Found problems: 85335
How many of the following capital English letters look the same when rotated $180^\circ$ about their center?
[center]A B C D E F G H I J K L M N O P Q R S T U V W X Y Z[/center]
[i]Proposed by William Yue[/i]
Let $X = 123456789$. Find the sum of the tens digits of all integral multiples of $11$ that can be obtained by interchanging two digits of $X$.
There are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a straight line to each one of the other points, visiting each point exactly once and stopping at the final point. This creates a trail of 7 line segments. What is the maximum number of times the ant can cross its own path as it walks?
[i]Proposed by Gabriel Wu[/i]
Jon and Steve ride their bicycles on a path that parallels two side-by-side train tracks running in the east/west direction. Jon rides east at 20 miles per hour, and Steve rides west at 20 miles per hour. Two trains of equal length traveling in opposite directions at constant but different speeds, each pass the two riders. Each train takes exactly 1 minute to go past Jon. The westbound train takes 10 times as long as the eastbound train to go past Steve. The length of each train is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Let $DE$ the diameter of a circunference $\Gamma$. Let $B, C$ on $\Gamma$ such that $BC$ is perpendicular to $DE$, and let $Q$ the intersection of $BC$ with $DE$. Let $P$ a point on segment $BC$ such that $BP=4PQ$. Let $A$ the second intersection of $PE$ with $\Gamma$. If $DE=2$ and $EQ=\frac{1}{2}$, find all possible values of the sides of triangle $ABC$.
Let be a tetahedron $ OABC $ with $ OA\perp OB\perp OC\perp OA. $ Show that
$$ OH\le r\left( 1+\sqrt 3 \right) , $$
where $ H $ is the orthocenter of $ ABC $ and $ r $ is radius of the inscribed spere of $ OABC. $
[i]Valentin Vornicu[/i]
A nonempty set $S$ is called [i]Bally[/i] if for every $m\in S$, there are fewer than $\frac{1}{2}m$ elements of $S$ which are less than $m$. Determine the number of Bally subsets of $\{1, 2, . . . , 2020\}$.
Let a point $M$ not lying on coordinates axes be given. Points $Q$ and $P$ move along $Y$ - and $X$-axis respectively so that angle $P M Q$ is always right. Find the locus of points symmetric to $M$ wrt $P Q$.
A square is divided into congruent rectangles with sides of integer lengths. A rectangle is important if it has at least one point in common with a given diagonal of the square. Prove that this diagonal bisects the total area of the important rectangles
Find all real pairs $(x,y)$ that solve the system of equations \begin{align} x^5 &= 21x^3+y^3
\\ y^5 &= x^3+21y^3. \end{align}
Let $S$ denote the set of rational numbers different from $ \{ -1, 0, 1 \} $. Define $f: S \rightarrow S $ by $f(x)=x-1/x$. Prove or disprove that \[ \cap_{n=1}^{\infty} f^{(n)} (S) = \emptyset \] where $f^{(n)}$ denotes $f$ composed with itself $n$ times.
Let $ O$ be the circumcenter of acute triangle $ ABC$. Point $ P$ is in the interior of triangle $ AOB$. Let $ D,E,F$ be the projections of $ P$ on the sides $ BC,CA,AB$, respectively. Prove that the parallelogram consisting of $ FE$ and $ FD$ as its adjacent sides lies inside triangle $ ABC$.
Given three circles $c_1,c_2,c_3$ with $r_1,r_2,r_3$ radiuses, $r_1 > r_2, r_1 > r_3$. Each lies outside of two others. The A point -- an intersection of the outer common tangents to $c_1$ and $c_2$ -- is outside $c_3$. The $B$ point -- an intersection of the outer common tangents to $c_1$ and $c_3$ -- is outside $c_2$. Two pairs of tangents -- from $A$ to $c_3$ and from $B$ to $c_2$ -- are drawn. Prove that the quadrangle, they make, is circumscribed around some circle and find its radius.
Given the tetrahedron $ABCD$, whose opposite edges are equal, that is, $AB=CD, AC=BD$ and $BC=AD$. Prove that exist exactly $6$ planes intersecting the triangular angles of the tetrahedron and dividing the total surface and volume of this tetrahedron in half.
Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$.
Points $D,E$ lie on segments $AB,AC$ of $\triangle ABC$ such that $DE\parallel BC$. Let $O_1,O_2$ be the circumcenters of $\triangle ABE, \triangle ACD$ respectively. Line $O_1O _2$ meets $AC$ at $P$, and $AB$ at $Q$. Let $O$ be the circumcenter of $\triangle APQ$, and $M$ be the intersection of $AO$ extended and $BC$. Prove that $M$ is the midpoint of $BC$.
On February 13 [i]The Oshkosh Northwester[/i] listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57 \textsc{am}$, and the sunset as $8:15 \textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?
$\textbf{(A)}\hspace{.05in}5:10 \textsc{pm} \quad \textbf{(B)}\hspace{.05in}5:21 \textsc{pm} \quad \textbf{(C)}\hspace{.05in}5:41\textsc{pm} \quad \textbf{(D)}\hspace{.05in}5:57 \textsc{pm} \quad \textbf{(E)}\hspace{.05in}6:03 \textsc{pm} $
There are $30$ children, $a_1,a_2,...,a_{30}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.
Let $\Gamma_1$ and $\Gamma_2$ be two circles externally tangent to each other at $N$ that are both internally tangent to $\Gamma$ at points $U$ and $V$ , respectively. A common external tangent of $\Gamma_1$ and $\Gamma_2$ is tangent to $\Gamma_1$ and $\Gamma_2$ at $P$ and $Q$, respectively, and intersects $\Gamma$ at points $X$ and $Y$ . Let $M$ be the midpoint of the arc $XY$ that does not contain $U$ and $V$ . Let $Z$ be on $\Gamma$ such $MZ \perp NZ$, and suppose the circumcircles of $QVZ$ and $PUZ$ intersect at $T\ne Z$. Find, with proof, the value of $T U + T V$ , in terms of $R$, $r_1$, and $r_2$, the radii of $\Gamma$, $\Gamma_1$, and $\Gamma_2$, respectively.
Given $f_1(x)=2x-2$ and, for $k\ge2$, defined $f_k(x)=f(f_{k-1}(x))$ to be a real-valued function of $x$. Find the remainder when $f_{2013}(2012)$ is divided by the prime $2011$.
Let $E$ denote the set of $25$ points $(m,n)$ in the $\text{xy}$-plane, where $m,n$ are natural numbers, $1\leq m\leq5,1\leq n\leq5$. Suppose the points of $E$ are arbitrarily coloured using two colours, red and blue. SHow that there always exist four points in the set $E$ of the form $(a,b),(a+k,b),(a+k,b+k),(a,b+k)$ for some positive integer $k$ such that at least three of these four points have the same colour. (That is, there always exist four points in the set $E$ which form the vertices of a square with sides parallel to the axes and having at least three points of the same colour.)
It is known that for any point $P$ on the circumcircle of a triangle $ABC$, the orthogonal projections of $P$ onto $AB,BC,CA$ lie on a line, called a [i]Simson line[/i] of $P$. Show that the Simson lines of two diametrically opposite points $P_1$ and $P_2$ are perpendicular.
Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the from $\pi(a-b\sqrt{c})$, where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.
[asy]
real r=2-sqrt(3);
draw(Circle(origin, 1));
int i;
for(i=0; i<12; i=i+1) {
draw(Circle(dir(30*i), r));
dot(dir(30*i));
}
draw(origin--(1,0)--dir(30)--cycle);
label("1", (0.5,0), S);[/asy]
A square was split into several rectangles so that the centers of rectangles form a convex polygon.
a) Is it true for sure that each rectangle adjoins to a side of the square?
b) Can the number of rectangles equal 23 ?
Alexandr Shapovalov
$n\ge2$ is a given positive integer. $i\leq a_i \leq n$ satisfies for all $1\leq i\leq n$, and $S_i$ is defined as $a_1+a_2+...+a_i(S_0=0)$. Show that there exists such $1\leq k\leq n$ that satisfies $a_k^2+S_{n-k}<2S_n-\frac{n(n+1)}{2}$.