Found problems: 85335
Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.
$ABCD$ is a rectangle with $AB = 20$ and $BC = 3$. A circle with radius $5$, centered at the midpoint of $DC$, meets the rectangle at four points: $W, X, Y$ , and $Z$. Find the area of quadrilateral $WXY Z$.
In the Cartesian plane $\mathbb{R}^2,$ each triangle contains a Mediterranean point on its sides or in its interior, even if the triangle is degenerated into a segment or a point. The Mediterranean points have the following properties:
[b](i)[/b] If a triangle is symmetric with respect to a line which passes through the origin $(0,0)$, then the Mediterranean point lies on this line.
[b](ii)[/b] If the triangle $DEF$ contains the triangle $ABC$ and if the triangle $ABC$ contains the Mediterranean points $M$ of $DEF,$ then $M$ is the Mediterranean point of the triangle $ABC.$
Find all possible positions for the Mediterranean point of the triangle with vertices $(-3,5),\ (12,5),\ (3,11).$
A triangle has sides $a, b,c$, radius of the incircle $r$ and radii of the excircles $r_a, r_b, r_c$: Prove that:
a) The triangle is right-angled if and only if: $r + r_a + r_b + r_c = a + b + c$.
b) The triangle is right-angled if and only if: $r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2$.
Is it possible to divide a circle by three chords, different from diameters, into several equal parts?
Q.A bug travels in the co-ordinate plane moving along only the lines that are parallel to the $X$ and $Y$ axes.Let $A=(-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$.How many lattice points lie on at least one of these paths.
My answer ($87$)
In a circular ring with radii $11r$ and $9r$, we put circles of radius $r$ which are tangent to the boundary circles and do not overlap. Determine the maximum number of circles that can be put this way. (You may use that $9.94<\sqrt{99}<9.95$)
Let $n>1$ be an integer. Find all $r\in \mathbb{R}$ so that the system of equations in real variables $x_1, x_2, \dots, x_n$:
\begin{align*}
&(r\cdot x_1-x_2)(r\cdot x_1-x_3)\dots (r\cdot x_1-x_n)=\\
=&(r\cdot x_2-x_1)(r\cdot x_2-x_3)\dots (r\cdot x_2-x_n)=\\
&\qquad \qquad \qquad \qquad \vdots \\
=&(r\cdot x_n-x_1)(r\cdot x_n-x_2)\dots (r\cdot x_n-x_{n-1})
\end{align*}
has a solution where the numbers $x_1, x_2, \dots, x_n$ are pairwise distinct.
Find all positive integers $n$ which satisfy the following tow conditions:
(a) $n$ has at least four different positive divisors;
(b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$.
[i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]
You have a four-liter jug and a six-liter pot (both of cylindrical shape), and a big barrel of water. Can you measure exactly one liter of water?
Let \(a\), \(b\), and \(n\) be positive integers. A lemonade stand owns \(n\) cups, all of which are initially empty. The lemonade stand has a [i]filling machine[/i] and an [i]emptying machine[/i], which operate according to the following rules: [list] [*]If at any moment, \(a\) completely empty cups are available, the filling machine spends the next \(a\) minutes filling those \(a\) cups simultaneously and doing nothing else. [*]If at any moment, \(b\) completely full cups are available, the emptying machine spends the next \(b\) minutes emptying those \(b\) cups simultaneously and doing nothing else. [/list] Suppose that after a sufficiently long time has passed, both the filling machine and emptying machine work without pausing. Find, in terms of \(a\) and \(b\), the least possible value of \(n\).
[i]Proposed by Raymond Feng[/i]
Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$?
$ \textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11 $
At least how many weighings of a balanced scale are needed to order four stones with distinct weights from the lightest to the heaviest?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ 8
$
For which finite group G does there exist natural number s with the following property: for any subgroup H of a finite direct power of G, each subgroup of H is produced as an intersection of subgroups of H with index at most s.
not sure of translation.
Prove that for every $n \in \mathbb{N}$ the following proposition holds: $7|3^n +n^3$ if and only if $7|3^{n} n^3 +1$.
Charlie noticed his golden ticket was golden in two ways! In addition to being gold, it was a rectangle whose side lengths had ratio the golden ratio $\varphi = \tfrac{1+\sqrt{5}}{2}$. He then folds the ticket so that two opposite corners (vertices connected by a diagonal) coincide and makes a sharp crease (the ticket folds just as any regular piece of paper would). The area of the resulting shape can be expressed as $a + b \varphi$. What is $\tfrac{b}{a}$?
Fix positive integers $m$ and $n$. Suppose that $a_1, a_2, \dots, a_m$ are reals, and that pairwise distinct vectors $v_1, \dots, v_m\in \mathbb{R}^n$ satisfy $$\sum_{j\neq i} a_j \frac{v_j-v_i}{||v_j-v_i||^3}=0$$ for $i=1,2,\dots,m$.
Prove that $$\sum_{1\le i<j\le m} \frac{a_ia_j}{||v_j-v_i||}=0.$$
Let $A$ and $B$ be two cubes. Numbers $1,2,...,14$, are assigned in any order, to the faces and vertices of cube $A$. Then each edge of cube $A$ is assigned the average of the numbers assigned to the two faces that contain it. Finally assigned to each face of the cube $B$ the sum of the numbers associated with the vertices, the face and the edges on the corresponding face of cube $A$. If $S$ is the sum of the numbers assigned to the faces of $B$, find the largest and smallest value that $S$ can take.
The national debt of the United States is on track to reach $5 \cdot 10^{13}$ dollars by $2033$. How many digits does this number of dollars have when written as a numeral in base $5$? (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem.)
$
\textbf{(A) }18 \qquad
\textbf{(B) }20 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }24 \qquad
\textbf{(E) }26 \qquad
$
A deck consists of 13 types of cards: \( A > K > Q > J > 10 > 9 > \dots > 3 > 2 \), each card appearing 4 times. In total, there are 52 cards.
Marius and Alexandru each receive half of the standard deck of cards, placing them face down. On each turn, the players simultaneously draw the top card from their hands, and the player with the most valuable card takes both cards and places them under all of his other cards, with Marius deciding the order in which the two cards are placed. In case of a tie, each player places their own card under the rest of their cards. The game ends when one of the players runs out of cards.
Is it possible that, although Alexandru initially has all four \( A \) cards, the game lasts forever?
Let $ABC$ be points such that $AB=7, BC=5, AC=10$, and $M$ be the midpoint of $AC$. Let $\omega$, $\omega_1$ be the circumcircles of $ABC$ and $BMC$. $\Omega$, $\Omega_1$ are circles through $A$ and $M$ such that $\Omega$ is tangent to $\omega_1$ and $\Omega_1$ is tangent to the line through the centers of $\omega_1$ and $\Omega$. $D, E$ be the intersection of $\Omega$ with $\omega$ and $\Omega_1$ with $\omega_1$. If $F$ is the intersection of the circumcircle of $DME$ with $BM$, find $FB$.
[i]Proposed by David Tang[/i]
The unit of a screw is listed as $0.2$ cents. When a group of screws is sold to a customer, the total cost of the screws is computed with the listed price and then rounded to the nearest cent. If Al has $50$ cents and wishes to only make one purchase, what is the maximum possible number of screws he can buy?
The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?
What is the smallest number of nodes that can be marked in a rectangular $n \times k$ grid so that each cell contains at least two marked nodes?
Three diagonals of a regular $n$-gon prism intersect at an interior point $O$. Show that $O$ is the center of the prism.
(The diagonal of the prism is a segment joining two vertices not lying on the same face of the prism.)