Found problems: 85335
A closed, non-self-intersecting broken line $L$ is drawn over a $(2n+1) \times (2n+1)$ chessboard in such a way that the set of L's vertices coincides with the set of the vertices of the board’s squares and every edge in $L$ is a side of some board square. All board squares lying in the interior of $L$ are coloured in red. Prove that the number of neighbouring pairs of red squares in every row of the board is even.
$ \log 125$ equals:
$ \textbf{(A)}\ 100 \log 1.25 \qquad \textbf{(B)}\ 5 \log 3 \qquad \textbf{(C)}\ 3 \log 25$
$ \textbf{(D)}\ 3 \minus{} 3\log 2 \qquad \textbf{(E)}\ (\log 25)(\log 5)$
Points $G$ and $N$ are chosen on the interiors of sides $ED$ and $DO$ of unit square $DOME$, so that pentagon $GNOME$ has only two distinct side lengths. The sum of all possible areas of quadrilateral $NOME$ can be expressed as $\frac{a-b\sqrt{c}}{d}$, where $a,b,c,d$ are positive integers such that $\gcd(a,b,d) = 1$ and $c$ is square-free (i.e. no perfect square greater than $1$ divides $c$). Compute $1000a+100b+10c+d$.
In a school with some students, for any three student, there exists at least one student who are friends with all these three students.(Friendships are mutual) For any friends $A$ and $B$, any two of their common friends are also friends with each other. It's not possible to partition these students into two groups, such that every student in each group are friends with all the students in the other gruop. Prove that any two people who aren't friends with each other, has the same number of common friends.(Each person is a friend of him/herself.)
Let $x,y$ be distinct real numbers such that $\frac{x^n-y^n}{x-y}$ is an integer for $4$ consecutive positive integer $n$. Prove that $\frac{x^n-y^n}{x-y}$ is an integer for all positive integers $n$.
Given the positive numbers $a$ and $b$ and the natural number $n$, find the greatest among the $n + 1$ monomials in the binomial expansion of $\left(a+b\right)^n$.
Suppose in triangle $ABC$ incircle touches the side $BC$ at $P$ and $\angle APB=\alpha$. Prove that : \[\frac1{p-b}+\frac1{p-c}=\frac2{rtg\alpha}\]
Acute triangle $ABC$ is inscribed in circle $O$. $P$ is the foot of altitude from $A$ to $BC$, and $D$ is the intersection of $O$ and line $AP$. $M, N$ are midpoint of $AB, AC$ respectively. $MP$ and $CD$ intersects at $Q$, and $NP$ and $BD$ intersects at $R$. Show that $AD, BQ, CR$ meet at one point if and only if $AB=AC$.
Let $a, b_1, b_2, \dots, b_n, c_1, c_2, \dots, c_n$ be real numbers such that \[x^{2n} + ax^{2n - 1} + ax^{2n - 2} + \dots + ax + 1 = \prod_{i = 1}^{n}{(x^2 + b_ix + c_i)}\]
Prove that $c_1 = c_2 = \dots = c_n = 1$.
As a consequence, all complex zeroes of this polynomial must lie on the unit circle.
Let $ABC$ a triangle with circumcircle $\Gamma$ and $R$ a point inside $ABC$ such that $\angle ABR=\angle RBC$. Let $\Gamma_1$ and $\Gamma_2$ the circumcircles of triangles $ARB$ and $CRB$ respectly. The parallel to $AC$ that pass through $R$, intersect $\Gamma$ in $D$ and $E$, with $D$ on the same side of $BR$ that $A$ and $E$ on the same side of $BR$ that $C$. $AD$ intersect $\Gamma_1$ in $P$ and $CE$ intersect $\Gamma_2$ in $Q$. Prove that $APQC$ is cyclic if and only if $AB=BC$
Pixar Prison, for Pixar villains, is shaped like a 600 foot by 1000 foot rectangle with a 300 foot by 500 foot rectangle removed from it, as shown below. The warden separates the prison into three congruent polygonal sections for villains from The Incredibles, Finding Nemo, and Cars. What is the perimeter of each of these sections?
