Found problems: 85335
Given a triangle $ABC$, let $D$ be a point different from $A$ on the external bisectrix $\ell$ of the angle $BAC$, and let $E$ be an interior point of the segment $AD$. Reflect $\ell$ in the internal bisectrices of the angles $BDC$ and $BEC$ to obtain two lines that meet at some point $F$. Show that the angles $ABD$ and $EBF$ are congruent.
Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$.
Determine the size of the set $\{ \det A : A \in M \}$.
[i]Here $\det A$ denotes the determinant of the matrix $A$.[/i]
Every cell of a $2017\times 2017$ grid is colored either black or white, such that every cell has at least one side in common with another cell of the same color. Let $V_1$ be the set of all black cells, $V_2$ be the set of all white cells. For set $V_i (i=1,2)$, if two cells share a common side, draw an edge with the centers of the two cells as endpoints, obtaining graphs $G_i$. If both $G_1$ and $G_2$ are connected paths (no cycles, no splits), prove that the center of the grid is one of the endpoints of $G_1$ or $G_2$.
Let $ABC$ a triangle, $I$ the incenter, $D$ the contact point of the incircle with the side $BC$ and $E$ the foot of the bisector of the angle $A$. If $M$ is the midpoint of the arc $BC$ which contains the point $A$ of the circumcircle of the triangle $ABC$ and $\{F\} = DI \cap AM$, prove that $MI$ passes through the midpoint of $[EF]$.
Let $a, b, c$ be sides of a triangle. Prove that
$$\frac{1}{3} \leq \frac{a^2 +b^2 +c^2 }{(a+b+c)^2 } < \frac{1}{2}$$
and show that $\frac{1}{2}$ cannot be replaced with a smaller number.
For which numbers $ n$ is there a positive integer $ k$ with the following property: The sum of digits for $ k$ is $ n$ and the number $ k^2$ has sum of digits $ n^2.$
Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of points that lie on closed segments joining pairs of points of $A$ (a one-point set should be considered to be a special case of a closed segment). For a given nonempty set $A_0$, define $A_n =S(A_{n-1})$ for $n=1,2,\ldots$ Prove that $A_2 =A_3 =\ldots.$
If $ P(x),Q(x),R(x)$, and $ S(x)$ are all polynomials such that \[ P(x^5)\plus{}xQ(x^5)\plus{}x^2R(x^5)\equal{}(x^4\plus{}x^3\plus{}x^2\plus{}x\plus{}1)S(x),\] prove that $ x\minus{}1$ is a factor of $ P(x)$.
Let $x_1,x_2,\dots,x_n$ be positive numbers with product equal to 1. Prove that there exists a $k\in\{1,2,\dots,n\}$ such that
\[\frac{x_k}{k+x_1+x_2+\cdots+x_k}\ge 1-\frac{1}{\sqrt[n]{2}}.\]
On a square board with $9$ squares (three by three), nine elements of the set $S=\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ must be placed, different from each other, so that each one is in a box and the following conditions are met:
$\bullet$ The sums of the numbers in the second and third rows are, respectively, double and triple the sum of the numbers in the first row.
$\bullet$ The sum of the numbers in the second and third columns are, respectively, double and triple the sum of the numbers in the first column.
Show all the possible ways to place elements of $S$ on the board, fulfilling the indicated conditions.
The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$
You can inscribe a circle in the pentagon $ABCDE$. It is also known that $\angle ABC = \angle BAE = \angle CDE = 90^o$. Find the measure of the angle $ADB$.
Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$.
Given a finite graph $G$, let $f(G)$ be the number of triangles in graph $G$, $g(G)$ be the number of edges in graph $G$, find the minimum constant $c$, so that for each graph $G$, there is $f^ 2(G)\le c \cdot g^3(G)$.
Find the following limit.
\[\lim_{n\to\infty} \frac{1}{n} \log \left(\sum_{k=2}^{2^n} k^{1/n^2} \right)\]
Find all sequences ${{a}_{1}},{{a}_{2}},...,{{a}_{2000}}$ of real numbers such that $\sum\limits_{n=1}^{2000}{{{a}_{n}}=1999}$ and such that $\frac{1}{2}<{{a}_{n}}<1$ and ${{a}_{n+1}}={{a}_{n}}(2-{{a}_{n}})$ for all $n\ge 1$.
Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$, for which the following statement is true:
If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function
and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that
$f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$ then there is $h\in (x_{0}, x_{n})$ for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]
(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality.
\[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \]
(2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$
Let $T_1$ and $T_2$ internally tangent circumferences at $P$, with radius $R$ and $2R$, respectively.
Find the locus traced by $P$ as $T_1$ rolls tangentially along the entire perimeter of $T_2$
Triangle $\triangle PQR$, with $PQ=PR=5$ and $QR=6$, is inscribed in circle $\omega$. Compute the radius of the circle with center on $\overline{QR}$ which is tangent to both $\omega$ and $\overline{PQ}$.
Given a prime number $p$, how many $4$-tuples $(a, b, c, d)$ of positive integers with $0 \le a, b, c, d \le p-1$ satisfy $ad = bc$ mod $p$?
(a) Let $z_1,z_2$ be complex numbers such that $z_1z_2=1$ and $|z_1-z_2|=2$. Let $A,B,M_1,M_2$ denote the points in complex plane corresponding to $-1,1,z_1,z_2$, respectively. Show that $AM_1BM_2$ is a trapezoid and compute the lengths of its non-parallel sides. Specify the particular cases.
(b) Let $\mathcal C_1$ and $\mathcal C_2$ be circles in the plane with centers $O_1$ and $O_2$, respectively, and with radius $d\sqrt2$, where $2d=O_1O_2$. Let $P$ and $Q$ be two variable points on $\mathcal C_1$ and $\mathcal C_2$ respectively, both on $O_1O_2$ on on different sides of $O_1O_2$, such that $PQ=2d$. Prove that the locus of midpoints $I$ of segments $PQ$ is the same as the locus of points $M$ with $MO_1\cdot MO_2=m$ for some $m$.
In an acute-angled triangle $ABC$ with circumcenter $O$, points $P$ and $Q$ are taken on sides $AC$ and $BC$ respectively such that $\frac{AP}{PQ} = \frac{BC}{AB}$ and $\frac{BQ}{PQ} =\frac{AC}{AB}$ . Prove that the points $O, P,Q,C$ lie on a circle.
Let $ABCD$ be a convex quadrilateral. Take an arbitrary point $M$ on the line $AB$, and let $N$ be the point of intersection of the circumcircles of triangles $MAC$ and $MBC$ (different from $M$). Prove that:
a) The point $N$ lies on a fixed circle;
b) The line $MN$ passes though a fixed point.
If $b>1$, $x>0$ and $(2x)^{\log_b 2}-(3x)^{\log_b 3}=0$, then $x$ is
$\text{(A)}\ \frac{1}{216} \qquad \text{(B)}\ \frac{1}{6} \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ \text{not uniquely determined}$