Found problems: 85335
Let \(ABC\) be a triangle and \(D\) be the mid-point of \(BC\). Suppose the angle bisector of \(\angle ADC\) is tangent to the circumcircle of triangle \(ABD\) at \(D\). Prove that \(\angle A=90^{\circ}\).
A snail crawls $2\frac12$ centimeters in $4\frac14$ minutes. At this rate, how many centimeters can the snail crawl is 85 minutes?
Let $ABC$ be a triangle such that $\measuredangle BAC = 60^{\circ}$. Let $D$ and $E$ be the feet of the perpendicular from $A$ to the bisectors of the external angles of $B$ and $C$ in triangle $ABC$, respectively. Let $O$ be the circumcenter of the triangle $ABC$. Prove that circumcircle of the triangle $BOC$ has exactly one point in common with the circumcircle of $ADE$.
Determine all functions $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ that satisfies the equation
$$f\left(\frac{x+y+z}{3},\frac{a+b+c}{3}\right)=f(x,a)f(y,b)f(z,c)$$
for any real numbers $x,y,z,a,b,c$ such that $az+bx+cy\neq ay+bz+cx$.
Incircle of a triangle $ABC$ touches $AB$ and $AC$ at $D$ and $E$, respectively. Point $J$ is the excenter of $A$. Points $M$ and $N$ are midpoints of $JD$ and $JE$. Lines $BM$ and $CN$ cross at point $P$. Prove that $P$ lies on the circumcircle of $ABC$.
A square of side length $s$ is inscribed in circle $C_1$ and circumscribed about circle $C_2$. The area of the region in $C_1$ but outside $C_2$ is $25\pi$. What is $s$?
[i]2017 CCA Math Bonanza Team Round #2[/i]
Given triangle $ ABC$. The incircle of triangle $ ABC$ is tangent to $ BC,CA,AB$ at $ D,E,F$ respectively. Construct point $ G$ on $ EF$ such that $ DG$ is perpendicular to $ EF$. Prove that $ \frac{FG}{EG} \equal{} \frac{BF}{CE}$.
Compute the smallest positive integer $n$ such that $2016^n$ does not divide $2016!$.
Let $S=\{(x,y)| x,y\in \mathbb{Q} , 0\le x,y\le 1\}$, where $\mathbb{Q}$ is the set of all rational numbers. Given a set of lines and a set of marked points in $S$, Euclid can do one of two moves:
(i) Draw a line connecting two marked points, or
(ii) Mark a point in $S$ which lies on at least two drawn lines.
At first, the five distinct points $A(0,0), B(1,0), C(1,1), D(0,1)$ and $P\in S$ are marked. Find all such points $P$ such that Euclid can mark any point in $S$ after finitely many moves.
[i]Glen Lim[/i]
Prove that there are no integers $a, b, c, d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$ and $2$ at $x=62$.
Does there exist a set $ M$ with the following properties?
[i](i)[/i] The set $ M$ consists of 1992 natural numbers.
[i](ii)[/i] Every element in $ M$ and the sum of any number of elements have the form $ m^k$ $ (m, k \in \mathbb{N}, k \geq 2).$
$ABCD$ is a rectangle $\overline{AB} = 5\sqrt{3}$, $\overline{AD} = 30$. Extend $\overline{BC}$ past $C$ and construct point $P$ on this extension such that $\angle APD = 60^{\circ}$. Point $H$ is on $\overline{AP}$ such that $\overline{DH} \perp \overline{AP}$. Find the length of $\overline{DH}$.
[i]Proposed by Kevin Wu[/i]
Find the least value of $k\in \mathbb{N}$ with the following property: There doesn’t exist an arithmetic progression with 2019 members, from which exactly $k$ are integers.
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.
Find all triples $(x,y,z)$ of positive rationals such that $x + y + z$, $\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ and $xyz$ are all integers.
For a fixed $\theta \in \lbrack 0, \frac{\pi}{2} \rbrack$, find the smallest $a \in \mathbb{R}^{+}$ which satisfies the following conditions:
[b]I. [/b] $\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} >
1$.
[b]II.[/b] There exists $x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta},
\frac{\sqrt a}{\cos \theta} \rbrack$ such that $\lbrack (1 -
x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} +
\lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta}
\rbrack^{2} \leq a$.
