Found problems: 85335
1966 IMO Shortlist, 27
Given a point $P$ lying on a line $g,$ and given a circle $K.$ Construct a circle passing through the point $P$ and touching the circle $K$ and the line $g.$
2016 NIMO Problems, 6
Let $ABC$ be a triangle with $AB=20$, $AC=34$, and $BC=42$. Let $\omega_1$ and $\omega_2$ be the semicircles with diameters $\overline{AB}$ and $\overline{AC}$ erected outwards of $\triangle ABC$ and denote by $\ell$ the common external tangent to $\omega_1$ and $\omega_2$. The line through $A$ perpendicular to $\overline{BC}$ intersects $\ell$ at $X$ and $BC$ at $Y$. The length of $\overline{XY}$ can be written in the form $m+\sqrt n$ where $m$ and $n$ are positive integers. Find $100m+n$.
[i]Proposed by David Altizio[/i]
2016 ASDAN Math Tournament, 3
Real numbers $x,y,z$ form an arithmetic sequence satisfying
\begin{align*}
x+y+z&=6\\
xy+yz+zx&=10.
\end{align*}
What is the absolute value of their common difference?
2001 Greece Junior Math Olympiad, 4
Let $ABC$ be a triangle with altitude $AD$ , angle bisectors $AE$ and $BZ$ that intersecting at point $I$. From point $I$ let $IT$ be a perpendicular on $AC$. Also let line $(e)$ be perpendicular on $AC$ at point $A$. Extension of $ET$ intersects line $(e)$ at point $K$. Prove that $AK=AD$.
1996 Baltic Way, 17
Using each of the eight digits $1,3,4,5,6,7,8$ and $9$ exactly once, a three-digit number $A$, two two-digit numbers $B$ and $C$, $B<C$, and a one digit number $D$ are formed. The numbers are such that $A+D=B+C=143$. In how many ways can this be done?
1984 Tournament Of Towns, (065) A3
An infinite (in both directions) sequence of rooms is situated on one side of an infinite hallway. The rooms are numbered by consecutive integers and each contains a grand piano. A finite number of pianists live in these rooms. (There may be more than one of them in some of the rooms.) Every day some two pianists living in adjacent rooms (the Arth and ($k +1$)st) decide that they interfere with each other’s practice, and they move to the ($k - 1$)st and ($k + 2$)nd rooms, respectively. Prove that these moves will cease after a finite number of days.
(VG Ilichev)
2022 Kyiv City MO Round 1, Problem 3
Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$.
[i](Proposed by Oleksii Masalitin)[/i]
2011 Federal Competition For Advanced Students, Part 2, 1
Every brick has $5$ holes in a line. The holes can be filled with bolts (fi tting in one hole) and braces (fi tting into two neighboring holes). No hole may remain free.
One puts $n$ of these bricks in a line to form a pattern from left to right. In this line no two braces and no three bolts may be adjacent.
How many diff erent such patterns can be produced with $n$ bricks?
1951 AMC 12/AHSME, 32
If $ \triangle ABC$ is inscribed in a semicircle whose diameter is $ AB$, then $ AC \plus{} BC$ must be
$ \textbf{(A)}\ \text{equal to }AB \qquad\textbf{(B)}\ \text{equal to }AB\sqrt {2} \qquad\textbf{(C)}\ \geq AB\sqrt {2}$
$ \textbf{(D)}\ \leq AB\sqrt {2} \qquad\textbf{(E)}\ AB^2$
2021 Iran MO (2nd Round), 1
There are two distinct Points $A$ and $B$ on a line. We color a point $P$ on segment $AB$, distinct from $A,B$ and midpoint of segment $AB$ to red. In each move , we can reflect one of the red point wrt $A$ or $B$ and color the midpoint of the resulting point and the point we reflected from ( which is one of $A$ or $B$ ) to red. For example , if we choose $P$ and the reflection of $P$ wrt to $A$ is $P'$ , then midpoint of $AP'$ would be red. Is it possible to make the midpoint of $AB$ red after a finite number of moves?
