Found problems: 85335
The hexagon $ABCDEF$ is convex and opposite sides are parallel. Show that the triangles $ACE$ and $BDF$ have equal area
Consider a rectangle $ABCD$ with $AB = a$ and $AD = b.$ Let $l$ be a line through $O,$ the center of the rectangle, that cuts $AD$ in $E$ such that $AE/ED = 1/2$. Let $M$ be any point on $l,$ interior to the rectangle.
Find the necessary and sufficient condition on $a$ and $b$ that the four distances from M to lines $AD, AB, DC, BC$ in this order form an arithmetic progression.
Determine the smallest and the greatest possible values of the expression
$$\left( \frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right)\left( \frac{a^2}{a^2+1}+\frac{b^2}{b^2+1}+\frac{c^2}{c^2+1}\right)$$ provided $a,b$ and $c$ are non-negative real numbers satisfying $ab+bc+ca=1$.
[i]Proposed by Walther Janous, Austria [/i]
What is the largest integer $n < 2018$ such that for all integers $b > 1$, $n$ has at least as many $1$'s in its base-$4$ representation as it has in its base-$b$ representation?
For each $n\in\mathbb{N}$ let $d_n$ denote the gcd of $n$ and $(2019-n)$.
Find value of $d_1+d_2+\cdots d_{2018}+d_{2019}$
In a unit circle where $O$ is your circumcenter, let $A$ and $B$ points in the circle with $\angle BOA = 90$. In the arc $AB$(minor arc) we have the points $P$ and $Q$ such that $PQ$ is parallel to $AB$. Let $X$ and $Y$ be the points of intersections of the line $PQ$ with $OA$ and $OB$ respectively. Find the value of $PX^2 + PY^2$
In a triangle $ABC$ the altitude $AD$ is drawn. Points $M$ on side $AC$ and $N$ on side $AB$ are taken so that $\angle MDA=\angle NDA$. Prove that the lines $AD,BM$ and $CN$ are concurrent.
Let $AA_1, BB_1, CC_1$ be the altitudes of triangle $ABC$, and $A0, C0$ be the common points of the circumcircle of triangle $A_1BC_1$ with the lines $A_1B_1$ and $C_1B_1$ respectively. Prove that $AA_0$ and $CC_0$ meet on the median of ABC or are parallel to it
An alien with four feet wants to wear four identical socks and four identical shoes, where on each foot a sock must be put on before a shoe. How many ways are there for the alien to wear socks and shoes?
Given a regular $26$-gon. Prove that for any $9$ vertices of that regular $26$-gon, then there exists three vertices that forms an isosceles triangle.
The polynomial $f(x) = x^3 + rx^2 + sx + t$ has $r, s$, and $t$ as its roots (with multiplicity), where $f(1)$ is rational and $t \ne 0$. Compute $|f(0)|$.
1) Prove that there are infinitely many positive integers $n$ such that there exists a permutation of $1, 2, 3, . . . , n$ with the property that the difference between any two adjacent numbers is equal to either $2015$ or $2016$.
2) Let $k$ be a positive integer. Is the statement in 1) still true if we replace the numbers $2015$ and $2016$ by $k$ and $k + 2016$, respectively?
Evaluate the sum $$\sum_{n=0}^{\infty}\frac{5n+7}{6^n}$$
Let $ABCD$ be a trapezoid such that $AD\parallel BC$. The interior angle bisectors of the corners $A$ and $B$ meet on $[DC]$. If $|BC|=9$ and $|AD|=4$, find $|AB|$.
For a positive number $n$, we write $d (n)$ for the number of positive divisors of $n$.
Determine all positive integers $k$ for which exist positive integers $a$ and $b$ with the property $k = d (a) = d (b) = d (2a + 3b)$.
$ABCD$ 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]
Show how to cut an isosceles right triangle into a number of triangles similar to it in such a way that every two of these triangles is of different size.
(AV Savkin)
Let $A$ and $B$ be sets with $2011^2$ and $2010$ elements, respectively. Show that there is a function $f:A \times A \to B$ satisfying the condition $f(x,y)=f(y,x)$ for all $(x,y) \in A \times A$ such that for every function $g:A \to B$ there exists $(a_1,a_2) \in A \times A$ with $g(a_1)=f(a_1,a_2)=g(a_2)$ and $a_1 \neq a_2.$
\[\int_1^\infty \frac{1}{x\sqrt{x^{2023}-1}}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
A non-Hookian spring has force $F = -kx^2$ where $k$ is the spring constant and $x$ is the displacement from its unstretched position. For the system shown of a mass $m$ connected to an unstretched spring initially at rest, how far does the spring extend before the system momentarily comes to rest? Assume that all surfaces are frictionless and that the pulley is frictionless as well.
