Found problems: 85335
On the market one seller is selling watermelons, melons and young corn cobs. Total number of watermelons, melons and corn cobs is $239$. One buyer bought $\frac{2}{3}$ of all watermelons, $\frac{3}{5}$ of all melons and $\frac{5}{7}$ of all corn cobs. Other buyer bought $\frac{1}{13}$ of all watermelons, $\frac{1}{4}$ of all melons and $\frac{1}{5}$ of all corn cobs. How many pieces in total bought second buyer and how many seller had at the beggining of each watermelons, melons and corn cobs?
Let $ABC$ be an acute triangle and let $\omega$ be its circumcircle. Let the tangents to $\omega$ through $B,C$ meet each other at point $P$. Prove that the perpendicular bisector of $AB$ and the parallel to $AB$ through $P$ meet at line $AC$.
Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?
Find all pairs of naturals $a,b$ with $a <b$, such that the sum of the naturals greater than $a$ and less than $ b$ equals $1998$.
Let $P(x)$ be a polynomial of degree at most $2018$ such that $P(i)=\binom{2018}i$ for all integer $i$ such that $0\le i\le 2018$. Find the largest nonnegative integer $n$ such that $2^n\mid P(2020)$.
[i]Proposed by Michael Ren
The radii of the circumscribed circle and the inscribed circle of a regular $n$-gon, $n\ge 3$ are denoted by $R_n$ and $r_n$, respectively. Prove that
\[\frac{r_n}{R_n}\ge\left(\frac{r_{n+1}}{R_{n+1}}\right)^2.\]
An equilateral triangle is divided into $9000000$ congruent equilateral triangles by lines parallel to its sides. Each vertex of the small triangles is coloured in one of three colours. Prove that there exist three points of the same colour being the vertices of a triangle with its sides parallel to the lines of the original triangle.
Find all functions $f:\ \mathbb{R}\rightarrow\mathbb{R}$ such that for any real number $x$ the equalities are true:
$f\left(x+1\right)=1+f(x)$ and $f\left(x^4-x^2\right)=f^4(x)-f^2(x).$
[url=http://matol.kz/comments/3373/show]source[/url]
Does there exist an operation $*$ on the set of all integers such that the following conditions hold simultaneously:
$(1)$ for all integers $x,y,z$, $(x*y)*z=x*(y*z)$;
$(2)$ for all integers $x$ and $y$, $x*x*y=y*x*x=y$?
Let $P$ be an arbitrary variable point in the plane of a triangle $ABC. A_1$ is the projection of $P$ onto $BC, A_2$ is the midpoint of line segment $PA_1, A_2P$ meets $BC$ at $A_3, A_4$ is the reflection of $P$ about $A_3$. Prove that $PA_4$ has a fixed point.
Trần Quang Hùng
[b]a)[/b] Solve the system
$$ \left\{\begin{matrix} 3^y-4^x=11\\ \log_4{x} +\log_3 y =3/2\end{matrix}\right. $$
[b]b)[/b] Solve the equation $ \quad 9^{\log_5 (x-2)} -5^{\log_9 (x+2)} = 4. $
For how many nonnegative integer values of $k$ does the equation $7x^2 +kx +11 = 0$ have no real solutions?
Start with a positive integer. Double it, subtract $4,$ halve it, then subtract the original integer to get $x.$ What is $x?$
Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$, where $r$ is a positive integer less than $15$ and relatively prime to $15$, satisfy
\[x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.\]
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $AC = 15$. Let$ D$ and $E$ be the feet of the altitudes from $A$ and $B$, respectively. Find the circumference of the circumcircle of $\vartriangle CDE$
In the triangle $ABC$, the bisector of angle $\angle B$ meets $AC$ at $D$ and the bisector of angle $\angle C$ meets $AB$ at $E$. The bisectors meet each other at $O$. Furthermore, $OD = OE$. Prove that either $ABC$ is isosceles or $\angle BAC = 60^\circ$.
A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$.
Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.
Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?
Let $A$ be the set of all sequences from 0’s or 1’s with length 4. What’s the minimal number of sequences that can be chosen, so that an arbitrary sequence from $A$ differs at most in 1 position from one of the chosen?
The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
[asy]
size(200);
defaultpen(linewidth(0.7));
path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin;
path laceR=reflect((75,0),(75,-240))*laceL;
draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray);
for(int i=0;i<=3;i=i+1)
{
path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5);
unfill(circ1); draw(circ1);
unfill(circ2); draw(circ2);
}
draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]
Let $k$ and $s$ be positive integers such that $s<(2k + 1)^2$. Initially, one cell out of an $n \times n$ grid is coloured green. On each turn, we pick some green cell $c$ and colour green some $s$ out of the $(2k + 1)^2$ cells in the $(2k + 1) \times (2k + 1)$ square centred at $c$. No cell may be coloured green twice. We say that $s$ is $k-sparse$ if there exists some positive number $C$ such that, for every positive integer $n$, the total number of green cells after any number of turns is always going to be at most $Cn$. Find, in terms of $k$, the least $k$-sparse integer $s$.
[I]Proposed by Nikolai Beluhov.[/i]
Are there positive integers $m,n$ such that there exist at least $2012$ positive integers $x$ such that both $m-x^2$ and $n-x^2$ are perfect squares?
[i]David Yang.[/i]
Is there an equilateral triangle in the coordinate plane, both coordinates of each vertex of which are integers?
Circles $\omega_a, \omega_b, \omega_c$ have centers $A, B, C$, respectively and are pairwise externally tangent at points $D, E, F$ (with $D\in BC, E\in CA, F\in AB$). Lines $BE$ and $CF$ meet at $T$. Given that $\omega_a$ has radius $341$, there exists a line $\ell$ tangent to all three circles, and there exists a circle of radius $49$ tangent to all three circles, compute the distance from $T$ to $\ell$.
[i]Proposed by Andrew Gu.[/i]