Let point $I$ be the center of the inscribed circle of an acute-angled triangle $ABC$. Rays $AI$ and $BI$ intersect the circumcircle $k$ of triangle $ABC$ at points $D$ and $E$ respectively. The segments $DE$ and $CA$ intersect at point $F$, the line through point $E$ parallel to the line $FI$ intersects the circle $k$ at point $G$, and the lines $FI$ and $DG$ intersect at point $H$. Prove that the lines $CA$ and $BH$ touch the circumcircle of the triangle $DFH$ at the points $F$ and $H$ respectively.
An integer $a$ is given. Find all real-valued functions $f (x)$ defined on integers $x \ge a$, satisfying the equation $f (x+y) = f (x) f (y)$ for all $x,y \ge a$ with $x + y \ge a$.
Find all positive integers $m$ and $n$ such that $1 + 5 \cdot 2^m = n^2$.
Complex numbers $\omega_1, \ldots, \omega_n$ each have magnitude $1.$ Let $z$ be a complex number distinct from $\omega_1, \ldots, \omega_n$ such that $$\frac{z+\omega_1}{z-\omega_1}+\ldots+\frac{z+\omega_n}{z-\omega_n}=0.$$ Prove that $|z|=1.$
Let $a_1,\ldots,a_n,b_1,\ldots,b_n$ be positive numbers. Show that
\[\sqrt{\left(a_1+\cdots+a_n\right)\left(b_1+\cdots+b_n\right)}\ge\sqrt{a_1b_1}+\cdots+\sqrt{a_nb_n}\]
and prove that equality holds if and only if
\[\frac{a_1}{b_1}=\cdots=\frac{a_n}{b_n}.\]
Let $g:[2013,2014]\to\mathbb{R}$ a function that satisfy the following two conditions:
i) $g(2013)=g(2014) = 0,$
ii) for any $a,b \in [2013,2014]$ it hold that $g\left(\frac{a+b}{2}\right) \leq g(a) + g(b).$
Prove that $g$ has zeros in any open subinterval $(c,d) \subset[2013,2014].$
If $x, y, z, t > 1$ then: $$\left(\log _{zxt}x\right)^2+\left(\log _{xyt}y\right)^2+\left(\log _{xyz}z\right)^2+\left(\log _{yzt}t\right)^2>\frac{1}{4}$$
Let $A$ and $B$ be two subsets of $S = \{1, 2, . . . , 2000\}$ with $|A| \cdot |B| \geq 3999$. For a set $X$ , let $X-X$ denotes the set $\{s-t | s, t \in X, s \not = t\}$. Prove that $(A-A) \cap (B-B)$ is nonempty.
Bamal, Halvan, and Zuca are playing [i]The Game[/i]. To start, they‘re placed at random distinct vertices on regular
hexagon $ABCDEF$. Two or more players collide when they‘re on the same vertex. When this happens, all the colliding players lose and the game ends. Every second, Bamal and Halvan teleport to a random vertex adjacent to their current position (each with probability $\dfrac{1}{2}$), and Zuca teleports to a random vertex adjacent to his current position, or to the vertex directly opposite him (each with probability $\dfrac{1}{3}$). What is the probability that when [i]The Game[/i] ends Zuca hasn‘t lost?
[i]Proposed by Edwin Zhao[/i]
[hide=Solution][i]Solution.[/i] $\boxed{\dfrac{29}{90}}$
Color the vertices alternating black and white. By a parity argument if someone is on a different color than the other
two they will always win. Zuca will be on opposite parity from the others with probability $\dfrac{3}{10}$. They will all be on the same parity with probability $\dfrac{1}{10}$.
At this point there are $2 \cdot 2 \cdot 3$ possible moves. $3$ of these will lead to the same arrangement, so we disregard those. The other $9$ moves are all equally likely to end the game. Examining these, we see that Zuca will win in exactly $2$ cases (when Bamal and Halvan collide and Zuca goes to a neighboring vertex). Combining all of this, the answer is
$$\dfrac{3}{10}+\dfrac{2}{9} \cdot \dfrac{1}{10}=\boxed{\dfrac{29}{90}}$$
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