Found problems: 85335
2020 MIG, 22
Jane's uncle gives her a "$4$-balance." The $4$-balance acts like a normal balance scale, but it compares four masses instead of two, tilting towards the weight that is heaviest (if all four are equal, it stays balanced). He then gives her $25$ coins, one of which is a counterfeit heavier than the rest. What is the minimum number of uses of the $4$-balance needed to ensure she identifies the counterfeit?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
1994 AIME Problems, 8
The points $(0,0),$ $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.
2022 Kyiv City MO Round 1, Problem 3
In triangle $ABC$ $\angle B > 90^\circ$. Tangents to this circle in points $A$ and $B$ meet at point $P$, and the line passing through $B$ perpendicular to $BC$ meets the line $AC$ at point $K$. Prove that $PA = PK$.
[i](Proposed by Danylo Khilko)[/i]
1949-56 Chisinau City MO, 2
What is the last digit of $777^{777}$?
2011 Kosovo National Mathematical Olympiad, 3
Find maximal value of the function $f(x)=8-3\sin^2 (3x)+6 \sin (6x)$
2016 Iran MO (2nd Round), 1
If $0<a\leq b\leq c$ prove that
$$\frac{(c-a)^2}{6c}\leq \frac{a+b+c}{3}-\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$
2017 International Zhautykov Olympiad, 3
Rectangle on a checked paper with length of a unit square side being $1$ Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with $3$ colours such that for any two corners with distance $1$ the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.
2015 Canadian Mathematical Olympiad Qualification, 4
Given an acute-angled triangle $ABC$ whose altitudes from $B$ and $C$ intersect at $H$, let $P$ be any point on side $BC$ and $X, Y$ be points on $AB, AC$, respectively, such that $PB = PX$ and $PC = PY$. Prove that the points $A, H, X, Y$ lie on a common circle.
2003 CentroAmerican, 6
Say a number is [i]tico[/i] if the sum of it's digits is a multiple of $2003$.
$\text{(i)}$ Show that there exists a positive integer $N$ such that the first $2003$ multiples, $N,2N,3N,\ldots 2003N$ are all tico.
$\text{(ii)}$ Does there exist a positive integer $N$ such that all it's multiples are tico?
2005 AMC 12/AHSME, 14
A circle having center $ (0,k)$, with $ k > 6$, is tangent to the lines $ y \equal{} x, y \equal{} \minus{} x$ and $ y \equal{} 6$. What is the radius of this circle?
$ \textbf{(A)}\ 6 \sqrt 2 \minus{} 6\qquad
\textbf{(B)}\ 6\qquad
\textbf{(C)}\ 6 \sqrt 2\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 6 \plus{} 6 \sqrt 2$
2023 Romanian Master of Mathematics, 1
Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$
1997 Belarusian National Olympiad, 3
$$Problem3;$$If distinct real numbers x,y satisfy $\{x\} = \{y\}$ and $\{x^3\}=\{y^3\}$
prove that $x$ is a root of a quadratic equation with integer coefficients.
2009 All-Russian Olympiad Regional Round, 11.5
We drew several straight lines on the plane and marked all of them intersection points. How many lines could be drawn? if one point is marked on one of the drawn lines, on the other - three, and on the third - five? Find all possible options and prove that there are no others.
2002 Turkey Team Selection Test, 2
Two circles are internally tangent at a point $A$. Let $C$ be a point on the smaller circle other than $A$. The tangent line to the smaller circle at $C$ meets the bigger circle at $D$ and $E$; and the line $AC$ meets the bigger circle at $A$ and $P$. Show that the line $PE$ is tangent to the circle through $A$, $C$, and $E$.
1999 ITAMO, 5
There is a village of pile-built dwellings on a lake, set on the gridpoints of an $m \times n$ rectangular grid. Each dwelling is connected by exactly $p$ bridges to some of the neighboring dwellings (diagonal connections are not allowed, two dwellings can be connected by more than one bridge). Determine for which values $m,n, p$ it is possible to place the bridges so that from any dwelling one can reach any other dwelling.
1997 All-Russian Olympiad, 1
Let $P(x)$ be a quadratic polynomial with nonnegative coeficients. Show that for any real numbers $x$ and $y$, we have the inequality $P(xy)^2 \leqslant P(x^2)P(y^2)$.
[i]E. Malinnikova[/i]
2004 Tournament Of Towns, 1
Three circles pass through point X. Their intersection points (other than X) are denoted A, B, C. Let A' be the second point of intersection of line AX and the circle circumscribed around triangle BCX, and define similarly points B', C'. Prove that triangles ABC', AB'C, and A'BC are similar.
LMT Theme Rounds, 6
How many functions $f:\{1,2,3,4\}\rightarrow \{1,2,3\}$ are surjective?
[i]Proposed by Nathan Ramesh
2011 Germany Team Selection Test, 1
Two circles $\omega , \Omega$ intersect in distinct points $A,B$ a line through $B$ intersects $\omega , \Omega$ in $C,D$ respectively such that $B$ lies between $C,D$ another line through $B$ intersects $\omega , \Omega$ in $E,F$ respectively such that $E$ lies between $B,F$ and $FE=CD$. Furthermore $CF$ intersects $\omega , \Omega$ in $P,Q$ respectively and $M,N$ are midpoints of the arcs $PB,QB$. Prove that $CNMF$ is a cyclic quadrilateral.
2006 Bulgaria Team Selection Test, 1
[b]Problem 4.[/b] Let $k$ be the circumcircle of $\triangle ABC$, and $D$ the point on the arc $\overarc{AB},$ which do not pass through $C$. $I_A$ and $I_B$ are the centers of incircles of $\triangle ADC$ and $\triangle BDC$, respectively. Proove that the circumcircle of $\triangle I_AI_BC$ touches $k$ iff \[ \frac{AD}{BD}=\frac{AC+CD}{BC+CD}. \]
[i] Stoyan Atanasov[/i]
2010 Victor Vâlcovici, 1
[b]a)[/b] Let be real numbers $ s,t\ge 0 $ and $ a,b\ge 1. $ Show that for any real $ x, $ it holds:
$$ a^{s\sin x+t\cos x}b^{s\cos x+t\sin x}\le 10^{(s+t)\sqrt{\text{tg}^2 a+\text{tg}^2 b}} $$
[b]b)[/b] For $ a,b>0 $ is the above inequality still true?
[i]Ilie Diaconu[/i]
2020 Brazil EGMO TST, 4
Determine all positive integers $n$ such that $\frac{n(n-1)}{2}-1$ divides $1^7+2^7+\dots +n^7$.
2020 Yasinsky Geometry Olympiad, 1
Given an acute triangle $ABC$. A circle inscribed in a triangle $ABC$ with center at point $I$ touches the sides $AB, BC$ at points $C_1$ and $A_1$, respectively. The lines $A_1C_1$ and $AC$ intersect at the point $Q$. Prove that the circles circumscribed around the triangles $AIC$ and $A_1CQ$ are tangent.
(Dmitry Shvetsov)
2011 239 Open Mathematical Olympiad, 3
Positive reals $a,b,c,d$ satisfy $a+b+c+d=4$. Prove that
$\sum_{cyc}\frac{a}{a^3 + 4} \le \frac{4}{5}$
1982 IMO Longlists, 10
Let $r_1, \ldots , r_n$ be the radii of $n$ spheres. Call $S_1, S_2, \ldots , S_n$ the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that
\[\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.\]