Found problems: 85335
On the side $AB$ of a triangle $ABC$ is chosen a point $P$. Let $Q$ be the midpoint of $BC$ and let $CP$ and $AQ$ intersect at $R$. If $AB + AP = CP$, prove that $CR = AB$.
Two circles with centers $ S_{1}$ and $ S_{2}$ are externally tangent at point $ K$. These circles are also internally tangent to circle $ S$ at points $ A_{1}$ and $ A_{2}$, respectively. Denote by $ P$one of the intersection points of $ S$ and common tangent to $ S_{1}$ and $ S_{2}$ at $ K$.Line $ PA_{1}$ intersects $ S_{1}$ at $ B_{1}$ while $ PA_{2}$ intersects $ S_{2}$ at $ B_{2}$.
Prove that $ B_{1}B_{2}$ is common tangent of circles $ S_{1}$ and $ S_{2}$.
Find all positive real numbers $\lambda$ such that for all integers $n\geq 2$ and all positive real numbers $a_1,a_2,\cdots,a_n$ with $a_1+a_2+\cdots+a_n=n$, the following inequality holds:
$\sum_{i=1}^n\frac{1}{a_i}-\lambda\prod_{i=1}^{n}\frac{1}{a_i}\leq n-\lambda$.
On the party every boy gave $1$ candy to every girl and every girl gave $1$ candy to every boy. Then every boy ate $2$ candies and every girl ate $3$ candies. It is known that $\frac 14$ of all candies was eaten. Find the greatest possible number of children on the party.
For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that
\[\sum_{t \mid n} d(t)^3\]is divisible by $35$?
Segments $AB$ and $CD$ of length $1$ intersect at point $O$ and angle $AOC$ is equal to sixty degrees. Prove that $AC+BD \ge 1$.
Draw on the plane a set of points whose coordinates $(x,y)$ satisfy the equation $x^3 + y^3 = x^2y^2 + xy$.
$n$ players took part in badminton tournament, where $n$ is positive and odd integer. Each two players played two matches with each other. There were no draws. Each player has won as many matches as he has lost. Prove that you can cancel half of the matches s.t. each player still has won as many matches as he has lost.
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
Solve the equation
\[ 3^x \minus{} 5^y \equal{} z^2.\]
in positive integers.
[i]Greece[/i]
Let $ABC$ be a triangle with medians $m_a$ , $m_b$, $m_c$. Prove that:
(a) There is a triangle with side lengths $m_a$ ,$m_b$, $m_c$.
(b) This triangle is similar to $ABC$ if and only if the squares of the side lengths of triangle $ABC$ form an arithmetical sequence.
Find all functions f : $R \to R $such that for all $x, y \in R$:
$$f(x + yf(x)) = f(xf(y)) - x + f(y + f(x)).$$
Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?
$ \textbf{(A)}\ \left(\frac{1}{12}\right)^{12}\qquad
\textbf{(B)}\ \left(\frac{1}{6}\right)^{12}\qquad
\textbf{(C)}\ 2\left(\frac{1}{6}\right)^{11}\qquad
\textbf{(D)}\ \frac{5}{2}\left(\frac{1}{6}\right)^{11}\qquad
\textbf{(E)}\ \left(\frac{1}{6}\right)^{10}$
If $y=(\log_23)(\log_34)\cdots(\log_n[n+1])\cdots(\log_{31}32)$ then
$\textbf{(A) }4<y<5\qquad\textbf{(B) }y=5\qquad\textbf{(C) }5<y<6\qquad$
$\textbf{(D) }y=6\qquad \textbf{(E) }6<y<7$
If the Highest Common Divisor of $ 6432$ and $ 132$ is diminished by $ 8$, it will equal:
$ \textbf{(A)}\ \minus{}6 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ \minus{}2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.
Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.
In convex cyclic quadrilateral $ABCD$, we know that lines $AC$ and $BD$ intersect at $E$, lines $AB$ and $CD$ intersect at $F$, and lines $BC$ and $DA$ intersect at $G$. Suppose that the circumcircle of $\triangle ABE$ intersects line $CB$ at $B$ and $P$, and the circumcircle of $\triangle ADE$ intersects line $CD$ at $D$ and $Q$, where $C,B,P,G$ and $C,Q,D,F$ are collinear in that order. Prove that if lines $FP$ and $GQ$ intersect at $M$, then $\angle MAC = 90^\circ$.
[i]Proposed by Kada Williams[/i]
Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.
Let $ABC$ be a triangle. Draw circles $\omega_A$, $\omega_B$, and $\omega_C$ such that $\omega_A$ is tangent to $AB$ and $AC$, and $\omega_B$ and $\omega_C$ are defined similarly. Let $P_A$ be the insimilicenter of $\omega_B$ and $\omega_C$. Define $P_B$ and $P_C$ similarly. Prove that $AP_A$, $BP_B$, and $CP_C$ are concurrent.
[i]Tom Lu.[/i]
In a triangle two altitudes are not smaller than the sides on to which they are dropped. Find the angles of the triangle.
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.
Let $ABC$ be an acute triangle, with $AB <AC$. Let $M$ be the midpoint of $AB$, $H$ the foot of the altitude from $A$, and $Q$ be point on side $AC$ such that $\angle ABQ = \angle BCA$. Show that the circumcircles of the triangles $ABQ$ and $BHM$ are tangent.
The planar diagram below, with equilateral triangles and regular hexagons, sides length $2$ cm, is folded along the dashed edges of the polygons, to create a closed surface in three-dimensional Euclidean spaces. Edges on the periphery of the planar diagram are identified (or glued) with precisely one other edge on the periphery in a natural way. Thus, for example, $BA$ will be joined to $QP$ and $AC$ will be joined to $DC$. Find the volume of the three-dimensional region enclosed by the resulting surface.
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMy9jL2ZiZjc1ZjY5Nzk5YzRiMjhjODNlZDBiZjU1MzljYzZkNTVhOGQ3LnBuZw==&rn=VlRSTUMgMjAxNS5wbmc=[/img]
Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each other at $D$. Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$. Show that the line $KD$ bisects the angle $\angle EKF$.