Found problems: 85335
Let $ABC$ a triangle and let $\omega$ be its circumcircle. Let $E{}$ be the midpoint of the minor arc $BC$ of $\omega$, and $M{}$ the midpoint of $BC$. Let $V$ be the other point of intersection of $AM$ with $\omega$, $F{}$ the point of intersection of $AE$ with $BC$, $X{}$ the other point of intersection of the circumcircle of $FEM$ with $\omega$, $X'$ the reflection of $V{}$ with respect to $M{}$, $A'{}$ the foot of the perpendicular from $A{}$ to $BC$ and $S{}$ the other point of intersection of $XA'$ with $\omega$. If $Z \in \omega$ with $Z\neq X$ is such that $AX = AZ$, then prove that $S, X'$ and $Z{}$ are collinear.
Consider a circle and a point $P$ outside the circle. The angle of given measure with vertex at $P$ subtends a diameter of the circle. Construct the circle’s diameter with ruler and compass.
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
Let $k$ be the smallest positive integer for which there exist distinct integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ such that the polynomial $$p(x)=\left(x-m_{1}\right)\left(x-m_{2}\right)\left(x-m_{3}\right)\left(x-m_{4}\right)\left(x-m_{5}\right)$$ has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ for which this minimum $k$ is achieved.
Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, of a square $ABCD$. If $|BM|=21$, $|DN|=4$, and $|NC|=24$, what is $m(\widehat{MAN})$?
$
\textbf{(A)}\ 15^\circ
\qquad\textbf{(B)}\ 30^\circ
\qquad\textbf{(C)}\ 37^\circ
\qquad\textbf{(D)}\ 45^\circ
\qquad\textbf{(E)}\ 60^\circ
$
$1001$ marbles are drawn at random and without replacement from a jar of $2020$ red marbles and $n$ blue marbles. Find the smallest positive integer $n$ such that the probability that there are more blue marbles chosen than red marbles is strictly greater than $\frac{1}{2}$.
[i]Proposed by Taiki Aiba[/i]
Let $ABCD$ be a cyclic quadrilateral with $|AB|=26$, $|BC|=10$, $m(\widehat{ABD})=45^\circ$,$m(\widehat{ACB})=90^\circ$. What is the area of $\triangle DAC$ ?
$ \textbf{(A)}\ 120
\qquad\textbf{(B)}\ 108
\qquad\textbf{(C)}\ 90
\qquad\textbf{(D)}\ 84
\qquad\textbf{(E)}\ 80
$
Let $H$ be the orthocenter of $\triangle ABC$. The points $A_{1} \not= A$, $B_{1} \not= B$ and $C_{1} \not= C$ lie, respectively, on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$ and satisfy $A_{1}H=B_{1}H=C_{1}H$. Denote by $H_{1}$, $H_{2}$ and $H_{3}$ the orthocenters of $\triangle A_{1}BC$, $\triangle B_{1}CA$ and $\triangle C_{1}AB$, respectively. Prove that $\triangle A_{1}B_{1}C_{1}$ and $\triangle H_{1}H_{2}H_{3}$ have the same orthocenter.
Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying
(1) for each $n_i$, its digits belong to the set $\{1,2\}$;
(2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right.
Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.
Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.
We will assume that the sides of a square are reflective and we will designate them with the names of the four cardinal points. Marking a point on the side $N$ , determine in which direction a ray of light should exit (into the interior of the square) so that it returns to it after having undergone $n$ reflections on the side $E$ , another $n$ on the side $W$ , $m$ on the $S$ and $m - 1$ on the $N$, where $n$ and $m$ are known natural numbers. What happens if m and $n$ are not prime to each other? Calculate the length of the light ray considered as a function of $m$ and $n$, and of the length of the side of the square.
Point $H$ is the orthocenter of the scalene triangle $ABC.$ A line, which passes through point $H$, intersect the sides $AB$ and $AC$ at points $D$ and $E$, respectively, such that $AD=AE.$ Let $M$ be the midpoint of side $BC.$ Line $MH$ intersects the circumscribed circle of triangle $ABC$ at point $K$, which is on the smaller arc $AB$. Prove that Nibab can draw a circle through $A, D, E$ and $K.$
A cake is prepared for a dinner party to which only $p$ or $q$ persons will come ($p$ and $q$ are given co-prime integers). Find the minimum number of pieces (not necessarily equal) into which the cake must be cut in advance so that the cake may be equally shared between the persons in either case.
(D. Fomin, Leningrad)
$\forall n \in \mathbb{N}$, $P(n)$ denotes the number of the partition of $n$ as the sum of positive integers (disregarding the order of the parts), e.g. since $4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4$, so $P(4)=5$. "Dispersion" of a partition denotes the number of different parts in that partitation. And denote $q(n)$ is the sum of all the dispersions, e.g. $q(4)=1+2+2+1+1=7$. $n \geq 1$. Prove that
(1) $q(n) = 1 + \sum^{n-1}_{i=1} P(i).$
(2) $1 + \sum^{n-1}_{i=1} P(i) \leq \sqrt{2} \cdot n \cdot P(n)$.
