Found problems: 85335
Let $d$ be a positive number. On the parabola, whose equation has the coefficient $1$ at the quadratic term, points $A, B$ and $C$ are chosen in such a way that the difference of the $x$-coordinates of points $A$ and $B$ is $d$ and the difference of the $x$-coordinates of points $B$ and $C$ is also $d$. Find the area of the triangle $ABC$.
A regular tetrahedron has side $L$. What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$?
Given $10$ points arranged in a equilateral triangular grid of side length $4$, how many ways are there to choose two distinct line segments, with endpoints on the grid, that intersect in exactly one point (not necessarily on the grid)?
Find all pairs $(k,n)$ of positive integers such that \[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]
[i]Proposed by Gabriel Chicas Reyes, El Salvador[/i]
Let $ABC$ be a triangle, $D$ be a point on side $BC$, and let $\mathcal{O}$ be the circumcircle of triangle $ABC$. Show that the circles tangent to $\mathcal{O},AD,BD$ and to $\mathcal{O},AD,DC$ are tangent to each other if and only if $\angle BAD=\angle CAD$.
[i]Dan Branzei[/i]
In a ABCD cyclic quadrilateral 4 points K, L ,M, N are taken on AB , BC , CD and DA , respectively such that KLMN is a parallelogram. Lines AD, BC and KM have a common point. And also lines AB, DC and NL have a common point. Prove that KLMN is rhombus.
Given six points on a circle, $A, a, B, b, C, c$, show that the Pascal lines of the hexagrams $AaBbCc, AbBcCa, AcBaCb$ are concurrent.
When $ 15$ is appended to a list of integers, the mean is increased by $ 2$. When $ 1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $ 1$. How many integers were in the original list?
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
Given the natural numbers $a$ and $b$, with $1 \le a <b$, prove that there exist natural numbers $n_1<n_2< ...<n_k$, with $k \le a$ such that $$\frac{a}{b}=\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}$$
Sherlock and Mycroft are playing Battleship on a $4\times4$ grid. Mycroft hides a single $3\times1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser?
Let $x, y$ and $z$ be positive real numbers such that $x \geq y+z$.
Proof that
$$\frac{x+y}{z} + \frac{y+z}{x} +\frac{z+x}{y} \geq 7$$
When does equality occur?
(Walther Janous)
The [i]fibboican[/i] sequence $a_1,\ a_2,\ \dots$, is defined by $a_1 = a_2 = 1$, and for integers $k \geq 3$,
[list]
[*] $a_k = a_{k-1} + a_{k-2}$ if $k$ is odd
[*] $\frac {1}{a_k} = \frac {1}{a_{k-1}} + \frac {1}{a_{k-2}}$ if $k$ is even.
[/list]
Prove that, for each integer $m\ge 1$, the numerator of $a_m$ (when written in simplest form) is a power of $2$.
[i]Eric Shen (CAN)[/i]
Let $ABCD$ be a square and let $S$ be the point of intersection of its diagonals $AC$ and $BD$. Two circles $k,k'$ go through $A,C$ and $B,D$; respectively. Furthermore, $k$ and $k'$ intersect in exactly two different points $P$ and $Q$. Prove that $S$ lies on $PQ$.
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$, and the afternoon class’s mean score is $70$. The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$. What is the mean of the scores of all the students?
$\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78$
Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$.
[i]Proposed by Evan Chen[/i]
At a position $O$ of an airport in a plateau there is a gun which can rotate arbitrarily. Two tanks moving along two given segments $AB$ and $CD$ attack the airport. Determine, by a ruler and a compass, the reach of the gun, knowing that the total length of the parts of the trajectories of the two tanks reachable by the gun is equal to a given length $\ell$.
Let $a$ be a fixed integer. Find all integer solutions $x, y, z$ of the system
\[5x + (a + 2)y + (a + 2)z = a,\]\[(2a + 4)x + (a^2 + 3)y + (2a + 2)z = 3a - 1,\]\[(2a + 4)x + (2a + 2)y + (a^2 + 3)z = a + 1.\]
Assume that a face of a convex polyhedron $ P$ has a common edge with every other face. Show that there exists a simple closed polygon that consists of edges of $ P$ and passes through all vertices.
[i]L .Lovasz[/i]
Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2+y^2+z^2$.
Given triangle $ PQR$ with $ \overline{RS}$ bisecting $ \angle R$, $ PQ$ extended to $ D$ and $ \angle n$ a right angle, then:
[asy]path anglemark2(pair A, pair B, pair C, real t=8, bool flip=false)
{
pair M,N;
path mark;
M=t*0.03*unit(A-B)+B;
N=t*0.03*unit(C-B)+B;
if(flip)
mark=Arc(B,t*0.03,degrees(C-B)-360,degrees(A-B));
else
mark=Arc(B,t*0.03,degrees(A-B),degrees(C-B));
return mark;
}
unitsize(1.5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
pair P=(0,0), R=(3,2), Q=(4,0);
pair S0=bisectorpoint(P,R,Q);
pair Sp=extension(P,Q,S0,R);
pair D0=bisectorpoint(R,Sp), Np=midpoint(R--Sp);
pair D=extension(Np,D0,P,Q), M=extension(Np,D0,P,R);
draw(P--R--Q);
draw(R--Sp);
draw(P--D--M);
draw(anglemark2(Sp,P,R,17));
label("$p$",P+(0.35,0.1));
draw(anglemark2(R,Q,P,11));
label("$q$",Q+(-0.17,0.1));
draw(anglemark2(R,Np,D,8,true));
label("$n$",Np+(+0.12,0.07));
draw(anglemark2(R,M,D,13,true));
label("$m$",M+(+0.25,0.03));
draw(anglemark2(M,D,P,29));
label("$d$",D+(-0.75,0.095));
pen f=fontsize(10pt);
label("$R$",R,N,f);
label("$P$",P,S,f);
label("$S$",Sp,S,f);
label("$Q$",Q,S,f);
label("$D$",D,S,f);[/asy]$ \textbf{(A)}\ \angle m \equal{} \frac {1}{2}(\angle p \minus{} \angle q) \qquad \textbf{(B)}\ \angle m \equal{} \frac {1}{2}(\angle p \plus{} \angle q)$
$ \textbf{(C)}\ \angle d \equal{} \frac {1}{2} (\angle q \plus{} \angle p) \qquad \textbf{(D)}\ \angle d \equal{} \frac {1}{2}\angle m \qquad \textbf{(E)}\ \text{none of these is correct}$
Given a line segment $AB=7$, $C$ is constructed on $AB$ so that $AC=5$. Two equilateral triangles are constructed on the same side of $AB$ with $AC$ and $BC$ as a side. Find the length of the segment connecting their two circumcenters.
Consider the isosceles right triangle$ ABC, \angle A = 90^o$, and point $D \in (AB)$ such that $AD = \frac13 AB$. In the half-plane determined by the line $AB$ and point $C$ , consider a point $E$ such that $\angle BDE = 60^o$ and $\angle DBE = 75^o$. Lines $BC$ and $DE$ intersect at point $G$, and the line passing through point $G$ parallel to the line $AC$ intersects the line $BE$ at point $H$. Prove that the triangle $CEH$ is equilateral.
Given two points, $P$ and $Q$, on the same side of a line $L$, the problem is to find a third point $R$ so that $PR+ RQ+RS$ is minimal, where $S$ is the unique point on $L$ such that $RS$ is perpendicular to $L.$ Consider all cases.
Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.
Does there exist a positive integer $n$ such that all its digits (in the decimal system) are greather than 5, while all the digits of $n^2$ are less than 5?