Found problems: 85335
A circle passes through the points $A,C$ of triangle $ABC$ intersects with the sides $AB,BC$ at points $D,E$ respectively. Let $ \frac{BD}{CE}=\frac{3}{2}$, $BE=4$, $AD=5$ and $AC=2\sqrt{7} $.
Find the angle $ \angle BDC$.
In an acute-angled triangle $ ABC,$ let $ AD,BE$ be altitudes and $ AP,BQ$ internal bisectors. Denote by $ I$ and $ O$ the incenter and the circumcentre of the triangle, respectively. Prove that the points $ D, E,$ and $ I$ are collinear if and only if the points $ P, Q,$ and $ O$ are collinear.
Evaluate \[ \sqrt{\binom82+\binom92+\binom{15}2+\binom{16}2}. \]
[i] Proposed by Evan Chen [/i]
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.
Positive integers $a, b$ satisfy both of the following conditions.
For a positive integer $m$, if $m^2 \mid ab$, then $m = 1$.
There exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 = z^2 + w^2$ and $z^2 + w^2 > 0$.
Prove that there exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 + n = z^2 + w^2$, for each integer $n$.
Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?
$(\textbf{A})\: 1.6\qquad(\textbf{B}) \: 1.8\qquad(\textbf{C}) \: 2.0\qquad(\textbf{D}) \: 2.2\qquad(\textbf{E}) \: 2.4$
Let $ n $ be a natural number, and $ A $ the set of the first $ n $ natural numbers. Find the number of nondecreasing functions $ f:A\longrightarrow A $ that have the property
$$ x,y\in A\implies |f(x)-f(y)|\le |x-y|. $$
In terms of real parameter $a$ solve inequality:
$\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}$
in set of real numbers
Let $x,y$ be integer number with $x,y\neq-1$ so that $\frac{x^{4}-1}{y+1}+\frac{y^{4}-1}{x+1}\in\mathbb{Z}$. Prove that $x^{4}y^{44}-1$ is divisble by $x+1$
A class of $10$ children is divided into $5$ pairs of partners. Each pair of partners sits next to each other and works together during class. One day, the teacher decides he wants to divide the class into two groups. In order to make sure the students work with new people, he makes sure not to put any student in the same group as his or her partner. How many different ways can he divide the class into these two groups?
$\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }10\qquad\textbf{(D) }16\qquad\textbf{(E) }32$
Let $ABCD$ be a regular tetrahedron with side length $1$. Let $X$ be the point in the triangle $BCD$ such that $[XBC]=2[XBD]=4[XCD]$, where $[\overline{\omega}]$ denotes the area of figure $\overline{\omega}$. Let $Y$ lie on segment $AX$ such that $2AY=YX$. Let $M$ be the midpoint of $BD$. Let $Z$ be a point on segment $AM$ such that the lines $YZ$ and $BC$ intersect at some point. Find $\frac{AZ}{ZM}$.
Let $a,b$ and $c$ be positive reals such that $abc=\frac{1}{8}$.Then prove that \[a^2+b^2+c^2+a^2b^2+a^2c^2+b^2c^2\ge\frac{15}{16}\]
How namy subsets with $3$ elements of set $S=\{1,2,3,...,19,20\}$ exist, such that their product is divisible by $4$.
Call a pair of integers $(a,b)$ [i]primitive[/i] if there exists a positive integer $\ell$ such that $(a+bi)^\ell$ is real. Find the smallest positive integer $n$ such that less than $1\%$ of the pairs $(a, b)$ with $0 \le a, b \le n$ are primitive.
[i]Proposed by Mehtaab Sawhney[/i]
Find all $f:\mathbb{R}\to\mathbb{R}$ functions such that
$$f(x+y)^3=(x+2y)f(x^2)+f(f(y))(x^2+3xy+y^2)$$
for all real numbers $x,y$
[b]Q1.[/b] Assum that $a-b=-(a-b).$ Then:
$(A) \; a=b; \qquad (B) \; a<b; \qquad (C) \; a>b \qquad (D) \; \text{ It is impossible to compare those of a and b.}$
Let triangle $ABC$ be an isosceles triangle with $AB = AC$. Suppose that the angle bisector of its angle $\angle B$ meets the side $AC$ at a point $D$ and that $BC = BD+AD$.
Determine $\angle A$.
Let $300$ students participate to the Olympiad. Between each $3$ participants there is a pair that are not friends. Hamza enumerates participants in some order and denotes by $x_i$ the number of friends of $i$-th participant. It occurs that $\{x_1,x_2,...,x_{299},x_{300}\} = \{1, 2,..., N - 1,N\}$ Find the biggest possible value for $N$.
Find $ \inf_{z\in\mathbb{C}} \left( |z^2+z+1|+|z^2-z+1| \right) . $
[i]Gheorghe Andrei[/i] and [i]Doru Constantin Caragea[/i]
Given an acute triangle $ABC$ with their incenter $I$. Point $X$ lies on $BC$ on the same side as $B$ wrt $AI$. Point $Y$ lies on the shorter arc $AB$ of the circumcircle $ABC$. It is given that $$\angle AIX = \angle XYA = 120^\circ.$$
Prove that $YI$ is the angle bisector of $XYA$.
Alex and Bruno play the following game: each one, in your turn, the player writes, exactly one digit, in the right of the last number written. The game finishes if we have a number with $6$ digits( distincts ) and Alex starts the game. Bruno wins if the number with $6$ digits is a prime number, otherwise Alex wins.
Which player has the winning strategy?
Find all functions $f$ and $g$ defined from $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ such that for all $x, y > 0$ the two equations hold
$$ (f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2 $$
$$ (-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1) $$
[i]Note: $\mathbb{R}_{>0}$ denotes the set of positive real numbers.[/i]
Alex, Bertha, Cameron, Dylan, and Ellen each have a different toy. Each kid puts each of their own toys into a large bag. The toys are then randomly distributed such that each kid receives a toy. How many ways are there for exactly one kid to get the same toy that they put in?
[i]2018 CCA Math Bonanza Lightning Round #2.4[/i]
Points $E$ and $F$ are taken on the diagonals $AB_1$ and $CA_1$ of the lateral faces $ABB_1A_1$ and $CAA_1C_1$ of a triangular prism $ABCA_1B_1C_1$ so that $EF\parallel BC_1$. Find the ratio of the lengths of $EF$ and $BC_1$.
Let $ABCD$ be a rectangle and the circle of center $D$ and radius $DA$, which cuts the extension of the side $AD$ at point $P$. Line $PC$ cuts the circle at point $Q$ and the extension of the side $AB$ at point $R$. Show that $QB = BR$.