Found problems: 85335
Starting at $12:00:00$ AM on January $1,$ $2022,$ after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365.$ Find $y + d.$
Let $n$ be a positive integer. Find the maximal real number $k$, such that the following holds:
For any $n$ real numbers $x_1,x_2,...,x_n$, we have $\sqrt{x_1^2+x_2^2+\dots+x_n^2}\geq k\cdot\min(|x_1-x_2|,|x_2-x_3|,...,|x_{n-1}-x_n|,|x_n-x_1|)$
Let $S$ be the set $\{1, 2, ..., 10\}$. Let $A$ be a subset of $S$.
We arrange the elements of $A$ in increasing order, that is, $A = \{a_1, a_2, ...., a_k\}$ with $a_1 < a_2 < ... < a_k$.
Define [i]WSUM [/i] for this subset as $3(a_1 + a_3 +..) + 2(a_2 + a_4 +...)$ where the first term contains the odd numbered terms and the second the even numbered terms.
(For example, if $A = \{2, 5, 7, 8\}$, [i]WSUM [/i] is $3(2 + 7) + 2(5 + 8)$.)
Find the sum of [i]WSUMs[/i] over all the subsets of S.
(Assume that WSUM for the null set is $0$.)
Segments connecting an inner point of a convex non-equilateral n-gon to its vertices divide the n-gon into n equal triangles. What is the least possible n?
Sofia summed all the page numbers from a book starting at $1$ and getting $2013$. Pablo saw how she did this and realized Sofia skipped a page. How many pages does the book have, and what page did Sofia skip?
For a given positive integer $n$, there are a total of $5n$ balls labeled with numbers $1$, $2$, $3$, $\cdots$, $n$, with 5 balls for each number. The balls are put into $n$ boxes, with $5$ balls in each box. Show that you can color two balls red and one ball blue in each box so that the sum of the numbers on the red balls is twice the sum of the numbers on the blue balls.
Let $N$ be a positive integer. The sequence $x_1, x_2, \ldots$ of non-negative reals is defined by $$x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}$$ for all positive integers $n>N$. Show that there exists a constant $c>0$, such that $x_n \leq \frac{n} {2}+c$ for all positive integers $n$.
Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?
Let $k$ be a natural number. For each natural number $n$ we define $f_k (n)$ to be the least number, greater than $kn$, for which $nf_k (n)$ is a perfect square. Prove that $f_k (n)$ is injective.
Compute
$$\int_0^1\frac{2xe^x-1}{2x^2e^x+2}dx.$$
Prove that $ \left( \left.\left\{\begin{pmatrix} a & b & c \\ 3c & a & b \\ 3b & 3c & a\end{pmatrix} \right| a,b,c\in\mathbb{Q}\right\} ,+,\cdot\right) $ is a field.
Let $AB$ be a diameter of semicircle $h$. On this semicircle there is point $C$, distinct from points $A$ and $B$. Foot of perpendicular from point $C$ to side $AB$ is point $D$. Circle $k$ is outside the triangle $ADC$ and at the same time touches semicircle $h$ and sides $AB$ and $CD$. Touching point of $k$ with side $AB$ is point $E$, with semicircle $h$ is point $T$ and with side $CD$ is point $S$
$a)$ Prove that points $A$, $S$ and $T$ are collinear
$b)$ Prove that $AC=AE$
Let be a finite set $ S. $ Determine the number of functions $ f:S\rightarrow S $ that satisfy $ f\circ f=f. $
Given $30$ equal cups with milk. An elf tries to make the amount of milk equal in all the cups. He takes a pair of cups and aligns the milk level in two cups. Can there be such an initial distribution of milk in the cups, that the elf will not be able to achieve his goal in a finite number of operations?
Two circles with radii 1 and 2 touch each other and a line as in the figure. In the region between the circles and the line, there is a circle with radius $ r$ which touches the two circles and the line. What is $ r$?
[img]http://i250.photobucket.com/albums/gg265/geometry101/GeometryImage2.jpg[/img]
A. 1/3
B. $ \frac {1}{\sqrt {5}}$
C. $ \sqrt {3} \minus{} \sqrt {2}$
D. $ 6 \minus{} 4 \sqrt {2}$
E. None of these
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:
\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]
[i]Proposed by Juhan Aru, Estonia[/i]
Three circles through the point $O$ and of radius $r$ intersect pairwise in the additional points $A$,$B$,$C$. Prove that the circle through the points $A$, $B$, and $C$ also has radius $r$.
We say that $A$$=${$a_1,a_2,a_3\cdots a_n$} consisting $n>2$ distinct positive integers is $good$ if for every $i=1,2,3\cdots n$ the number ${a_i}^{2015}$ is divisible by the product of all numbers in $A$ except $a_i$. Find all integers $n>2$ such that exists a $good$ set consisting of $n$ positive integers.
Find the number of real solutions of the equation $a e^x = x^3$, where $a$ is a real parameter.
$\frac{n(n + 1)}{2}$ distinct numbers are arranged at random into $n$ rows. The first row has $1$ number, the second has $2$ numbers, the third has $3$ numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers.
Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.
Diagonals $ AC $ and $ BD $ of a convex quadrilateral $ ABCD $ intersect a point $ O. $ Prove that if triangles $ OAB,OBC,OCD $ and $ ODA $ have the same perimeter, then $ ABCD $ is a rhombus. What happens if $ O $ is some other point inside the quadrilateral?
In $\triangle ABC$, $\angle ABC=90^{\circ}$. Points $D,G$ lie on side $AC$. Points $E, F$ lie on segment $BD$, such that $AE \perp BD $ and $GF \perp BD$. Show that if $BE=EF$, then $\angle ABG=\angle DFC$.
Let $AA_1$ be an altitude in $\Delta ABC$. Let $H_a$ be the orthocenter of the triangle with vertices the tangential points of the excircle to $\Delta ABC$, opposite to $A$. The points $B_1$, $C_1$, $H_b$, and $H_c$ are defined analogously. Prove that $A_1 H_a$, $B_1 H_b$, and $C_1 H_c$ are concurrent.
Positive rational numbers $x<y<z$ sum to $1$ and satisfy the equation
$$(x^2+y^2+z^2-1)^3+8xyz=0.$$
Given that $\sqrt{z}$ is also rational, it can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. If $m+n < 1000$, find the maximum value of $m+n$.
[I]Proposed by [b] Th3Numb3rThr33 [/b][/I]