Found problems: 85335
2010 APMO, 3
Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?
1991 AMC 12/AHSME, 1
If for any three distinct numbers $a$, $b$ and $c$ we define \[\boxed{a,b,c} = \frac{c + a}{c - b},\] then $\boxed{1,-2,-3}=$
$ \textbf{(A)}\ -2\qquad\textbf{(B)}\ -\frac{2}{5}\qquad\textbf{(C)}\ -\frac{1}{4}\qquad\textbf{(D)}\ \frac{2}{5}\qquad\textbf{(E)}\ 2 $
2003 National Olympiad First Round, 13
Let $ABC$ be a triangle such that $|AB|=8$ and $|AC|=2|BC|$. What is the largest value of altitude from side $[AB]$?
$
\textbf{(A)}\ 3\sqrt 2
\qquad\textbf{(B)}\ 3\sqrt 3
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ \dfrac {16}3
\qquad\textbf{(E)}\ 6
$
2013 Vietnam National Olympiad, 3
Let $ABC$ be a triangle such that $ABC$ isn't a isosceles triangle. $(I)$ is incircle of triangle touches $BC,CA,AB$ at $D,E,F$ respectively. The line through $E$ perpendicular to $BI$ cuts $(I)$ again at $K$. The line through $F$ perpendicular to $CI$ cuts $(I)$ again at $L$.$J$ is midpoint of $KL$.
[b]a)[/b] Prove that $D,I,J$ collinear.
[b]b)[/b] $B,C$ are fixed points,$A$ is moved point such that $\frac{AB}{AC}=k$ with $k$ is constant.$IE,IF$ cut $(I)$ again at $M,N$ respectively.$MN$ cuts $IB,IC$ at $P,Q$ respectively. Prove that bisector perpendicular of $PQ$ through a fixed point.
1986 India National Olympiad, 2
Solve
\[ \left\{ \begin{array}{l}
\log_2 x\plus{}\log_4 y\plus{}\log_4 z\equal{}2 \\
\log_3 y\plus{}\log_9 z\plus{}\log_9 x\equal{}2 \\
\log_4 z\plus{}\log_{16} x\plus{}\log_{16} y\equal{}2 \\
\end{array} \right.\]
1989 AMC 8, 3
Which of the following numbers is the largest?
$\text{(A)}\ .99 \qquad \text{(B)}\ .9099 \qquad \text{(C)}\ .9 \qquad \text{(D)}\ .909 \qquad \text{(E)}\ .9009$
2003 Romania National Olympiad, 1
Find the maximum number of elements which can be chosen from the set $ \{1,2,3,\ldots,2003\}$ such that the sum of any two chosen elements is not divisible by 3.
2022 Czech-Austrian-Polish-Slovak Match, 4
Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.
2025 District Olympiad, P4
Find all triplets of matrices $A,B,C\in\mathcal{M}_2(\mathbb{R})$ which satisfy \begin{align*}
A=BC-CB \\
B=CA-AC \\
C=AB-BA
\end{align*}
[i]Proposed by David Anghel[/i]
III Soros Olympiad 1996 - 97 (Russia), 11.3
A chord $AB$ is drawn in a certain circle. The smaller of the two arcs $AB$ corresponds to a central angle of $120^o$. A tangent $p$ to this arc is drawn. Two circles with radii $R$ and $r$ are constructed, touching this smaller arc $AB$ and straight lines $AB$ and $p$. Find the radius of the original circle.
2020 Brazil Team Selection Test, 1
Consider an $n\times n$ unit-square board. The main diagonal of the board is the $n$ unit squares along the diagonal from the top left to the bottom right. We have an unlimited supply of tiles of this form:
[asy]
size(1.5cm);
draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0));
[/asy]
The tiles may be rotated. We wish to place tiles on the board such that each tile covers exactly three unit squares, the tiles do not overlap, no unit square on the main diagonal is covered, and all other unit squares are covered exactly once. For which $n\geq 2$ is this possible?
