Found problems: 85335
2016 IFYM, Sozopol, 3
The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.
2018 Regional Olympiad of Mexico Center Zone, 2
Let $\vartriangle ABC$be a triangle and let $\Gamma$ its circumscribed circle. Let $M$ be the midpoint of the side $BC$ and let $D$ be the point of intersection of the line $AM$ with $\Gamma$. By $D$ a straight line is drawn parallel to $BC$, which intersects $\Gamma$ at a point $E$. Let $N$ be the midpoint of the segment $AE$ and let $P$ be the point of intersection of $CN$ with $AM$. Show that $AP = PC$.
2023 UMD Math Competition Part II, 1
An Indian raga has two kinds of notes: a short note, which lasts for $1$ beat and a long note, which lasts for $2$ beats. For example, there are $3$ ragas which are $3$ beats long; $3$ short notes, a short note followed by a long note, and a long note followed by a short note. How many Indian ragas are 11 beats long?
2022 Iranian Geometry Olympiad, 4
Let $ABCD$ be a trapezoid with $AB\parallel CD$. Its diagonals intersect at a point $P$. The line passing through $P$ parallel to $AB$ intersects $AD$ and $BC$ at $Q$ and $R$, respectively. Exterior angle bisectors of angles $DBA$, $DCA$ intersect at $X$. Let $S$ be the foot of $X$ onto $BC$. Prove that if quadrilaterals $ABPQ$, $CDQP$ are circumcribed, then $PR=PS$.
[i]Proposed by Dominik Burek, Poland[/i]
2023 Czech and Slovak Olympiad III A., 1
Alice and Bob are playing a game on a plane consisting of $72$ cells arranged in circle. At the beginning of the game, Bob places a stone on some of the cells. Then, in every round first Alice picks one empty cell and then Bob must move a stone from one of the two neighboring cells on this cell. If he is unable to do that, game ends. Determine the smallest number of stones he has to place in the beginning so he has a strategy to make the game last for at least $2023$ rounds.
2011 China Northern MO, 5
If the positive integers $a, b, c$ satisfy $a^2+b^2=c^2$, then $(a, b, c)$ is called a Pythagorean triple. Find all Pythagorean triples containing $30$.
2003 Italy TST, 1
The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. The line through $A$ parallel to $DF$ meets the line through $C$ parallel to $EF$ at $G$.
$(a)$ Prove that the quadrilateral $AICG$ is cyclic.
$(b)$ Prove that the points $B,I,G$ are collinear.
2019 India IMO Training Camp, P2
Let $ABC$ be a triangle with $\angle A=\angle C=30^{\circ}.$ Points $D,E,F$ are chosen on the sides $AB,BC,CA$ respectively so that $\angle BFD=\angle BFE=60^{\circ}.$ Let $p$ and $p_1$ be the perimeters of the triangles $ABC$ and $DEF$, respectively. Prove that $p\le 2p_1.$
Ukrainian TYM Qualifying - geometry, I.5
The heights of a triangular pyramid intersect at one point. Prove that all flat angles at any vertex of the surface are either acute, or right, or obtuse.
2008 China Girls Math Olympiad, 4
Equilateral triangles $ ABQ$, $ BCR$, $ CDS$, $ DAP$ are erected outside of the convex quadrilateral $ ABCD$. Let $ X$, $ Y$, $ Z$, $ W$ be the midpoints of the segments $ PQ$, $ QR$, $ RS$, $ SP$, respectively. Determine the maximum value of
\[ \frac {XZ\plus{}YW}{AC \plus{} BD}.
\]
2010 Contests, 1
Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.
1998 Slovenia Team Selection Test, 2
A semicircle with center $O$ and diameter $AB$ is given. Point $M$ on the extension of $AB$ is taken so that $AM > BM$. A line through $M$ intersects the semicircle at $C$ and $D$ so that $CM < DM$. The circumcircles of triangles $AOD$ and $OBC$ meet again at point $K$. Prove that $OK$ and $KM$ are perpendicular
1995 Turkey Team Selection Test, 3
The sequence $\{x_n\}$ of real numbers is defined by
\[x_1=1 \quad\text{and}\quad x_{n+1}=x_n+\sqrt[3]{x_n} \quad\text{for}\quad n\geq 1.\]
Show that there exist real numbers $a, b$ such that $\lim_{n \rightarrow \infty}\frac{x_n}{an^b} = 1$.