On a plane are given finitely many red and blue lines, no two parallel, such that any intersection point of two lines of the same color also lies on another line of the other color. Prove that all the lines pass through a single point.
Find the least positive multiple of 999 that does not have a 9 as a digit.
Let $ABCD$ be a convex quadrilateral. If $M, N, K$ are the midpoints of the segments $AB, BC$, and $CD$, respectively, and there is also a point $P$ inside the quadrilateral $ABCD$ such that, $\angle BPN= \angle PAD$ and $\angle CPN=\angle PDA$. Show that $AB \cdot CD=4PM\cdot PK$.
A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove
that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive
integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.
Let $ABC$ be a scalene, non-right triangle. Let $\omega$ be the incircle and let $\gamma$ be the nine-point circle (the circle through the feet of the altitudes) of $\triangle ABC$, with centers $I$ and $N$, respectively. Suppose $\omega$ and $\gamma$ are tangent at a point $F$. Let $D$ be the foot of the perpendicular from $A$ to line $BC$ and let $M$ be the midpoint of side $\overline{BC}$. The common tangent to $\omega$ and $\gamma$ at $F$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let lines $DP$ and $DQ$ intersect $\gamma$ at points $P_1 \neq D$ and $Q_1 \neq D$, respectively. Suppose that point $Z$ lies on line $P_1Q_1$ such that $\angle MFZ=90^{\circ}$ and $MZ \perp DF$. Suppose that $\gamma$ has radius 11 and $\omega$ has radius 5. Then $DI=\frac{k\sqrt{m}}{n}$, where $k,m,$ and $n$ are positive integers such that $\gcd(k,n)=1$ and $m$ is not divisible by the square of any prime. Compute $100k+10m+n$.
[i]Proposed by Luke Robitaille[/i]
[b]a)[/b] Let $ z_1,z_2,z_3 $ be three complex numbers of same absolute value, and $ 0=z_1+z_2+z_3. $ Show that these represent the affixes of an equilateral triangle.
[b]b)[/b] Find all subsets formed by roots of the same unity that have the property that any three elements of every such, doesn’t represent the vertices of an equilateral triangle.
Let $n$ be a positive integer. On a blackboard is a circle, and around it $n$ non-negative integers are written. Raphinha plays a game in which an operation consists of erasing a number $a$ neighboring $b$ and $c$, with $b \geq c$, and writing in its place $b + c$ if $b + c \leq 5a/4$ and $b - c$ otherwise.
Your goal is to make all the numbers on the board equal $0$. Find all $n$ such that Raphinha always manages to reach her goal.
The computer "Intel stump-V" can do only one operation with a number: add 1 to it, then rearrange all the zeros in the decimal representation to the end and rearrenge the left digits in any order. (For example from 1004 you could get 1500 or 5100). The number $12345$ was written on the computer and after performing 400 operations, the number 100000 appeared on the screen. How many times has a number with the last digit 0 appeared on the screen?
The diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected and edges are added between the trisection points as shown. Then the shaded corner regions are removed, leaving the octagon which is not shaded in the diagram. Find the perimeter of this octagon.
[asy]
size(4cm);
dotfactor=3.5;
pair A,B,C,D,E,F,G,H,W,X,Y,Z;
A=(0,12);
B=(0,24);
C=(16,36);
D=(32,36);
E=(48,24);
F=(48,12);
G=(32,0);
H=(16,0);
W=origin;
X=(0,36);
Y=(48,36);
Z=(48,0);
filldraw(W--A--H--cycle^^B--X--C--cycle^^D--Y--E--cycle^^F--Z--G--cycle,rgb(.76,.76,.76));
draw(W--X--Y--Z--cycle,linewidth(1.2));
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
[/asy]
Let $ABCD$ be a rhombus with $\angle BAD < 90^o$. The circle passing through $D$ with center $A$ intersects the line $CD$ a second time in point $E$. Let $S$ be the intersection of the lines $BE$ and $AC$. Prove that the points $A$, $S$, $D$ and $E$ lie on a circle.
[i](Karl Czakler)[/i]
The triangle $ABC$ ($AB > BC$) is inscribed in the circle $\Omega$. On the sides $AB$ and $BC$, the points $M$ and $N$ are chosen, respectively, so that $AM = CN$, The lines $MN$ and $AC$ intersect at point $K$. Let $P$ be the center of the inscribed circle of triangle $AMK$, and $Q$ the center of the excircle of the triangle $CNK$ tangent to side $CN$. Prove that the midpoint of the arc $ABC$ of the circle $\Omega$ is equidistant from the $P$ and $Q$.