Found problems: 85335
2001 Federal Math Competition of S&M, Problem 2
Vertices of a square $ABCD$ of side $\frac{25}4$ lie on a sphere. Parallel lines passing through points $A,B,C$ and $D$ intersect the sphere at points $A_1,B_1,C_1$ and $D_1$, respectively. Given that $AA_1=2$, $BB_1=10$, $CC_1=6$, determine the length of the segment $DD_1$.
2009 Greece Team Selection Test, 3
Find all triples $(x,y,z)\in \mathbb{R}^{3}$ such that $x,y,z>3$ and $\frac{(x+2)^2}{y+z-2}+\frac{(y+4)^2}{z+x-4}+\frac{(z+6)^2}{x+y-6}=36$
2005 Poland - Second Round, 1
Find all positive integers $n$ for which $n^n+1$ and $(2n)^{2n}+1$ are prime numbers.
2022-2023 OMMC, 24
Define acute $\triangle ABC$ with circumcenter $O$. The circumcircle of $\triangle ABO$ meets segment $BC$ at $D \ne B$, segment $AC$ at $F \ne A$, and the Euler line of $\triangle ABC$ at $P \ne O$. The circumcircle of $\triangle ACO$ meets segment $BC$ at $E \ne C$. Let $\overline{BC}$ and $\overline{FP}$ intersect at $X$, with $C$ between $B$ and $X$. If $BD=13$, $EC=8$, and $CX=27$, find $DE$.
$\emph{(The Euler line of a triangle passes through its orthocenter, circumcenter, and centroid.)}$
2021 Austrian MO National Competition, 5
Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie.
(a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it.
(b) Show that in all other cases the four points thus obtained lie on one circle.
(Theresia Eisenkölbl)
Indonesia MO Shortlist - geometry, g6.6
Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.
I Soros Olympiad 1994-95 (Rus + Ukr), 9.3
Is there a quadrilateral in which the position of any vertex can be changed, leaving the other three in place, so that the resulting four points serve as the vertices of a quadrilateral equal to the original one?
2007 Moldova Team Selection Test, 4
We are given $n$ distinct points in the plane. Consider the number $\tau(n)$ of segments of length 1 joining pairs of these points. Show that $\tau(n)\leq \frac{n^{2}}3$.
2005 IMO Shortlist, 5
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that
\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]
[i]Hojoo Lee, Korea[/i]
2011 Dutch IMO TST, 3
Let $\Gamma_1$ and $\Gamma_2$ be two intersecting circles with midpoints respectively $O_1$ and $O_2$, such that $\Gamma_2$ intersects the line segment $O_1O_2$ in a point $A$. The intersection points of $\Gamma_1$ and $\Gamma_2$ are $C$ and $D$. The line $AD$ intersects $\Gamma_1$ a second time in $S$. The line $CS$ intersects $O_1O_2$ in $F$. Let $\Gamma_3$ be the circumcircle of triangle $AD$. Let $E$ be the second intersection point of $\Gamma_1$ and $\Gamma_3$. Prove that $O_1E$ is tangent to $\Gamma_3$.
2003 Alexandru Myller, 3
Let be three elements $ a,b,c $ of a nontrivial, noncommutative ring, that satisfy $ ab=1-c, $ and such that there exists an element $ d $ from the ring such that $ a+cd $ is a unit. Prove that there exists an element $ e $ from the ring such that $ b+ec $ is a unit.
[i]Andrei Nedelcu[/i] and [i] Lucian Ladunca [/i]
2023 Ecuador NMO (OMEC), 6
Let $DE$ the diameter of a circunference $\Gamma$. Let $B, C$ on $\Gamma$ such that $BC$ is perpendicular to $DE$, and let $Q$ the intersection of $BC$ with $DE$. Let $P$ a point on segment $BC$ such that $BP=4PQ$. Let $A$ the second intersection of $PE$ with $\Gamma$. If $DE=2$ and $EQ=\frac{1}{2}$, find all possible values of the sides of triangle $ABC$.