2010 CHMMC Winter, 1
A matrix $M$ is called idempotent if $M^2 = M$. Find an idempotent $2 \times 2$ matrix with distinct rational entries or write “none” if none exist.
2003 National Olympiad First Round, 7
Starting with the sequence $\text{AAAIEE}$, we replace $\text{AIE}$ with $\text{EA}$, $\text{AE}$ with $\text{IE}$, $\text{E}$ with $\text{AI}$. After repeating replace operations many times, which of the following cannot be got?
$
\textbf{(A)}\ \text{AIAIIAI}
\qquad\textbf{(B)}\ \text{AIAIAI}
\qquad\textbf{(C)}\ \text{AIAAA}
\qquad\textbf{(D)}\ \text{AIAA}
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2015 Harvard-MIT Mathematics Tournament, 5
Let $I$ be the set of points $(x,y)$ in the Cartesian plane such that $$x>\left(\frac{y^4}{9}+2015\right)^{1/4}$$ Let $f(r)$ denote the area of the intersection of $I$ and the disk $x^2+y^2\le r^2$ of radius $r>0$ centered at the origin $(0,0)$. Determine the minimum possible real number $L$ such that $f(r)<Lr^2$ for all $r>0$.
1999 All-Russian Olympiad Regional Round, 8.6
Given triangle $ABC$. Point $A_1$ is symmetric to vertex $A$ wrt line $BC$, and point $C_1$ is symmetric to vertex $C$ wrt line $AB$. Prove that if points $A_1$, $B$ and $C_1$ lie on the same line and $C_1B = 2A_1B$, then angle $\angle CA_1B$ is right.
2022 South Africa National Olympiad, 2
Find all pairs of real numbers $x$ and $y$ which satisfy the following equations:
\begin{align*}
x^2 + y^2 - 48x - 29y + 714 & = 0 \\
2xy - 29x - 48y + 756 & = 0
\end{align*}
2005 All-Russian Olympiad, 2
Do there exist 12 rectangular parallelepipeds $P_1,\,P_2,\ldots,P_{12}$ with edges parallel to coordinate axes $OX,\,OY,\,OZ$ such that $P_i$ and $P_j$ have a common point iff $i\ne j\pm 1$ modulo 12?
1999 Greece JBMO TST, 2
For $a,b,c>0$, prove that
(i) $\frac{a+b+c}{2}-\frac{ab}{a+b}-\frac{bc}{b+c}-\frac{ca}{c+a}\ge 0$
(ii) $a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}$
2019 Junior Balkan Team Selection Tests - Romania, 4
The numbers from $1$ through $100$ are written in some order on a circle.
We call a pair of numbers on the circle [i]good [/i] if the two numbers are not neighbors on the circle and if at least one of the two arcs they determine on the circle only contains numbers smaller then both of them. What may be the total number of good pairs on the circle.
2003 Manhattan Mathematical Olympiad, 3
Assume $a,b,c$ are positive numbers, such that
\[ a(1-b) = b(1-c) = c(1-a) = \dfrac14 \]
Prove that $a=b=c$.
1994 Balkan MO, 1
An acute angle $XAY$ and a point $P$ inside the angle are given. Construct (using a ruler and a compass) a line that passes through $P$ and intersects the rays $AX$ and $AY$ at $B$ and $C$ such that the area of the triangle $ABC$ equals $AP^2$.
[i]Greece[/i]
2011 Miklós Schweitzer, 8
Given a nonzero real number $a\leq 1/e$, let $z_1, ..., z_n \in C$ be non-real numbers for which $ze^z + a = 0$ holds, and let $c_1, ..., c_n \in C$ be arbitrary. Show that the function $f(x)=Re(\sum_{j=1}^n c_j e^{z_j x})$ ($x \in R$) has a zero in every closed interval of length 1.
2019 Turkey Team SeIection Test, 4
For an integer $n$ with $b$ digits, let a [i]subdivisor[/i] of $n$ be a positive number which divides a number obtained by removing the $r$ leftmost digits and the $l$ rightmost digits of $n$ for nonnegative integers $r,l$ with $r+l<b$ (For example, the subdivisors of $143$ are $1$, $2$, $3$, $4$, $7$, $11$, $13$, $14$, $43$, and $143$). For an integer $d$, let $A_d$ be the set of numbers that don't have $d$ as a subdivisor. Find all $d$, such that $A_d$ is finite.