Found problems: 85335
Triangle $\vartriangle ABC$ has side lengths $AB = 5$, $BC = 8$, and $CA = 7$. Let the perpendicular bisector of $BC$ intersect the circumcircle of $\vartriangle ABC$ at point $D$ on minor arc $BC$ and point $E$ on minor arc $AC$, and $AC$ at point $F$. The line parallel to $BC$ passing through $F$ intersects $AD$ at point $G$ and $CE$ at point $H$. Compute $\frac{[CHF]}{[DGF]}$ . (Given a triangle $\vartriangle ABC$, $[ABC]$ denotes its area.)
Calculate $$\sum_{k=5}^{k=49}\frac{11_(k}{2\sqrt[3]{1331_(k}}$$ knowing that the numbers $11$ and $1331$ are written in base $k \ge 4$.
In tetrahedron $ABCD,$ as shown below, compute the number of ways to start at $A,$ walk along some path of edges, and arrive back at $A$ without walking over the same edge twice.
[Insert Diagram]
[i]Proposed by Richard Chen[/i]
Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$.
[i]Demetres Christofides, Cyprus[/i]
Let $\Delta ABC$ be an acute triangle with $a>b$, center $O$ of its circumscribed circle and middle point $M$ of $AC$. Let $K$ be the reflection of $O$ in $M$. Point $E\in BC$ is such that $EO\perp AB$. Point $F\in MK$ is such that $FK=OE$ and $K$ lies between $F$ and $M$. The altitude through $C$ and the angle bisector of $\angle CAB$ intersect in $D$. Let $BD$ intersect the circumscribed circle of $\Delta ABC$ for a second time in $P$. Prove that $AP\perp CF$.
Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?
Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which
\[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\]
Find the number of elements of the set $A_n$.
[i]Proposed by Vidan Govedarica, Serbia[/i]
Among any $79$ consecutive natural numbers there exists one whose sum of digits is divisible by $13$. Find a sequence of $78$ consecutive natural numbers for which the above statement fails.
What is the least value of $n$ such that $n!$ is a multiple of $2024$?
$
\textbf{(A) }11 \qquad
\textbf{(B) }21 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }23 \qquad
\textbf{(E) }253 \qquad
$
Let $\Gamma$ be a circumference and $A$ a point outside it. Let $B$ and $C$ be points in $\Gamma$ such that $AB$ and $AC$ are tangent to $\Gamma$. Let $P$ be a point in $\Gamma$. Let $D$, $E$ and $F$ be points in $BC$, $AC$ and $AB$ respectively, such that $PD \perp BC$, $PE \perp AC$, and $PF \perp AB$.
Show that $PD^2 = PE \cdot PF$
A bug starts at vertex $A$ of triangle $ABC$. Six times, the bug travels to a randomly chosen adjacent vertex. For example, the bug could go from $A$, to $B$, to $C$, back to $B$, and back to $C$. What is the probability that the bug ends up at $A$ after its six moves?
Find all triples \( (x, y, z) \) of positive integers that satisfy the equation
\[
x + xy + xyz = 31.
\]
Determine the sum of all positive integers $n<100$ satisfying the following expression. \[\sum_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod \;{10^{k+1})}-n \;(\bmod \;{10^k)}\right)=\prod_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod\; 10^{k+1})-n \;(\bmod\; 10^k)\right).\] Here, $\textstyle\sum$ and $\textstyle\prod$ represent sum and product, respectively.
[i]2022 CCA Math Bonanza Lightning Round 3.3[/i]
$A, B$ are $n \times n$ matrices such that $\text{rank}(AB-BA+I) = 1.$ Prove that $\text{tr}(ABAB)-\text{tr}(A^2 B^2) = \frac{1}{2}n(n-1).$
Let $f(x)=\sin^4x-\sin x\cos x+cos^4 x$. Find the range of $f(x)$.
Show that for all positive reals $a,b,c$ with $a+b+c=3$,
$$\sum_{\text{cyc}}\sqrt{a+3b+\frac2c}\ge3\sqrt6.$$
Let $ A\equal{}A_1A_2A_3A_4$ be a tetrahedron, and suppose that for each $ j \not\equal{} k, [A_j,A_{jk}]$ is a segment of length $ \rho$ extending from $ A_j$ in the direction of $ A_k$. Let $ p_j$ be the intersection line of the planes $ [A_{jk}A_{jl}A_{jm}]$ and $ [A_kA_lA_m]$. Show that there are infinitely many straight lines that intersect the straight lines $ p_1,p_2,p_3,p_4$ simultaneously.
Angles are equal in a hexagon, three main diagonals are equal and the other six diagonals are also equal. Is it true that it has equal sides?
Given the three numbers $x, y=x^x, z=x^{(x^x)}$ with $.9<x<1.0$. Arranged in order of increasing magnitude, they are:
$\textbf{(A)}\ x, z, y \qquad\textbf{(B)}\ x, y, z \qquad\textbf{(C)}\ y, x, z \qquad\textbf{(D)}\ y, z, x \qquad\textbf{(E)}\ z, x, y$
We say that two triangles are oriented similarly if they are similar and have the same orientation. Prove that if $ALT, ARM, ORT, $ and $ULM$ are four triangles which are oriented similarly, then $A$ is the midpoint of the line segment $OU.$
Find all positive integers $x, y, z$ such that $45^x-6^y=2019^z$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this?
[i]Proposed by Nathan Xiong[/i]
Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.
A square cake of uniform height is evenly covered with frosting on the top and all four sides. Find a way to cut the cake into five portions such that:
(a) All portions contain the same amount of cake.
(b) All portions contain the same amount of cake and frosting.