Found problems: 85335
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?
2022 Belarus - Iran Friendly Competition, 2
Let $P(x)$ be a polynomial with rational coefficients such that $P(n)$ is integer for all
integers $n$. Moreover: $gcd(P(1), \ldots , P(k), \ldots) = 1$. Prove that every integer $k$ can be represented
in infinitely many ways of the form
$\pm P(1) \pm P(2) \pm \ldots \pm P(m)$,
for some positive integer $m$ and certain choices of $\pm$.
2004 AMC 12/AHSME, 12
Let $ A \equal{} (0,9)$ and $ B \equal{} (0,12)$. Points $ A'$ and $ B'$ are on the line $ y \equal{} x$, and $ \overline{AA'}$ and $ \overline{BB'}$ intersect at $ C \equal{} (2,8)$. What is the length of $ \overline{A'B'}$?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 2 \plus{} \sqrt 2\qquad \textbf{(E)}\ 3\sqrt 2$
2009 Tournament Of Towns, 4
Three planes dissect a parallelepiped into eight hexahedrons such that all of their faces are quadrilaterals (each plane intersects two corresponding pairs of opposite faces of the parallelepiped and does not intersect the remaining two faces). One of the hexahedrons has a circumscribed sphere. Prove that each of these hexahedrons has a circumscribed sphere.
2007 Indonesia TST, 3
Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})\equal{}1$.
2004 VTRMC, Problem 5
Let $f(x)=\int^x_0\sin(t^2-t+x)dt$. Compute $f''(x)+f(x)$ and deduce that $f^{(12)}(0)+f^{(10)}(0)=0$.
2019 Kyiv Mathematical Festival, 1
A bunch of lilac consists of flowers with 4 or 5 petals. The number of flowers and the total number of petals are perfect squares. Can the number of flowers with 4 petals be divisible by the number of flowers with 5 petals?