Found problems: 85335
In the following diagram, $AB = 1$. The radius of the circle with center $C$ can be expressed as $\frac{p}{q}$. Determine $p+q$.
[i]2022 CCA Math Bonanza Lightning Round 3.2[/i]
Let $ AD$ be height of triangle $ \triangle ABC$ and $ R$ circumradius. Denote by $ E$ and $ F$ feet of perpendiculars from point $ D$ to sides $ AB$ and $ AC$.
If $ AD\equal{}R\sqrt{2}$, prove that circumcenter of triangle $ \triangle ABC$ lies on line $ EF$.
A [b] bijective[/b] mapping from a plane to itself maps every circle to a circle.
Prove that it maps every line to a line.
An acute triangle has area $84$ and perimeter $42$, with each side being at least $10$ units long. Let $S$ be the set of points that are within $5$ units of some vertex of the triangle. What fraction of the area of $S$ lies outside the triangle?
[i]Proposed by Nathan Ramesh
Can one cover a cube by three paper triangles (without overlapping)?
If $\log_{10}2=a$ and $\log_{10}3=b$, then $\log_{5}12=?$
${{ \textbf{(A)}\ \frac{a+b}{a+1} \qquad\textbf{(B)}\ \frac{2a+b}{a+1} \qquad\textbf{(C)}\ \frac{a+2b}{1+a} \qquad\textbf{(D)}\ \frac{2a+b}{1-a} }\qquad\textbf{(E)}\ \frac{a+2b}{1-a}} $
Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.
The set $M= \{1;2;3;\ldots ; 29;30\}$ is divided in $k$
subsets such that if $a+b=n^2, (a,b \in M, a\neq b, n$ is an
integer number $)$, then $a$ and $b$ belong different subsets.
Determine the minimum value of $k$.
Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$
Mia has $2$ green sticks of $\textbf{3cm}$ each one, $2$ blue sticks of $\textbf{4cm}$ each one and $2$ red sticks of $\textbf{5cm}$ each one. She wants to make a triangle using the $6$ sticks as it´s perimeter, all at once and without overlapping them. How many non-congruent triangles can make?
Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that:
$\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$
Find all natural numbers, which cannot be expressed in the form $\frac{a}{b}+\frac{a+1}{b+1}$ where $a,b\in \mathbb{N}$.
Let $b$ be a positive integer such that $\gcd(b,6)=1$. Show that there are positive integers $x$ and $y$ such that $\frac1x+\frac1y=\frac3b$ if and only if $b$ is divisible by some prime number of form $6k-1$.
Let $\vartriangle ABC$ be a triangle such that $\angle ABC = 30^o$, $\angle ACB = 15^o$. Let $M$ be midpoint of segment $BC$ and point $N$ lies on segment $MC$, such that the length of $NC$ is equal to length of $AB$. Proce that $AN$ is the bisector of the angle $\angle MAC$.
[img]https://cdn.artofproblemsolving.com/attachments/2/7/4c554b53f03288ee69931fdd2c6fbd3e27ab13.png[/img]
A mathematician investigates methods of finding area of a convex quadrilateral obtains the following formula for the area $A$ of a quadrilateral with consecutive sides $a,b,c,d$:
$A =\frac{a+c}{2}\frac{b+d}{2}$ (1) and $A = \sqrt{(p-a)(p-b)(p-c)(p-d)}$ (2) where $p = (a+b+c+d)/2$.
However, these formulas are not valid for all convex quadrilaterals. Prove that (1) holds if and only if the quadrilateral is a rectangle, while (2) holds if and only if the quadrilateral is cyclic.
The number of all positive integers $n$ such that $n + s(n) = 2016$, where $s(n)$ is the sum of all digits of $n$ is
(A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.
Determine the maximum value of the sum
\[
S=\sum_{n=1}^{\infty}\frac{n}{2^n}(a_1 a_2 \dots a_n)^{\frac{1}{n}}
\]
over all sequences $a_1,a_2,a_3,\dots$ of nonnegative real numbers satisfying
\[
\sum_{k=1}^{\infty}a_k=1.
\]
Point $P$ is $9$ units from the center of a circle of radius $15$. How many different chords of the circle contain $P$ and have integer lengths?
$ \textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 29 $
The license plate of each automobile in Iran consists of a two-digit and a three-digit number as well as a letter of the Persian alphabet. The digit $0$ is not used in the two numbers. To each license plate, we assign the product of the two numbers on it. For example, if the two numbers are $12$ and $365$ on a license plate, the assigned number would be $12 \times 365 = 4380$. What is the average of all the assigned numbers to all possible license plates?
We are given a trapezoid $ABCD$ with $AB \parallel CD$, $CD=2AB$ and $DB \perp BC$. Let $E$ be the intersection of lines $DA$ and $CB$, and $M$ be the midpoint of $DC$.
(a) Prove that $ABMD$ is a rhombus.
(b) Prove that triangle $CDE$ is isosceles.
(c) If $AM$ and $BD$ meet at $O$, and $OE$ and $AB$ meet at $N,$ prove that the line $DN$ bisects segment $EB$.
Let $p$ be a prime number. Determine all positive integers $a$ such that the sequence $(a_n)_{n\geq 0}$ defined by $a_0=a$ and $a_{n+1}=pa_n-(p-1)\lfloor \sqrt[p]{a_ n} \rfloor^p$, for every $n\geq0$, is eventually constant.
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$:
(i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$
(ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$
[i]Proposed by Ashwin Sah[/i]
$\triangle KWU$ is an equilateral triangle with side length $12$. Point $P$ lies on minor arc $\overarc{WU}$ of the circumcircle of $\triangle KWU$. If $\overline{KP} = 13$, find the length of the altitude from $P$ onto $\overline{WU}$.
[i]Proposed by Bradley Guo[/i]
$\triangle ABC$ is an equilateral triangle. $D$ is a point on $\overline{AC}$, and $E$ is a point on $\overline{BD}$. Let $P$ and $Q$ be the circumcenters of $\triangle ABD$ and $\triangle AED$, respectively. Prove that $ \triangle EPQ$ is an equilateral triangle if and only if $ \overline{AB} \perp \overline{CE}$.
Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.