[asy]
draw((0,0)--(0,6)--(10,6)--(10,0)--(8,0)--(8,3)--(3,3)--(3,0)--(0,0));
label("600", (1,3.5));
label("1000", (5.5,6.5));
label("300", (4,1.5));
label("500", (5.5,3.5));
label("300", (1.5,-0.5));
[/asy]
[i]Proposed by Peter Rowley
Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.
$\textit{Proposed by Pablo Serrano, Ecuador}$
A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
[asy]
draw(Arc((0,0), 4, 0, 270));
draw((0,-4)--(0,0)--(4,0));
label("$4$", (2,0), S);
[/asy]
$\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7$
Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "prime" number (with justification!).
Which of the following are true?
$\textbf{(A)}~GL(n,\mathbb R)\text{ is connected}$
$\textbf{(B)}~GL(n,\mathbb C)\text{ is connected}$
$\textbf{(C)}~O(n,\mathbb R)\text{ is connected}$
$\textbf{(D)}~O(n,\mathbb C)\text{ is connected}$
For a positive real number $x$, find the minimum value of $f(x)=\int_x^{2x} (t\ln t-t)dt.$
Jay notices that there are $n$ primes that form an arithmetic sequence with common difference $12$. What is the maximum possible value for $n$?
[i]Proposed by James Lin[/i]
Chris has a bag with 4 black socks and 6 red socks (so there are $10$ socks in total). Timothy reaches into the bag and grabs two socks [i]without replacement[/i]. Find the probability that he will not grab two red socks.
[i]Proposed by Chris Tong[/i]
There are $100$ boxes that were labeled with the numbers $00$, $01$, $02$,$…$, $99$ . The numbers $000$, $001$, $002$, $…$, $999$ were written on a thousand cards, one on each card. Placing a card in a box is permitted if the box number can be obtained by removing one of the digits from the card number. For example, it is allowed to place card $037$ in box $07$, but it is not allowed to place the card $156$ in box $65$.Can it happen that after placing all the cards in the boxes, there will be exactly $50$ empty boxes?
If the answer is yes, indicate how the cards are placed in the boxes; If the answer is no, explain why it is impossible
The $2001 \times 2001$ trees in a park form a square grid. What is the largest number of trees that can be cut so that no tree stump can be seen from any other? (Each tree has zero width.)
Points $A$ and $B$ are given on a circle. With the help of a compass and a ruler, construct on this circle the points $C,$ $D$, $E$ that lie on one side of the straight line $AB$ and for which the pentagon with vertices $A$, $B$, $C$, $D$, $E$ has the largest possible area
Isosceles $\triangle{ABC}$ has interior point $O$ such that $AO = \sqrt{52}$, $BO = 3$, and $CO = 5$. Given that $\angle{ABC}=120^{\circ}$, find the length $AB$.
[i]Proposed by Powell Zhang[/i]
Find all the following functions $f:R\to R$ , which for arbitrary valid $x,y$ holds equality: $$f(xf(x+y))+f((x+y)f(y))=(x+y)^2$$
(Vadym Koval)
Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability it is tossed into bin $i$ is $2^{-i}$ for $i = 1, 2, 3, \ldots$. More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins $3$, $17$, and $10$.) What is $p+q$?
$\textbf{(A)}\ 55 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 57 \qquad\textbf{(D)}\ 58 \qquad\textbf{(E)}\ 59$
Several segments, which we shall call white, are given, and the sum of their lengths is $1$. Several other segments, which we shall call black, are given, and the sum of their lengths is $1$. Prove that every such system of segments can be distributed on the segment that is $1.51$ long in the following way: Segments of the same colour are disjoint, and segments of different colours are either disjoint or one is inside the other. Prove that there exists a system that cannot be distributed in that way on the segment that is $1.49$ long.