Found problems: 85335
Given an acute triangle $ABC$, in which $\angle BAC = 60^o$. On the sides $AC$ and $AB$ take the points $T$ and $Q$, respectively, such that $CT = TQ = QB$. Prove that the center of the inscribed circle of triangle $ATQ$ lies on the side $BC$.
(Dmitry Shvetsov)
Let $H$ be the orthocenter of triangle $ABC$. Let $D$ and $E$ be points on $AB$ and $AC$ such that $DE$ is parallel to $CH$. If the circumcircle of triangle $BDH$ passes through $M$, the midpoint of $DE$, then prove that $\angle ABM=\angle ACM$
In a plane, we are given $ 100 $ circles with radius $ 1 $ so that the area of any triangle whose vertices are circumcenters of those circles is at most $ 100 $. Prove that one may find a line that intersects at least $ 10 $ circles.
Determine all the two-digit numbers that satisfy the following condition: if we multiply their two digits, the result is equal to half the number. For example, $24$ does not satisfy the condition, because $2 \times 4 = 8$ and $8$ is not half of $24$.
In order to pass $ B$ going $ 40$ mph on a two-lane highway, $ A$, going $ 50$ mph, must gain $ 30$ feet. Meantime, $ C$, $ 210$ feet from $ A$, is headed toward him at $ 50$ mph. If $ B$ and $ C$ maintain their speeds, then, in order to pass safely, $ A$ must increase his speed by:
$ \textbf{(A)}\ \text{30 mph} \qquad
\textbf{(B)}\ \text{10 mph} \qquad
\textbf{(C)}\ \text{5 mph} \qquad
\textbf{(D)}\ \text{15 mph} \qquad
\textbf{(E)}\ \text{3 mph}$
Let $S=\{a+np : n=0,1,2,3,... \}$ where $a$ is a positive integer and $p$ is a prime. Suppose there exist positive integers $x$ and $y$ st $x^{41}$ and $y^{49}$ are in $S$. Determine if there exists a positive integer $z$ st $z^{2009}$ is in $S$.
A four-digit positive integer is called [i]virtual[/i] if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$.
A polygon can be transformed into a new polygon by making a straight cut, which creates two new pieces each with a new edge. One piece is then turned over and the two new edges are reattached. Can repeated transformations of this type turn a square into a triangle?
Prove that among any 18 consecutive positive 3-digit numbers, there is at least one that is divisible by the sum of its digits!
Find the number of ordered quadruples $(x,y,z,w)$ of integers with $0\le x,y,z,w\le 36$ such that ${{x}^{2}}+{{y}^{2}}\equiv {{z}^{3}}+{{w}^{3}}\text{ (mod 37)}$.
The sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{n}= 1+2^{2}+3^{3}+\cdots+n^{n}.\] Prove that there are infinitely many $n$ such that $a_{n}$ is composite.
Eight consecutive numbers are chosen from the Fibonacci sequence $1, 2, 3, 5, 8, 13, 21,...$. Prove that the sequence does not contain the sum of chosen numbers.
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?
Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that \[ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). \][i]Proposed by AJ Dennis[/i]
let $\omega_1$ be a circle with $O_1$ as its center , let $\omega_2$ be a circle passing through $O_1$ with center $O_2$ let $A$ be one of the intersection of $\omega_1$ and $\omega_2$ let $x$ be a line tangent line to $\omega_1$ passing from $A$ let $\omega_3$ be a circle passing through $O_1,O_2$ with its center on the line $x$ and intersect $\omega_2$ at $P$ (not $O_1$) prove that the reflection of $P$ through $x$ is on $\omega_1$
Given $a_0, a_1, ... , a_n$. It is known that $$a_0=a_n=0, a_{k-1}-2a_k+a_{k+1}\ge 0$$ for all $k = 1, 2, ... , k-1$.Prove that all the numbers are nonnegative.
$ABC$ is an obtuse triangle satisfying $\angle A>90^\circ$, and its circumcenter $O$ and circumcircle $\omega_1$. Let $\omega_2$ be a circle passing $C$ with center $B$. $\omega_2$ meets $BC$ at $D$. $\omega_1$ meets $AD$ and $\omega_2$ at $E$ and $F(\neq C)$, respectively. $AF$ meets $\omega_2$ at $G(\neq F)$. Prove that the intersection of $CE$ and $BG$ lies on the circumcircle of $AOB$.