2011 Math Prize For Girls Problems, 11
The sequence $a_0$, $a_1$, $a_2$, $\ldots\,$ satisfies the recurrence equation
\[
a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}
\]
for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?
2015 Tuymaada Olympiad, 7
In $\triangle ABC$ points $M,O$ are midpoint of $AB$ and circumcenter. It is true, that $OM=R-r$. Bisector of external $\angle A$ intersect $BC$ at $D$ and bisector of external $\angle C$ intersect $AB$ at $E$.
Find possible values of $\angle CED$
[i]D. Shiryaev [/i]
2023 Nordic, P4
Let $ABC$ be a triangle, and $M$ the midpoint of the side $BC$. Let $E$ and $F$ be points on the sides $AC$ and $AB$, respectively, so that $ME=MF$. Let $D$ be the second intersection of the circumcircle of $MEF$ and the side $BC$. Consider the lines $\ell_D$, $\ell_E$ and $\ell_F$ through $D, E$ and $F$, respectively, such that $\ell_D \perp BC$, $\ell_E \perp AC$ and $\ell_F \perp AB$. Show that $\ell_D, \ell_E$ and $\ell_F$ are concurrent.
1967 IMO, 4
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$
Kyiv City MO Seniors Round2 2010+ geometry, 2010.10.4
The points $A \ne B$ are given on the plane. The point $C$ moves along the plane in such a way that $\angle ACB = \alpha$ , where $\alpha$ is the fixed angle from the interval ($0^o, 180^o$). The circle inscribed in triangle $ABC$ has center the point $I$ and touches the sides $AB, BC, CA$ at points $D, E, F$ accordingly. Rays $AI$ and $BI$ intersect the line $EF$ at points $M$ and $N$, respectively. Show that:
a) the segment $MN$ has a constant length,
b) all circles circumscribed around triangle $DMN$ have a common point
2020 Germany Team Selection Test, 3
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
2013 Romania National Olympiad, 1
Given A, non-inverted matrices of order n with real elements, $n\ge 2$ and given ${{A}^{*}}$adjoin matrix A. Prove that $tr({{A}^{*}})\ne -1$ if and only if the matrix ${{I}_{n}}+{{A}^{*}}$ is invertible.
2017 India PRMO, 6
Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\frac{p}{q}$ . Find the value of $q - p$.
2007 AMC 10, 1
One ticket to a show costs $ \$20$ at full price. Susan buys 4 tickets using a coupon that gives her a $25\%$ discount. Pam buys 5 tickets using a coupon that gives her a $30\%$ discount. How many more dollars does Pam pay than Susan?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 20$
2020 Estonia Team Selection Test, 3
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
2010 Indonesia TST, 3
Given a non-isosceles triangle $ABC$ with incircle $k$ with center $S$. $k$ touches the side $BC,CA,AB$ at $P,Q,R$ respectively. The line $QR$ and line $BC$ intersect at $M$. A circle which passes through $B$ and $C$ touches $k$ at $N$. The circumcircle of triangle $MNP$ intersects $AP$ at $L$. Prove that $S,L,M$ are collinear.
2019 Dutch IMO TST, 2
Write $S_n$ for the set $\{1, 2,..., n\}$. Determine all positive integers $n$ for which there exist functions $f : S_n \to S_n$ and $g : S_n \to S_n$ such that for every $x$ exactly one of the equalities $f(g(x)) = x$ and $g(f(x)) = x$ holds.
2011 Junior Balkan Team Selection Tests - Moldova, 1
The absolute value of the difference of the solutions of the equation $x^2 + px + q = 0$, with $p, q \in R$, is equal to $4$. Find the solutions of the equation if it is known that $(q + 1) p^2 + q^2$ takes the minimum value.
2007 ITest, 43
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers: \begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} She notes that two of them have exactly $8$ positive divisors each. Find the common prime divisor of those two integers.
2007 All-Russian Olympiad, 3
$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear.
[i]V. Astakhov[/i]