Found problems: 85335
Point $S$ is the midpoint of arc $ACB$ of the circumscribed circle $k$ around triangle $ABC$ with $AC>BC$. Let $I$ be the incenter of triangle $ABC$. Line $SI$ intersects $k$ again at point $T$. Let $D$ be the reflection of $I$ across $T$ and $M$ be the midpoint of side $AB$. Line $IM$ intersects the line through $D$, parallel to $AB$, at point $E$. Prove that $AE=BD$.
We are given a finite set of segments of the same line. Prove that we can color each segment red or blue such that, for each point $p$ on the line, the number of red segments containing $p$ differs from the number of blue segments containing $p$ by at most $1$.
Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has
$$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$
where $\{ x \}$ denotes the fractional part of $x$.
Let $ K$ be a compact convex body in the $ n$-dimensional Euclidean space. Let $ P_1,P_2,...,P_{n\plus{}1}$ be the vertices of a simplex having maximal volume among all simplices inscribed in $ K$. Define the points $ P_{n\plus{}2},P_{n\plus{}3},...$ successively so that $ P_k \;(k>n\plus{}1)$ is a point of $ K$ for which the volume of the convex hull of $ P_1,...,P_k$ is maximal. Denote this volume by $ V_k$. Decide, for different values of $ n$, about the truth of the statement "the sequence $ V_{n\plus{}1},V_{n\plus{}2},...$ is concave."
[i]L. Fejes- Toth, E. Makai[/i]
Rectangle $ABCD$ has sides $AB = 10$ and $AD = 7$. Point $G$ lies in the interior of $ABCD$ a distance $2$ from side $\overline{CD}$ and a distance $2$ from side $\overline{BC}$. Points $H, I, J$, and $K$ are located on sides $\overline{BC}, \overline{AB}, \overline{AD}$, and $\overline{CD}$, respectively, so that the path $GHIJKG$ is as short as possible. Then $AJ = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\mathbb{R}^3$. For which distribution of $X$ will the expectation of the (Euclidean) distance of $X$ and $Y$ be maximal?
In an isosceles triangle $ABC~(AC=BC)$, let $O$ be its circumcenter, $D$ the midpoint of $AC$ and $E$ the centroid of $DBC$. Show that $OE$ is perpendicular to $BD$.
Determine all pairs $(m,n)$ of positive integers such that $m^2+n^2=2018(m-n)$
Find all natural numbers $x,y,z$ such that $a+2^x3^y=z^2$.
Find all functions $f: {\mathbb{R^\plus{}}}\to{\mathbb{R^\plus{}}}$ such that
\[ f(1\plus{}xf(y))\equal{}yf(x\plus{}y)\]
for all $x,y\in\mathbb{R^\plus{}}$.
Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be the point on the line $AB$, distinct from $B$, such that $CG = CB$. Let $H$ be the point on the line $BC$, distinct from $B$, such that $AB = AH$. Prove that triangle $DGH$ is isosceles.
Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.
Given are positive integers $a, b$ satisfying $a \geq 2b$. Does there exist a polynomial $P(x)$ of degree at least $1$ with coefficients from the set $\{0, 1, 2, \ldots, b-1 \}$ such that $P(b) \mid P(a)$?
Take two points $A\ (-1,\ 0),\ B\ (1,\ 0)$ on the $xy$-plane. Let $F$ be the figure by which the whole points $P$ on the plane satisfies $\frac{\pi}{4}\leq \angle{APB}\leq \pi$ and the figure formed by $A,\ B$.
Answer the following questions:
(1) Illustrate $F$.
(2) Find the volume of the solid generated by a rotation of $F$ around the $x$-axis.
Let $m$ and $n$ be two positive integers greater than $1$. Prove that there are $m$ positive integers $N_1$ , $\ldots$ , $N_m$ (some of them may be equal) such that \[\sqrt{m}=\sum_{i=1}^m{(\sqrt{N_i}-\sqrt{N_i-1})^{\frac{1}{n}}.}\]
Prove that any function that maps the integers to themselves is a sum of any finite number of injective functions that map the integers to themselves.
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
The point $ O$ is the center of the circle circumscribed about $ \triangle ABC$, with $ \angle BOC \equal{} 120^\circ$ and $ \angle AOB \equal{} 140^\circ$, as shown. What is the degree measure of $ \angle ABC$?
[asy]unitsize(2cm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
pair B=dir(80), A=dir(220), C=dir(320), O=(0,0);
draw(unitcircle);
draw(A--B--C--O--A--C);
draw(O--B);
draw(anglemark(C,O,A,2));
label("$A$",A,SW);
label("$B$",B,NNE);
label("$C$",C,SE);
label("$O$",O,S);
label("$140^{\circ}$",O,NW,fontsize(8pt));
label("$120^{\circ}$",O,ENE,fontsize(8pt));[/asy]$ \textbf{(A)}\ 35 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 60$
Let $ z_1,z_2,z_3\in\mathbb{C} , $ different two by two, having the same modulus $ \rho . $ Show that:
$$ \frac{1}{\left| z_1-z_2\right|\cdot \left| z_1-z_3\right|} +\frac{1}{\left| z_2-z_1\right|\cdot \left| z_2-z_3\right|} +\frac{1}{\left| z_3-z_1\right|\cdot \left| z_3-z_2\right|}\ge\frac{1}{\rho^2} . $$
A ten-level $2$-tree is drawn in the plane: a vertex $A_1$ is marked, it is connected by segments with two vertices $B_1$ and $B_2$, each of $B_1$ and $B_2$ is connected by segments with two of the four vertices $C_1, C_2, C_3, C_4$ (each $C_i$ is connected with one $B_j$ exactly); and so on, up to $512$ vertices $J_1, \ldots, J_{512}$. Each of the vertices $J_1, \ldots, J_{512}$ is coloured blue or golden. Consider all permutations $f$ of the vertices of this tree, such that (i) if $X$ and $Y$ are connected with a segment, then so are $f(X)$ and $f(Y)$, and (ii) if $X$ is coloured, then $f(X)$ has the same colour. Find the maximum $M$ such that there are at least $M$ permutations with these properties, regardless of the colouring.
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let
\[ c_n = \sum^8_{k=1} a^n_k\]
for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$
The $2001$ towns in a country are connected by some roads, at least one road from each town, so that no town is connected by a road to every other city. We call a set $D$ of towns [i]dominant[/i] if every town not in $D$ is connected by a road to a town in $D$. Suppose that each dominant set consists of at least $k$ towns. Prove that the country can be partitioned into $2001-k$ republics in such a way that no two towns in the same republic are connected by a road.
Let $a, b,c$ be positive integers such that $gcd(a, b, c) = 1$ and each product of two is divided by the third.
a) Prove that each of these numbers is equal to the least two remaining numbers the quotient of the coefficient and the highest coefficient.
b) Give an example of one of these larger numbers $a, b$ and $c$
Let $a,b,c \geq 0$ and $a+b+c=3.$ Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \geq \frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+a}$$
Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ be such that for every positive integer $n$, followings are satisfied.
i. $f(n+1) > f(n)$
ii. $f(f(n)) = 2n+2$
Find the value of $f(2013)$.
(Here, $\mathbb{N}$ is the set of all positive integers.)
Each of two similar triangles was cut into two triangles so that one of the resulting parts of one triangle is similar to one of the parts of the other triangle. Is it true that the remaining parts are also similar?
(D. Shnol)