$ and $AXY$ respectively. $AH$ meets $\omega$ again at $Q$ . If $AH'$ and $HH'$ intersect the circle with diameter $AH$ again at points $S$ and $T$, respectively, prove that the lines $AT , HS$ and $FQ$ are concurrent.
Pentagon $ANDD'Y$ has $AN \parallel DY$ and $AY \parallel D'N$ with $AN = D'Y$ and $AY = DN$. If the area of $ANDY$ is 20, the area of $AND'Y$ is 24, and the area of $ADD'$ is 26, the area of $ANDD'Y$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.
[i]Proposed by Andy Xu[/i]
Suppose $L_1$ is a circle with center $O$, and $L_2$ is a circle with center $O'$. The circles intersect at $ A$ and $ B$ such that $\angle OAO' = 90^o$. Suppose that point $X$ lies on the circumcircle of triangle $OAB$, but lies inside $L_2$. Let the extension of $OX$ intersect $L_1$ at $Y$ and $Z$. Let the extension of $O'X$ intersect $L_2$ at $W$ and $V$ . Prove that $\vartriangle XWZ$ is congruent with $\vartriangle XYV$.
Determine all polynomials $P$ with real coefficients satisfying the following condition: whenever $x$ and $y$ are real numbers such that $P(x)$ and $P(y)$ are both rational, so is $P(x + y)$.
What is the largest number of parts into which $n$ planes can divide space?
We assume that the set of planes is non-degenerate in the sense that any three planes intersect in one point and no four planes have a common point (and for n=2 it is necessary to require that the planes are not parallel).
Let $ABC$ be an acute triangle and $G$ be the intersection of the meadians of triangle $ABC$. Let $D $be the foot of the altitude drawn from $A$ to $BC$. Draw a parallel line such that it is parallel to $BC$ and one of the points of it is $A$.Donate the point $S$ as the intersection of the parallel line and circumcircle $ABC$. Prove that $S,G,D$ are co-linear
[asy]
size(6cm);
defaultpen(fontsize(10pt));
pair A = dir(50), S = dir(130), B = dir(200), C = dir(-20), G = (A+B+C)/3, D = foot(A, B, C);
draw(A--B--C--cycle, black+linewidth(1));
draw(A--S^^A--D, magenta);
draw(S--D, red+dashed);
draw(circumcircle(A, B, C), heavymagenta);
string[] names = {"$A$", "$B$", "$C$","$D$", "$G$","$S$"};
pair[] points = {A, B, C,D,G,S};
pair[] ll = {A, B, C,D, G,S};
int pt = names.length;
for (int i=0; i<pt; ++i)
dot(names[i], points[i], dir(ll[i]));
[/asy]
Let $ABC$ be a right triangle in $A$ , whose leg measures $1$ cm. The bisector of the angle $BAC$ cuts the hypotenuse in $R$, the perpendicular to $AR$ on $R$ , cuts the side $AB$ at its midpoint. Find the measurement of the side $AB$ .
Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
$\textbf{(A)}\ 2.0 \qquad \textbf{(B)}\ 2.2\qquad \textbf{(C)}\ 2.8 \qquad \textbf{(D)}\ 3.4 \qquad \textbf{(E)}\ 4.4$
Find all positive integers $k$ such that there exists positive integers $a, b$ such that
\[a^2+4=(k^2-4)b^2.\]
Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$
Let $a, b$, and $c$ denote the three sides of a billiard table in the shape of an equilateral triangle. A ball is placed at the midpoint of side $a$ and then propelled toward side $b$ with direction defined by the angle $\theta$. For what values of $\theta$ will the ball strike the sides $b, c, a$ in that order?
Find all positive integers $n$ such that there exist two permutations $a_0,a_1,\ldots,a_{n-1}$ and $b_0,b_1,\ldots,b_{n-1}$ of the set $\lbrace0,1,\ldots,n-1\rbrace$, satisfying the condition
$$ia_i\equiv b_i\pmod{n}$$
for all $0\le i\le n-1$.
[i]Proposed by Fysty[/i]
Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex).
Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.