Found problems: 85335
Points $X$ and $Y$ on the unit circle centered at $O = (0,0)$ are at $(-1,0)$ and $(0,-1)$ respectively. Points $P$ and $Q$ are on the unit circle such that $\angle P XO = \angle QY O = 30^o$. Let $Z$ be the intersection of line $X P$ and line $Y Q$. The area bounded by segment $Z P$, segment $ZQ$, and arc $PQ$ can be expressed as $a\pi -b$ where $a$ and $b$ are rational numbers. Find $\frac{1}{ab}$ .
Does there exist a cube in space such that the perpendiculars dropped from its eight vertices to a given plane are of length $0, 1, 2, 3, 4, 5, 6$ and $7$?
(V Proizvolov)
Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation
$$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$
for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$.
[i]Proposed by Maximiliano Sánchez[/i]
Find the sum of all distinct four-digit numbers that contain only the digits $1, 2, 3, 4,5$, each at most once.
If $125 + n + 135 + 2n + 145 = 900,$ find $n.$
Let $ABC$ be a triangle in which $\measuredangle{A}=135^{\circ}$. The perpendicular to the line $AB$ erected at $A$ intersects the side $BC$ at $D$, and the angle bisector of $\angle B$ intersects the side $AC$ at $E$.
Find the measure of $\measuredangle{BED}$.
In a group of $k$ people, some are acquainted with each other and some are not. Every evening, one person invites all his acquaintances to a party and introduces them to each other(if they have not already acquainted). Suppose that after each person has arranged at least one party, some two people do not know each other. Prove that they do not meet each other in the next party.
Given parallelogram $OABC$ in the coodinate with $O$ the origin and $A,B,C$ be lattice points. Prove that for all lattice point $P$ in the internal or boundary of $\triangle ABC$, there exists lattice points $Q,R$(can be the same) in the internal or boundary of $\triangle OAC$ with $\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{OR}$.
Only $A$ and $B$ have $n$ friends in a village of $2010$ people. The other $2008$ people have all different numbers of friends. How many possible values of $n$ are there?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \text{None}
$
Let $S$ denote a square of the side-length $7$, and let eight squares of the side-length $3$ be given. Show that $S$ can be covered by those eight small squares.
Seven chips are numbered $1, 2, 3, 4, 5, 6, 7$. Prove that none of the seven-digit numbers formed by these chips is divisible by any other of these seven-digit numbers.
The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine $r$ if $B,M$ and $N$ are collinear.
An integer $n$ is chosen uniformly at random from the set $\{1, 2, \ldots, 2023!\}.$ Compute the probability that $$\gcd(n^n+50, n+1)=1.$$
We say that a pile is a set of four or more nuts. Two persons play the following game. They start with one pile of $n \geq 4$ nuts. During a move a player takes one of the piles that they have and split it into two nonempty sets (these sets are not necessarily piles, they can contain arbitrary number of nuts). If the player cannot move, he loses. For which values of $n$ does the first player have a winning strategy?
Let $S$ be the set of integers that represent the number of intersections of some four distinct lines in the plane. List the elements of $S$ in ascending order.
Let $AD$ be the common chord of two circles $\Gamma_1$ and $\Gamma_2$. A line through $D$ intersects $\Gamma_1$ at $B$ and $\Gamma_2$ at $C$. Let $E$ be a point on the segment $AD$, different from $A$ and $D$. The line $CE$ intersect $\Gamma_1$ at $P$ and $Q$. The line $BE$ intersects $\Gamma_2$ at $M$ and $N$.
(i) Prove that $P,Q,M,N$ lie on the circumference of a circle $\Gamma_3$.
(ii) If the centre of $\Gamma_3$ is $O$, prove that $OD$ is perpendicular to $BC$.
Find the last five digits of
$1^{100}+2^{100}+3^{100}+...+999999^{100}$
Rectangle $EFGH$ with side lengths $8$, $9$ lies inside rectangle $ABCD$ with side lengths $13$, $14$, with their corresponding sides parallel. Let $\ell_A, \ell_B, \ell_C, \ell_D$ be the lines connecting $A,B,C,D$, respectively, with the vertex of $EFGH$ closest to them. Let $P = \ell_A \cap \ell_B$, $Q = \ell_B \cap \ell_C$, $R = \ell_C \cap \ell_D$, and $S = \ell_D \cap \ell_A$. Suppose that the greatest possible area of quadrilateral $PQRS$ is $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$.
[i]Proposed by Yannick Yao[/i]
How many triangles are there for which the perimeters are equal to $30$ cm and the lengths of sides are integers in centimeters?
A. $16$ B. $17$ C. $18$ D. $19$ E. $20$
The angle bisector of $A$ in triangle $ABC$ intersects $BC$ in the point $D$. The point $E$ lies on the side $AC$, and the lines $AD$ and $BE$ intersect in the point $F$. Furthermore, $\frac{|AF|}{|F D|}= 3$ and $\frac{|BF|}{|F E|}=\frac{5}{3}$. Prove that $|AB| = |AC|$.
[img]https://1.bp.blogspot.com/-evofDCeJWPY/XzT9dmxXzVI/AAAAAAAAMVY/ZN87X3Cg8iMiULwvMhgFrXbdd_f1f-JWwCLcBGAsYHQ/s0/2013%2BMohr%2Bp5.png[/img]
Let $a, b,c$ be positive real numbers.
Prove that $ \sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)$
(a) Prove that the area of a given convex quadrilateral is at least twice the area of an arbitrary convex quadrilateral inscribed in it whose sides are parallel to the diagonals of the original one.
(b) A tetrahedron with surface area $S$ is intersected by a plane perpendicular to two opposite edges. If the area of the cross-section is $Q$, prove that $S>4Q$.
Write $0.2010\overline{228}$ as a fraction.
In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$.
[b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$.
[b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.
Let $H $ be the orthocenter of $ABC $ ,$AB\neq AC $ ,and let $F $ be a point on circumcircle of $ABC $ such that $\angle AFH=90^{\circ} $.$K $ is the symmetric point of $H $ wrt $B $.Let $P $ be a point such that $\angle PHB=\angle PBC=90^{\circ} $,and $Q $ is the foot of $B $ to $CP $.Prove that $HQ $ is tangent to tge circumcircle of $FHK $.