[i]Proposed by Daniel Kohen[/i]
2016 All-Russian Olympiad, 8
Medians $AM_A,BM_B,CM_C$ of triangle $ABC$ intersect at $M$.Let $\Omega_A$ be circumcircle of triangle passes through midpoint of $AM$ and tangent to $BC$ at $M_A$.Define $\Omega_B$ and $\Omega_C$ analogusly.Prove that $\Omega_A,\Omega_B$ and $\Omega_C$ intersect at one point.(A.Yakubov)
[hide=P.S]sorry for my mistake in translation :blush: :whistling: .thank you jred for your help :coolspeak: [/hide]
2008 Peru IMO TST, 3
Given a positive integer $n$, consider the sequence $(a_i)$, $1 \leq i \leq 2n$, defined as follows:
$a_{2k-1} = -k, 1 \leq k \leq n$
$a_{2k} = n-k+1, 1 \leq k \leq n.$
We call a pair of numbers $(b,c)$ good if the following conditions are met:
$i) 1 \leq b < c \leq 2n,$
$ii) \sum_{j=b}^{c}a_j = 0$
If $B(n)$ is the number of good pairs corresponding to $n$, prove that there are infinitely many $n$ for which $B(n) = n$.
2011 China Second Round Olympiad, 6
In a tetrahedral $ABCD$, given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$, $AD=BD=3$, and $CD=2$. Find the radius of the circumsphere of $ABCD$.
2022 Bulgarian Autumn Math Competition, Problem 8.1
Solve the equation:
\[4x^2+|9-6x|=|10x-15|+6(2x+1)\]
2007 Hanoi Open Mathematics Competitions, 10
What is the smallest possible value of $x^2+2y^2-x-2y-xy$?
2009 Princeton University Math Competition, 6
In the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$. Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$. Find $\angle OPQ$ in degrees.
[asy]
pathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6)+ linewidth(0.7) + linetype("4 4"); real n = 10, start = 360/n*6-15;
pair O=(0,0), A=dir(start), B=dir(start+360/n), C=dir(start+2*360/n), D=dir(start+3*360/n), P=(A+C)/2, Q=(O+D)/2; D(D("O",O,NE)--D("A",A,W)--D("B",B,SW)--D("C",C,S)--D("D",D,SE)--O--D("P",P,1.6*dir(95))--D("Q",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2, dd); D(D--(D+dir(start+4*360/n))/2, dd);
[/asy]
2008 Princeton University Math Competition, A4/B7
What's the greatest integer $n$ for which the system $k < x^k < k + 1$ for $k = 1,2,..., n$ has a solution?
2009 Indonesia MO, 3
For every triangle $ ABC$, let $ D,E,F$ be a point located on segment $ BC,CA,AB$, respectively. Let $ P$ be the intersection of $ AD$ and $ EF$. Prove that:
\[ \frac{AB}{AF}\times DC\plus{}\frac{AC}{AE}\times DB\equal{}\frac{AD}{AP}\times BC\]
2022 Bulgarian Autumn Math Competition, Problem 11.4
The number $2022$ is written on the white board. Ivan and Peter play a game, Ivan starts and they alternate. On a move, Ivan erases the number $b$, written on the board, throws a dice which shows some number $a$, and writes the residue of $(a+b) ^2$ modulo $5$. Similarly, Peter throws a dice which shows some number $a$, and changes the previously written number $b$ to the residue of $a+b$ modulo $3$. The first player to write a $0$ wins. What is the probability of Ivan winning the game?
2001 USAMO, 3
Let $a, b, c \geq 0$ and satisfy \[ a^2+b^2+c^2 +abc = 4 . \] Show that \[ 0 \le ab + bc + ca - abc \leq 2. \]
2019 AMC 8, 15
On a beach 50 people are wearing sunglasses and 35 people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is $\frac{2}{5}$. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?
$\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}$
2019 Thailand TST, 2
A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold:
[list=1]
[*] each triangle from $T$ is inscribed in $\omega$;
[*] no two triangles from $T$ have a common interior point.
[/list]
Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
1993 National High School Mathematics League, 4
$C:(x-\arcsin a)(x-\arccos a)+(y-\arcsin a)(y+\arccos a)=0$. The length of string of $C$ cut by $l:x=\frac{\pi}{4}$ is $d$. When $a$ changes, the minumum value of $d$ is
$\text{(A)}\frac{\pi}{4}\qquad\text{(B)}\frac{\pi}{3}\qquad\text{(C)}\frac{\pi}{2}\qquad\text{(D)}\pi$
2021 239 Open Mathematical Olympiad, 7
Given $n$ lines on the plane, they divide the plane onto several
bounded or bounded polygonal regions. Define the rank of a region as
the number of vertices on its boundary (a vertex is a point which
belongs to at least two lines). Prove that the sum of squares of
ranks of all regions does not exceed $10n^2$.
(D. Fomin)