1954 AMC 12/AHSME, 2
The equation $ \frac{2x^2}{x\minus{}1}\minus{}\frac{2x\plus{}7}{3}\plus{}\frac{4\minus{}6x}{x\minus{}1}\plus{}1\equal{}0$ can be transformed by eliminating fractions to the equation $ x^2\minus{}5x\plus{}4\equal{}0$. The roots of the latter equation are $ 4$ and $ 1$. Then the roots of the first equation are:
$ \textbf{(A)}\ 4 \text{ and }1 \qquad
\textbf{(B)}\ \text{only }1 \qquad
\textbf{(C)}\ \text{only }4 \qquad
\textbf{(D)}\ \text{neither 4 nor 1} \qquad
\textbf{(E)}\ \text{4 and some other root}$
1998 Vietnam Team Selection Test, 2
Let $d$ be a positive divisor of $5 + 1998^{1998}$. Prove that $d = 2 \cdot x^2 + 2 \cdot x \cdot y + 3 \cdot y^2$, where $x, y$ are integers if and only if $d$ is congruent to 3 or 7 $\pmod{20}$.
2013 Saudi Arabia BMO TST, 1
In triangle $ABC$, $AB = AC = 3$ and $\angle A = 90^o$. Let $M$ be the midpoint of side $BC$. Points $D$ and $E$ lie on sides $AC$ and $AB$ respectively such that $AD > AE$ and $ADME$ is a cyclic quadrilateral. Given that triangle $EMD$ has area $2$, find the length of segment $CD$.
2017 AMC 10, 24
The vertices of an equilateral triangle lie on the hyperbola $xy=1,$ and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
$\textbf{(A)} \text{ 48} \qquad \textbf{(B)} \text{ 60} \qquad \textbf{(C)} \text{ 108} \qquad \textbf{(D)} \text{ 120} \qquad \textbf{(E)} \text{ 169}$
2025 Israel National Olympiad (Gillis), P6
Let $a$, $b$ and $c$ be non-negative numbers such that $ab+ac+bc+abc=4.$ . Prove that:
$$\sqrt{\frac{ab+ac+1}{a+2}}+\sqrt{\frac{ab+bc+1}{b+2}}+\sqrt{\frac{ac+bc+1}{c+2}}\leq3.$$
[hide="PS"]Dedicated to dear KhuongTrang :-D [/hide]
2019 USA TSTST, 1
Find all binary operations $\diamondsuit: \mathbb R_{>0}\times \mathbb R_{>0}\to \mathbb R_{>0}$ (meaning $\diamondsuit$ takes pairs of positive real numbers to positive real numbers) such that for any real numbers $a, b, c > 0$,
[list]
[*] the equation $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ holds; and
[*] if $a\ge 1$ then $a\,\diamondsuit\, a\ge 1$.
[/list]
[i]Evan Chen[/i]
2019 Romanian Master of Mathematics Shortlist, A2
Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that
$$
(a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n.
$$
[i](B. Serankou, M. Karpuk)[/i]
2021 AMC 12/AHSME Spring, 8
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34.$ What is the distance between two adjacent parallel lines?
$\textbf{(A)}\ 5\frac{1}{2} \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 6\frac{1}{2} \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 7\frac{1}{2}$
2018 MIG, 6
How many more hours are in $10$ years than seconds in $1$ day?
$\textbf{(A) }1000\qquad\textbf{(B) }1100\qquad\textbf{(C) }1150\qquad\textbf{(D) }1200\qquad\textbf{(E) }1300$
2022 New Zealand MO, 1
$ABCD$ is a rectangle with side lengths $AB = CD = 1$ and $BC = DA = 2$. Let $ M$ be the midpoint of $AD$. Point $P$ lies on the opposite side of line $MB$ to $A$, such that triangle $MBP$ is equilateral. Find the value of $\angle PCB$.
2012 Iran MO (3rd Round), 1
$P(x)$ is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers $q$ such that for some natural number $n$, $q|2^n+P(n)$.
[i]Proposed by Mohammad Gharakhani[/i]
2022 Durer Math Competition (First Round), 3
Paraflea makes jumps on the plane, starting from the origin $(0, 0)$. From point $(x, y)$ it may jump to another point of the form $(x + p, y + p^2)$, where $p$ is any positive real number. (The value of $p$ may differ for each jump.)
a) Is there any point in quadrant $I$ which cannot be reached by the flea? (Quadrant $I$ contains points $(x, y)$ for which $x$ and $y$ are positive real numbers.)
b) What is the minimum number of jumps that the flea must make from the origin so that it gets to the point $(100, 1)$?
2018 All-Russian Olympiad, 5
On the circle, 99 points are marked, dividing this circle into 99 equal arcs. Petya and Vasya play the game, taking turns. Petya goes first; on his first move, he paints in red or blue any marked point. Then each player can paint on his own turn, in red or blue, any uncolored marked point adjacent to the already painted one. Vasya wins, if after painting all points there is an equilateral triangle, all three vertices of which are colored in the same color. Could Petya prevent him?