[asy]
size(250);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((0,0)--(0,-1)--(2,-1)--(2+sqrt(3),-2));
draw((2.5,-2)--(4.5,-2),dashed);
draw(circle((2.2,-0.8),0.2));
draw((2.2,-0.8)--(1.8,-1.2));
draw((0,-0.6)--(0.6,-0.6)--(0.75,-0.4)--(0.9,-0.8)--(1.05,-0.4)--(1.2,-0.8)--(1.35,-0.4)--(1.5,-0.8)--(1.65,-0.4)--(1.8,-0.8)--(1.95,-0.6)--(2.2,-0.6));
draw((2+0.3*sqrt(3),-1.3)--(2+0.3*sqrt(3)+0.6/2,-1.3+sqrt(3)*0.6/2)--(2+0.3*sqrt(3)+0.6/2+0.2*sqrt(3),-1.3+sqrt(3)*0.6/2-0.2)--(2+0.3*sqrt(3)+0.2*sqrt(3),-1.3-0.2)); //super complex Asymptote code gg
draw((2+0.3*sqrt(3)+0.3/2,-1.3+sqrt(3)*0.3/2)--(2.35,-0.6677));
draw(anglemark((2,-1),(2+sqrt(3),-2),(2.5,-2)));
label("$30^\circ$",(3.5,-2),NW);
[/asy]
$ \textbf{(A)}\ \left(\frac{3mg}{2k}\right)^{1/2} $
$ \textbf{(B)}\ \left(\frac{mg}{k}\right)^{1/2} $
$ \textbf{(C)}\ \left(\frac{2mg}{k}\right)^{1/2} $
$ \textbf{(D)}\ \left(\frac{\sqrt{3}mg}{k}\right)^{1/3} $
$ \textbf{(E)}\ \left(\frac{3\sqrt{3}mg}{2k}\right)^{1/3} $
Let $ABC$ be an acute triangle. The interior angle bisectors of $\angle ABC$ and $\angle ACB$ meet the opposite sides in $L$ and $M$ respectively. Prove that there is a point $K$ in the interior of the side $BC$ such that the triangle $KLM$ is equilateral if and only if $\angle BAC = 60^\circ$.
Let $ABC$ be an acute triangle such that $\overline{AB}=\overline{AC}$. Let $M, L, N$ be the midpoints of segment $BC, AM, AC$, respectively. The circumcircle of triangle $AMC$, denoted by $\Omega$, meets segment $AB$ at $P(\neq A)$, and the segment $BL$ at $Q$. Let $O$ be the circumcenter of triangle $BQC$. Suppose that the lines $AC$ and $PQ$ meet at $X$, $OB$ and $LN$ meet at $Y$, and $BQ$ and $CO$ meets at $Z$. Prove that the points $X, Y, Z$ are collinear.
Find the least positive integer with the property that if its digits are reversed and then $450$ is added to this reversal, the sum is the original number. For example, $621$ is not the answer because it is not true that $621 = 126 + 450$.
Is it true that in any $2$-connected graph with a countably infinite number of vertices it's always possible to find a trail that is infinite in one direction?
[i]Submitted by Balázs Bursics and Anett Kocsis, Budapest[/i]
In the following $6\times 6$ matrix, one can choose any $k\times k$ submatrix, with $1<k\leq6 $ and add $1$ to all its entries. Is it possible to perform the operation a finite number of times so that all the entries in the $6\times 6$ matrix are multiples of $3$?
$ \begin{pmatrix}
2 & 0 & 1 & 0 & 2 & 0 \\
0 & 2 & 0 & 1 & 2 & 0 \\
1 & 0 & 2 & 0 & 2 & 0 \\
0 & 1 & 0 & 2 & 2 & 0 \\
1 & 1 & 1 & 1 & 2 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} $
Note: A $p\times q$ submatrix of a $m\times n$ matrix (with $p\leq m$, $q\leq n$) is a $p\times q$ matrix formed by taking a block of the entries of this size from the original matrix.