Found problems: 85335
Given the positive real numbers $a_{1},a_{2},\dots,a_{n},$ such that $n>2$ and $a_{1}+a_{2}+\dots+a_{n}=1,$ prove that the inequality
\[
\frac{a_{2}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{1}+n-2}+\frac{a_{1}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{2}+n-2}+\dots+\frac{a_{1}\cdot a_{2}\cdot\dots\cdot a_{n-1}}{a_{n}+n-2}\leq\frac{1}{\left(n-1\right)^{2}}\]
does holds.
The right triangle $ABC$ with a right angle at $C$. From all the rectangles $CA_1MB_1$, where $A_1\in BC, M\in AB$ and $B_1\in AC$ which one has the biggest area?
Find all solutions to
\[ (abcde)^2 = a^2+b^2+c^2+d^2+e^2+f^2. \]
in integers.
Proposed by [i]Seongjin Shim[/i]
Let $ X,Y,Z$ be the midpoints of the small arcs $ BC,CA,AB$ respectively (arcs of the circumcircle of $ ABC$). $ M$ is an arbitrary point on $ BC$, and the parallels through $ M$ to the internal bisectors of $ \angle B,\angle C$ cut the external bisectors of $ \angle C,\angle B$ in $ N,P$ respectively. Show that $ XM,YN,ZP$ concur.
Petrică, Vasile and Tudor participated at a math contest. At the contest $ 5$ problems where proposed, each worth distinct integer numbers of points. Petrică solved $4$ problems completely and got $21$ points and Vasile solved $3 $ problems completely and got $22$ points. Tudor solved all problems completely. What are the lowest and highest possible scores of Tudor?
Consider a point $ M$ found in the same plane with the triangle $ ABC$, but not found on any of the lines $ AB,BC$ and $ CA$. Denote by $ S_1,S_2$ and $ S_3$ the areas of the triangles $ AMB,BMC$ and $ CMA$, respectively. Find the locus of $ M$ satisfying the relation:
$ (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)$
Let $g$ be a real-valued function that is continuous on the closed interval $[0,1]$ and twice differentiable on the open interval $(0,1)$. Suppose that for some real number $r>1$,
\[
\lim_{x\to 0^+}\frac{g(x)}{x^r} = 0.
\]
Prove that either
\[
\lim_{x\to 0^+}g'(x) = 0\qquad\text{or}\qquad \limsup_{x\to 0^+}x^r|g''(x)|= \infty.
\]
Victor has a drawer with two red socks,
two green socks,
two blue socks,
two magenta socks,
two lavender socks,
two neon socks,
two mauve socks,
two wisteria socks,
and $2000$ copper socks,
for a total of $2016$ socks.
He repeatedly draws two socks at a time from the drawer at random, and stops if the socks are of the same color. However, Victor is red-green colorblind, so he also stops if he sees a red and green sock.
What is the probability that Victor stops with two socks of the same color? Assume Victor returns both socks to the drawer at each step.
Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
Show that for all positive real numbers $a, b$ and $c$, the inequality $(a+b)(a+c)\ge 2\sqrt{abc(a+b+c)}$ is true.
Let $(G,\cdot)$ be a grup with neutral element $e{}$, and let $H{}$ and $K$ be proper subgroups of $G$, satisfying $H\cap K=\{e\}$. It is known that $(G\setminus(H\cup K))\cup\{e\}$ is closed under the operation of $G$. Prove that $x^2=e$ for all the elements $x{}$ of $G{}$.
Let be a sequence $ \left( u_n \right)_{n\ge 1} $ given by the recurrence relation $ u_{n+1} =u_n+\sqrt{u_n^2-u_1^2} , $ and the constraints $ u_2\ge u_1>0. $
Calculate $ \lim_{n\to\infty }\frac{2^n}{u_n} . $
[i]Dan Negulescu[/i]
Let $ABC$ be a triangle with $AB = AC$ and $\angle BAC = 30^{\circ}$, Let $A'$ be the reflection of $A$ in the line $BC$; $B'$ be the reflection of $B$ in the line $CA$; $C'$ be the reflection of $C$ in line $AB$, Show that $A'B'C'$ is an equilateral triangle.
