Found problems: 85335
Raquel painted $650$ points in a circle with a radius of $16$ cm. Shows that there is a circular crown with $2$ cm of inner radius and $3$ cm of outer radius that contain at least $10$ of these points.
Three real numbers $ a,b,c$ satisfy the equations $ a\plus{}b\plus{}c\equal{}3, a^2\plus{}b^2\plus{}c^2\equal{}9, a^3\plus{}b^3\plus{}c^3\equal{}24.$ Find $ a^4\plus{}b^4\plus{}c^4$.
One player conceals a $10$ or $20$ copeck coin, and the other guesses its value. If he is right he gets the coin, if wrong he pays $15$ copecks. Is this a fair game? What are the optimal mixed strategies for both players?
Determine whether there exist $100$ disks $D_2,D_3,\ldots ,D_{101}$ in the plane such that the following conditions hold for all pairs $(a,b)$ of indices satisfying $2\le a< b\le 101$:
[list]
[*] If $a|b$ then $D_a$ is contained in $D_b$.
[*] If $\gcd (a,b)=1$ then $D_a$ and $D_b$ are disjoint.
[/list]
(A disk $D(O,r)$ is a set of points in the plane whose distance to a given point $O$ is at most a given positive real number $r$.)
If $a\ge 0$ prove that $a^4+ a^3-10 a^2+9 a+4>0$.
Determine all pairs of real numbers $(x,y)$ satisfying
\begin{align*} x^3+9x^2y&=10,\\
y^3+xy^2 &=2.
\end{align*}
Suppose the real number $\lambda \in \left( 0,1\right),$ and let $n$ be a positive integer. Prove that the modulus of all the roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{k}$$ are $1.$
Do there exist 2021 points with integer coordinates on the plane such that the pairwise distances between them are pairwise distinct consecutive integers?
Two points $E$ and $F$ are randomly chosen in the interior of unit square $ABCD$. Let the line through $E$ parallel to $AB$ hit $AD$ at $E_1$, the line through $E$ parallel to $AD$ hit $CD$ at $E_2$, the line through $F$ parallel to $AB$ hit $BC$ at $F_1$, and the line through $F$ parallel to $BC$ hit $AB$ at $F_2$. The expected value of the overlap of the areas of rectangles $EE_1DE_2$ and $FF_1BF_2$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
[i]Proposed by Kevin Zhao[/i]
In triangle $ABC$, the altitude through $B$ intersects $AC$ at $E$ and the altitude through $C$ intersects $AB$ at $F$. Point $T$ is such that $AETF$ is a parallelogram and points $ A$ ,$T$ lie on different half-planes wrt the line $EF$. Point $D$ is such that $ABDC$ is a parallelogram and points $ A$ ,$D$ lie in different half-planes wrt line $BC$. Prove that $T, D$ and the orthocenter of $ABC$ are collinear.
Suppose that $k \ge 2$ and $n_{1}, n_{2}, \cdots, n_{k}\ge 1$ be natural numbers having the property \[n_{2}\; \vert \; 2^{n_{1}}-1, n_{3}\; \vert \; 2^{n_{2}}-1, \cdots, n_{k}\; \vert \; 2^{n_{k-1}}-1, n_{1}\; \vert \; 2^{n_{k}}-1.\] Show that $n_{1}=n_{2}=\cdots=n_{k}=1$.
Find the minimum value of $\int_0^{\pi} (a\sin x+b\sin 2x+c\sin 3x-x)^2\ dx$
Find an $n<100$ such that $n\cdot 2^n-1$ is prime. Score will be $n-5$ for correct $n$, $5-n$ for incorrect $n$ ($0$ points for answer $<5$)
Prove that $$\frac{2}{3}+\frac{4}{5}+\dots +\frac{2010}{2011}$$ is not an integer.
Real numbers $a, b, c$ are such that
$$a^2+c-bc = b^2+a-ca = c^2+b-ab$$
Does it follow that $a=b=c$?
[i]Proposed by Mykhailo Shtandenko[/i]
Compute all ordered triples $(x, y, z)$ of real numbers satisfying the following system of equations:
$$xy + z = 40$$
$$xz + y = 51$$
$$x + y + z = 19.$$
Given is a real number $a \in (0,1)$ and positive reals $x_0, x_1, \ldots, x_n$ such that $\sum x_i=n+a$ and $\sum \frac{1}{x_i}=n+\frac{1}{a}$. Find the minimal value of $\sum x_i^2$.
On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$.
$(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$.
$(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.
Let $ABC$ be a triangle, let $M$ be the midpoint of $AB$, and let $N$ be the midpoint of $CM$. Let $X$ be a point satisfying both $\angle XMC = \angle MBC$ and $\angle XCM = \angle MCB$ such that $X$ and $B$ lie on opposite sides of $CM$. Let $\omega$ be the circumcircle of triangle $AMX$.
$(a)$ Show that $CM$ is tangent to $\omega$.
$(b)$ Show that the lines $NX$ and $AC$ intersect on $\omega$
Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.
For all positive numbers $a, b, c$ satisfying $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$, prove that:
$$ \frac{a}{a+bc} + \frac{b}{b+ca} + \frac{c}{c+ab} \geq \frac{3}{4} .$$
In a tournament there are six teams that play each other twice. A team earns 3 points for a win, 1 point for a draw, and 0 points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?
$\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30$
The triangle $ABC$ with $AB < AC$ has an altitude $AD$. The points $E$ and $A$ lie on opposite sides of $BC$, with $E$ on the circumcircle of $ABC$. Furthermore, $AD = DE$ and $\angle ADO=\angle CDE$, where $O$ is the circumcentre of $ABC$. Determine $\angle BAC$.
Let $\sqrt 3 = 1.b_1b_2b_3 \dots _{(2)}$ be the binary representation of $\sqrt 3$. Prove that for any positive integer $n$, at least one of the digits $b_n$, $b_{n+1}$, $\dots$, $b_{2n}$ equals $1$.
$C_1:\frac{x^2}{a^2}+y^2=1(a>0),C_2:y^2=2(x+m)$, one intersection of $C_1$ and $C_2$ is $P$, and $P$ is above the $x$-axis.
[b](a)[/b] Find the range value of $m$ (express with $a$).
[b](b)[/b] $O(0,0),A(-a,0)$. If $0<a<\frac{1}{2}$, find the maximum value of $S_{\triangle OAP}$.