Found problems: 85335
Circles $\Omega$ and $\omega$ are internally tangent at point $C$. Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$, where $E$ is the midpoint of $AB$. Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$. If $M$ is the midpoint of major arc $AB$, show that $\tan \angle ZEP = \tfrac{PE}{CM}$.
[i]Ray Li.[/i]
A card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared.
Determine all possible first moves of the first player after which he has a winning strategy.
[i]Proposed by Ilya Bogdanov & Vladimir Bragin, Russia[/i]
For $n \in N^*$ we will say that the non-negative integers $x_1, x_2, ... , x_n$ have property $(P)$ if
$$x_1x_2 ...x_n = x_1 + 2x_2 + 3x_3 + ...+ nx_n.$$
a) Show that for every $n \in N^*$ there exists $n$ positive integers with property $(P)$.
b) Find all integers $n \ge 2$ so that there exists $n$ positive integers $x_1, x_2, ... , x_n$ with $x_1< x_2<x_3< ... <x_n$, having property $(P)$.
If $a$ and $b$ are positive integers such that $\frac{1}{a}+\frac{1}{b}=\frac{1}{9}$, what is the greatest possible value of $a+b$?
Let $ABCD$ be a rectangle of center $O$, such that $\angle DAC = 60^o$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$ and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.
let $a_1,a_2,...,a_n$,$b_1,b_2,...,b_n$,$c_1,c_2,...,c_n$ be real numbers. prove that
$$ \sum_{cyc}{ \sqrt{\sum_{i \in \{1,...,n\} }{ (3a_i-b_i-c_i)^2}}} \ge \sum_{cyc}{\sqrt{\sum_{i \in \{1,2,...,n\}}{a_i^2}}}$$
When a block of wood with a weight of $\text{30 N}$ is completely submerged under water the buoyant force on the block of wood from the water is $\text{50 N}$. When the block is released it floats at the surface. What fraction of the block will then be visible above the surface of the water when the block is floating?
(A) $1/15$
(B) $1/5$
(C) $1/3$
(D) $2/5$
(E) $3/5$
Let $n$ be a positive integer. For a permutation $a_1, a_2, \dots, a_n$ of the numbers $1, 2, \dots, n$ we define
$$b_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j$$
We say that the permutation $a_1, a_2, \dots, a_n$ is [i]guadiana[/i] if the sequence $b_1, b_2, \dots, b_n$ does not contain two consecutive equal terms. How many guadiana permutations exist?
Prove that $x+\frac{1}{x^x}<2$ for $0<x<1$.
The board used for playing a game consists of the left and right parts. In each part there are several fields and there’re several segments connecting two fields from different parts (all the fields are connected.) Initially, there is a violet counter on a field in the left part, and a purple counter on a field in the right part. Lyosha and Pasha alternatively play their turn, starting from Pasha, by moving their chip (Lyosha-violet, and Pasha-purple) over a segment to other field that has no chip. It’s prohibited to repeat a position twice, i.e. can’t move to position that already been occupied by some earlier turns in the game. A player losses if he can’t make a move. Is there a board and an initial positions of counters that Pasha has a winning strategy?
The real numbers $a$ and $b$ satisfy the relation $a + b \ge 1$. Show that $8 (a^4 + b^4) \ge 1$.
On the side $AC$ of the triangle $ABC$ in the external side is constructed the parallelogram $ACDE$ . Let $O$ be the intersection point of its diagonals, $N$ and $K$ be midpoints of BC and BA respectively. Prove that lines $DK, EN$ and $BO$ intersect at one point.
What is the maximum possible volume of a cylinder inscribed in a cone and having the same axis of symmetry as the cone? What is the maximum possible volume of a ball inscribed in the cone with center on the axis of symmetry of the cone? Compare these three volumes.
Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win?
[i](Anant Mudgal)[/i]
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])
All sides and diagonals of a $25$-gon are drawn either red or white. Show that at least $500$ triangles, having all three sides are in same color and having all three vertices from the vertices of the $25$-gon, can be found.
Isabella has $6$ coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every 10 days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the 6 dates on her calender, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?
$\textbf{(A) }\text{Monday}\qquad\textbf{(B) }\text{Tuesday}\qquad\textbf{(C) }\text{Wednesday}\qquad\textbf{(D) }\text{Thursday}\qquad\textbf{(E) }\text{Friday}$
In a convex $12$-gon, all angles are equal. It is known that the lengths of some $10$ of its sides are equal to $1$, and the length of one more equals $2$.
What can be the area of this $12$-gon?
Let $\{a,b,c,d,e,f,g,h,i\}$ be a permutation of $\{1,2,3,4,5,6,7,8,9\}$ such that $\gcd(c,d)=\gcd(f,g)=1$ and
\[(10a+b)^{c/d}=e^{f/g}.\]
Given that $h>i$, evaluate $10h+i$.
[i]Proposed by James Lin[/i]
Two circles $C_1$ and $C_2$ intersect at two distinct points $P, Q$ in a plane. Let a line passing through $P$ meet the circles $C_1$ and $C_2$ in $A$ and $B$ respectively. Let $Y$ be the midpoint of $AB$ and let $QY$ meet the cirlces $C_1$ and $C_2$ in $X$ and $Z$ respectively. Show that $Y$ is also the midpoint of $XZ$.
Prove that there exists a uniqe $P(x)$ polynomial with real coefficients such that\\
$xy-x-y|(x+y)^{1000}-P(x)-P(y)$ for all real $x,y$.
In a galaxy far, far away there are $225$ inhabited planets. Between some pairs of inhabited planets there is a bidirectional space connection, and it is possible to reach any planet from any other (possibly with several transfers). The [i]influence[/i] of a planet is the number of other planets with which this planet has a direct connection. It is known that if two planets are not connected by a direct space flight, they have different influences. What is the smallest number of connections possible under these conditions?
[i]Proposed by Arsenii Nikolaev, Bogdan Rublov[/i]
Let $p$ be a fixed prime. Determine all the integers $m$, as function of $p$, such that there exist $a_1, a_2, \ldots, a_p \in \mathbb{Z}$ satisfying
\[m \mid a_1^p + a_2^p + \cdots + a_p^p - (p+1).\]
Circle $C$ (with center $O$) does not have common point with line $l$. Draw $OP$ perpendicular to $l$, $P \in l$. Let $Q$ be a point on $l$ ($Q$ is different from $P$), $QA$ and $QB$ are tangent to circle $C$, and intersect the circle at $A$ and $B$ respectively. $AB$ intersects $OP$ at $K$. $PM$, $PN$ are perpendicular to $QB$, $QA$, respectively, $M \in QB$, $N \in QA$. Prove that segment $KP$ is bisected by line $MN$.
Let $\triangle ABC$ be an acute triangle. Let us denote the foot of the altitudes from the vertices $A, B$ and $C$ to the opposite sides by $D, E$ and $F,$ respectively, and the intersection point of the altitudes of the triangle $ABC$ by $H.$ Let $P$ be the intersection of the line $BE$ and the segment $DF.$ A straight line passing through $P$ and perpendicular to $BC$ intersects $AB$ at $Q.$ Let $N$ be the intersection of the segment $EQ$ with the perpendicular drawn from $A.$
Prove that $N$ is the midpoint of segment $AH.$
In the plane there is a worm of length 1. Prove that it can be always covered by means of half of a circular disk of diameter 1.
[i]Note.[/i] Under a "worm", we understand a continuous curve. The "half of a circular disk" is a semicircle including its boundary.