The total number of languages used in KAUST is $n$. For each positive integer $k \le n$, let $A_k$ be the set of all those people in KAUST who can speak at least $k$ languages; and let $B_k$ be the set of all people $P$ in KAUST with the property that, for any $k$ pairwise different languages (used in KAUST), $P$ can speak at least one of these $k$ languages. Prove that
(a) If $2k \ge n + 1$ then $A_k \subseteq B_k$
(b) If $2k \le n + 1$ then $A_k \supseteq B_k.$
Nguyễn Duy Thái Sơn
Let $\mathcal{S}$ be a set of integers. Given a real positive $r$, we say that $\mathcal{S}$ is a $r$-discerning, if for any pair $m, n > 1$ of distinct integers such that $\left| \frac{ m-n}{m+n} \right| < r$, there exists $a \in \mathcal{S}$ and $k \geq 1$ such that $a^k$ divides $m$ but not $n$, or $a^k$ divides $n$ but not $m$
1. Show that for every $r > 0$ every $r$-discerning set contains an infinite number of primes.
2. For every $r > 0$ determine the maximal possible cardinality of $\mathcal{P} \backslash \mathcal{S}$ where $\mathcal{P}$ is the set of primes and $\mathcal{S} \subseteq \mathcal{P}$ is a $r$-discerning set.
Determine the smallest possible number $n> 1$ such that there exist positive integers $a_{1}, a_{2}, \ldots, a_{n}$ for which ${a_{1}}^{2}+\cdots +{a_{n}}^{2}\mid (a_{1}+\cdots +a_{n})^{2}-1$.
A circle with diameter $23$ is cut by a chord $AC$. Two different circles can be inscribed between the large circle and $AC$. Find the sum of the two radii.
Let $k$ be a circle with centre $O$. Let $AB$ be a chord of this circle with midpoint $M\neq O$. The tangents of $k$ at the points $A$ and $B$ intersect at $T$. A line goes through $T$ and intersects $k$ in $C$ and $D$ with $CT < DT$ and $BC = BM$. Prove that the circumcentre of the triangle $ADM$ is the reflection of $O$ across the line $AD$.
Signals used for communication are binary sequences of length $10$. Unfortunately, the receiving device got broken so that it cannot distinguish between two signals unless those differ in more than five places. What is the largest possible number of signals that can still be used to prevent ambiguities?
Function $f$ is defined on the positive integers by $f(1)=1$, $f(2)=2$ and
$$f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\enspace\text{for }n\ge1.$$
(a) Prove that $f(n+1)-f(n)\in\{0,1\}$ for each $n\ge1$.
(b) Show that if $f(n)$ is odd then $f(n+1)=f(n)+1$.
(c) For each positive integer $k$ find all $n$ for which $f(n)=2^{k-1}+1$.
Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?
[asy]
//diagram by pog
size(8.5cm);
usepackage("mathptmx");
defaultpen(mediumgray*0.5+gray*0.5+linewidth(0.63));
add(grid(6,6));
label(scale(0.7)*"$1$", (1,-0.3), black);
label(scale(0.7)*"$2$", (2,-0.3), black);
label(scale(0.7)*"$3$", (3,-0.3), black);
label(scale(0.7)*"$4$", (4,-0.3), black);
label(scale(0.7)*"$5$", (5,-0.3), black);
label(scale(0.7)*"$1$", (-0.3,1), black);
label(scale(0.7)*"$2$", (-0.3,2), black);
label(scale(0.7)*"$3$", (-0.3,3), black);
label(scale(0.7)*"$4$", (-0.3,4), black);
label(scale(0.7)*"$5$", (-0.3,5), black);
label(scale(0.75)*rotate(90)*"Price (dollars)", (-1,3.2), black);
label(scale(0.75)*"Weight (ounces)", (3.2,-1), black);
dot((1,1.2),black);
dot((1,1.7),black);
dot((1,2),black);
dot((1,2.8),black);
dot((1.5,2.1),black);
dot((1.5,3),black);
dot((1.5,3.3),black);
dot((1.5,3.75),black);
dot((2,2),black);
dot((2,2.9),black);
dot((2,3),black);
dot((2,4),black);
dot((2,4.35),black);
dot((2,4.8),black);
dot((2.5,2.7),black);
dot((2.5,3.7),black);
dot((2.5,4.2),black);
dot((2.5,4.4),black);
dot((3,2.5),black);
dot((3,3.4),black);
dot((3,4.2),black);
dot((3.5,3.8),black);
dot((3.5,4.5),black);
dot((3.5,4.8),black);
dot((4,3.9),black);
dot((4,5.1),black);
dot((4.5,4.75),black);
dot((4.5,5),black);
dot((5,4.5),black);
dot((5,5),black);
[/asy]
$\textbf{(A)} ~1\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~3\qquad\textbf{(D)} ~4\qquad\textbf{(E)} ~5\qquad$