Found problems: 85335
Transform by inversion two concentric and coplanar circles into two equal.
Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.
In a given disc, construct a subset such that its area equals the half of the disc area and its intersection with its reflection over an arbitrary diameter has the area equal to the quarter of the disc area.
The diagram below shows square $ABCD$ which has side length $12$ and has the same center as square $EFGH$ which has side length $6$. Find the area of quadrilateral $ABFE$.
[img]https://cdn.artofproblemsolving.com/attachments/a/a/8cfedf396cd2e86092c03fa0dcb1fb3c978965.png[/img]
A container is shaped like a right circular cone open at the top surmounted by a frustum which is open at
the top and bottom as shown below. The lower cone has a base with radius 2 centimeters and height 6
centimeters while the frustum has bases with radii 2 and 8 centimeters and height 6 centimeters. If there is
a rainfall measuring 2 centimeter of rain, the rain falling into the container will fill the container to a
height of $m + 3\sqrt{n}$ cm, where m and n are positive integers. Find m + n.
Let $a_1,a_2,\cdots$ be an infinity sequence of positive integers such that $a_1=2021$ and
$$a_{n+1}=(a_1+a_2+\cdots+a_n)^2-1$$
for all positive integers $n$. Prove that for any integer $n\ge 2$, $a_n$ is the product of at least $2n$ (not necessarily distinct) primes.
Let $n\ge 2$ be a positive integer. Fill up a $n\times n$ table with the numbers $1,2,...,n^2$ exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most $n$. Show that there exist a $2\times 2$ square of adjacent cells such that the diagonally opposite pairs sum to the same number.
For each integer $n > 3$ find all quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $n_1 +n_2 +n_3 +n_4 = n$ which maximize the expression $$\frac{n!}{n_1!n_2!n_3!n_4!}2^{ {n_1 \choose 2}+{n_2 \choose 2}+{n_3 \choose 2}+{n_4 \choose 2}+n_1n_2+n_2n_3+n_3n_4}$$
Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$.
($d(n)$ is the number of divisors of the number n including $1$ and $n$ ).
How many perfect squares are divisors of the product $ 1!\cdot 2!\cdot 3!\cdots 9!$?
$ \textbf{(A)}\ 504 \qquad \textbf{(B)}\ 672 \qquad \textbf{(C)}\ 864 \qquad \textbf{(D)}\ 936 \qquad \textbf{(E)}\ 1008$
Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown.
How large a fraction does the area of the small circle make up of that of the big one?
[img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]
On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$, it is possible to state that there is at least one rook in each $k\times k$ square ?
The equation $P(x) = 0$, where $P(x) = x^2+bx+c$, has a single root, and the equation $P(P(P(x))) = 0$ has exactly three different roots. Solve the equation $P(P(P(x))) = 0.$
Let $a_1,a_2,\ldots,a_{2n}$ be positive real numbers such that $a_ja_{n+j}=1$ for the values $j=1,2,\ldots,n$.
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a. Prove that either the average of the numbers $a_1,a_2,\ldots,a_n$ is at least 1 or the average of
the numbers $a_{n+1},a_{n+2},\ldots,a_{2n}$ is at least 1.
b. Assuming that $n\ge2$, prove that there exist two distinct numbers $j,k$ in the set $\{1,2,\ldots,2n\}$ such that
\[|a_j-a_k|<\frac{1}{n-1}.\]
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A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle?
$ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 36$
Call a positive integer number $n$ [i]poor[/i] if equation \[x_1x_2 \dots x_{101}=(n-x_1)(n-x_2)\dots (n-x_{101}) \] has no solutions in positive integers $1<x_i<n$. Does there exist poor number, which has more than $100 \ 000$ distinct prime divisors?
Prove that the interior of a convex pentagon whose sides are all equal, is not covered by the open disks having the sides of the pentagon as diameter.
Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.
The teacher drew a coordinate plane on the board and marked some points on this plane. Unfortunately, Vasya's second-grader, who was on duty, erased almost the entire drawing, except for two points $A (1, 2)$ and $B (3,1)$. Will the excellent Andriyko be able to follow these two points to construct the beginning of the coordinate system point $O (0, 0)$? Point A on the board located above and to the left of point $B$.
In a triangle $ ABC $ with $ AB\neq AC, $ let $ D $ be the midpoint of the side $ BC $ and denote with $ E $ the feet of the bisector of $ \angle BAC. $ Also, let $ M,N $ be two points situated in the exterior of $ ABC $ such that $ AMB\sim ANC. $ Prove the following propositions:
$ \text{(a)} MN\perp AD\iff MA\perp AB $
$ \text{(b)} MN\perp AE \iff\angle MAN=180^{\circ } $
Let $c$ be the probability that the cards are neither from the same suit or the same rank. Compute $\lfloor 1000c\rfloor$.
Let us define a $\textit{triangle equivalence}$ a group of numbers that can be arranged as shown
$a+b=c$
$d+e+f=g+h$
$i+j+k+l=m+n+o$
and so on...
Where at the $j$-th row, the left hand side has $j+1$ terms and the right hand side has $j$ terms.
Now, we are given the first $N^2$ positive integers, where $N$ is a positive integer. Suppose we eliminate any one number that has the same parity with $N$.
Prove that the remaining $N^2-1$ numbers can be formed into a $\textit{triangle equivalence}$.
For example, if $10$ is eliminated from the first $16$ numbers, the remaining numbers can be arranged into a $\textit{triangle equivalence}$ as shown.
$1+3=4$
$2+5+8=6+9$
$7+11+12+14=13+15+16$
An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is $21$. What is the value of the cube?
$\textbf{(A)}~42\qquad\textbf{(B)}~63\qquad\textbf{(C)}~84\qquad\textbf{(D)}~126\qquad\textbf{(E)}~252$
Three natural numbers are such that the product of any two of them is divisible by the sum of those two numbers. Prove that these numbers have a common divisor greater than $1$.
[i]Proposed by Evan Chang (squareman), USA[/i]
A baker with access to a number of different spices bakes ten cakes. He uses more than half of the different kinds of spices in each cake, but no two of the combinations of spices are exactly the same. Show that there exist three spices $a,b,c$ such that every cake contains at least one of these.