Find all positive integer triples $(x,y,z)$ such that
$x<y<z$, $\gcd (x,y)=6$, $\gcd (y,z)=10$, $\gcd (x,z)=8$, and lcm$(x,y,z)=2400$.
Note that the problems of the TST are not arranged in difficulty (Problem 1 of day 1 was probably the most difficult!)
A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A=(-3, 2)$ and $B=(3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths?
$ \textbf{(A)}\ 161 \qquad
\textbf{(B)}\ 185 \qquad
\textbf{(C)}\ 195 \qquad
\textbf{(D)}\ 227 \qquad
\textbf{(E)}\ 255 $
Find polynomials $f(x)$, $g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x)=\begin{cases}-1 & \text{if }x<-1\\3x+2 &\text{if }-1\leq x\leq 0\\-2x+2 & \text{if }x>0.\end{cases}\]
Let $m$ and $n$ be positive integers. Determine the number of ways to fill each cell of an $m \times n $ table with a number from $\{-2, -1, 1, 2\}$ so that the product of the numbers written in each row and column is $-2$.
A regular triangle $ABC$ is given. Points $K$ and $N$ lie in the segment $AB$, a point $L$ lies in the segment $AC$, and a point $M$ lies in the segment $BC$ so that $CL=AK$, $CM=BN$, $ML=KN$. Prove that $KL \parallel MN$.
Baron Münchhausen’s watch works properly, but has no markings on its face. The hour, minute and second hands have distinct lengths, and they move uniformly. The Baron claims that since none of the mutual positions of the hands is repeats twice in the period between 8:00 and 19:59, he can use his watch to tell the time during the day. Is his assertion true?
[i](5 points)[/i]
Janson found $2025$ dogs on a circle. Janson wants to select some (possibly none) of the dogs to take home, such that no two selected dogs have exactly two dogs (whether selected or not) in between them. Let $S_{1}$ be the number of ways for him to do so.
Ivan also found $2025$ cats on a circle. Ivan wants to select some (possibly none) of the cats to take home, such that no two selected cats have exactly five cats (whether selected or not) in between them. Let $S_{2}$ be the number of ways for him to do so.
a) Prove that $S_{1}=S_{2}$.
b) Prove that $S_{1}$ and $S_{2}$ are both perfect cubes.
Complex numbers $|z_1|=2,|z_2|=3$, and the intersection angle between the vectors corresponding to $z_1,z_2$ is $60^{\circ}$, then $\frac{|z_1+z_2|}{|z_1-z_2|}=$________.
Let \[T_0=2, T_1=3, T_2=6,\] and for $n\ge 3$, \[T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.\] The first few terms are \[2, 3, 6, 14, 40, 152, 784, 5158, 40576, 363392.\] Find a formula for $T_n$ of the form \[T_n=A_n+B_n,\] where $\{A_n\}$ and $\{B_n\}$ are well known sequences.
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
Let $a,b,c>0$ and $a^2+b^2+c^2=3$ . Prove that $$ \frac{a(a-b^2)}{a+b^2}+\frac{b(b-c^2)}{b+c^2}+\frac{c(c-a^2)}{c+a^2}\ge 0.$$
$ABC$ is a triangle with a circumscribed circle $k$, center $I$ of its inscribed circle $\omega$, and center $I_a$ of its excircle $\omega _a$, opposite to $A$. $\omega$ and $\omega _a$ are tangent to $BC$ in points $P$ and $Q$, respectively, and $S$ is the middle point of the arc $\widehat{BC}$ that doesn’t contain $A$. Consider a circle that is tangent to $BC$ in point $P$ and to $k$ in point $R$. Let $RI$ intersect $k$ for a second time in point $L$. Prove that, $LI_a$ and $SQ$ intersect in a point that lies on $k$.