Found problems: 85335
You’re given the complex number $\omega = e^{2i\pi/13} + e^{10i\pi/13} + e^{16i\pi/13} + e^{24i\pi/13}$, and told it’s a root of a unique monic cubic $x^3 +ax^2 +bx+c$, where $a, b, c$ are integers. Determine the value of $a^2 + b^2 + c^2$.
Let $ABC$ be an acute triangle, and its altitudes $AX,BY,CZ$ concurrent at $H$. Construct circles $(K_a), (K_b), (K_c)$ circumscribing the triangles $AY Z, BZX, CXY$ . Construct a circle $(K)$ that is internally tangent to all the three circles $(Ka), (K_b), (K_c)$. Prove that $(K)$ is tangent to the circumcircle $(O)$ of the triangle $ABC$.
Đỗ Thanh Sơn
Alice and Bob play a game on a connected graph with $2n$ vertices, where $n\in \mathbb{N}$ and $n>1$.. Alice and Bob have tokens named A and B respectively. They alternate their turns with Alice going first. Alice gets to decide the starting positions of A and B. Every move, the player with the turn moves their token to an adjacent vertex. Bob's goal is to catch Alice, and Alice's goal is to prevent this. Note that positions of A, B are visible to both Alice and Bob at every moment.
Provided they both play optimally, what is the maximum possible number of edges in the graph if Alice is able to evade Bob indefinitely?
[i]Proposed by Shashank Ingalagavi and Vighnesh Sangle[/i]
Suppose in a triangle $\triangle ABC$, $A$ , $B$ , $C$ are the three angles and $a$ , $b$ , $c$ are the lengths of the sides opposite to the angles respectively. Then prove that if $sin(A-B)= \frac{a}{a+b}\sin A \cos B - \frac{b}{a+b}\sin B \cos A$ then the triangle $\triangle ABC$ is isoscelos.
Find the value of $n\in{\mathbb{N}}$ satisfying the following inequality.
\[\left|\int_0^{\pi} x^2\sin nx\ dx\right|<\frac{99\pi ^ 2}{100n}\]
For $\triangle ABC$ and its circumcircle $\omega$, draw the tangents at $B,C$ to $\omega$ meeting at $D$. Let the line $AD$ meet the circle with center $D$ and radius $DB$ at $E$ inside $\triangle ABC$. Let $F$ be the point on the extension of $EB$ and $G$ be the point on the segment $EC$ such that $\angle AFB=\angle AGE=\angle A$. Prove that the tangent at $A$ to the circumcircle of $\triangle AFG$ is parallel to $BC$.
[i]Proposed by 61plus[/i]
Given an acute triangle $ABC$. A circle inscribed in a triangle $ABC$ with center at point $I$ touches the sides $AB, BC$ at points $C_1$ and $A_1$, respectively. The lines $A_1C_1$ and $AC$ intersect at the point $Q$. Prove that the circles circumscribed around the triangles $AIC$ and $A_1CQ$ are tangent.
(Dmitry Shvetsov)
The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves
the room. What is the average age of the four remaining people?
$\textbf{(A)}\ 25 \qquad
\textbf{(B)}\ 26 \qquad
\textbf{(C)}\ 29 \qquad
\textbf{(D)}\ 33 \qquad
\textbf{(E)}\ 36$
Let $a, b, c, d$ be real numbers for which $a^2 + b^2 + c^2 + d^2 = 1$. Show the following inequality:
$$a^2 + b^2 - c^2 - d^2 \le \sqrt{2 + 4(ac + bd)}.$$
$ \angle BAC \equal{} 80{}^\circ$, $ \left|AB\right| \equal{} \left|AC\right|$, $ K\in \left[AB\right]$, $ L\in \left[AB\right.$, $ \left|AB\right|^{2} \equal{} \left|AK\right|\cdot \left|AL\right|$, $ \left|BL\right| \equal{} \left|BC\right|$, $ \angle KCB \equal{} ?$
$\textbf{(A)}\ 20^\circ \qquad\textbf{(B)}\ 25^\circ \qquad\textbf{(C)}\ 30^\circ \qquad\textbf{(D)}\ 35^\circ \qquad\textbf{(E)}\ 40^\circ$
Let $x \geq y \geq z$ be real numbers such that $xy + yz + zx = 1$. Prove that $xz < \frac 12.$ Is it possible to improve the value of constant $\frac 12 \ ?$
Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .
Let $A(x)=\lfloor\frac{x^2-20x+16}{4}\rfloor$, $B(x)=\sin\left(e^{\cos\sqrt{x^2+2x+2}}\right)$, $C(x)=x^3-6x^2+5x+15$, $H(x)=x^4+2x^3+3x^2+4x+5$, $M(x)=\frac{x}{2}-2\lfloor\frac{x}{2}\rfloor+\frac{x}{2^2}+\frac{x}{2^3}+\frac{x}{2^4}+\ldots$, $N(x)=\textrm{the number of integers that divide }\left\lfloor x\right\rfloor$, $O(x)=|x|\log |x|\log\log |x|$, $T(x)=\sum_{n=1}^{\infty}\frac{n^x}{\left(n!\right)^3}$, and $Z(x)=\frac{x^{21}}{2016+20x^{16}+16x^{20}}$ for any real number $x$ such that the functions are defined. Determine $$C(C(A(M(A(T(H(B(O(N(A(N(Z(A(2016)))))))))))))).$$
[i]2016 CCA Math Bonanza Lightning #5.3[/i]
Let $P$ be a finite set of primes, $A$ an infinite set of positive integers, where every element of $A$ has a prime factor not in $P$. Prove that there exist an infinite subset $B$ of $A$, such that the sum of elements in any finite subset of $B$ has a prime factor not in $P$.
Find all primes $ p,q $ such that $ \alpha^{3pq} -\alpha \equiv 0 \pmod {3pq} $ for all integers $ \alpha $.
Each of the sides $ BC, CA, AB $ of the triangle $ ABC $ was divided into three equal parts and on the middle sections of these sides as bases, equilateral triangles were built outside the triangle $ ABC $, the third vertices of which were marked with the letters $ A', B' , C' $ respectively. In addition, points $ A'', B'', C'' $ were determined, symmetrical to $ A', B', C' $ respectively with respect to the lines $ BC, CA, AB $. Prove that the triangles $ A'B'C' $ and $ A''B''C'' $ are equilateral and have the same center of gravity as the triangle $ ABC $.
$32$ real and $32$ fake coins are given the same appearance. All fake coins weigh equally and less than the real ones, which also all weigh the same. How to determine the type of at least seven coins in six weighings on a scale with two bowls?
$|3 - \pi| =$
$ \textbf{(A)}\ \frac{1}{7}\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 3 - \pi\qquad\textbf{(D)}\ 3 + \pi\qquad\textbf{(E)}\ \pi - 3 $
On a chessboard $8\times 8$, $n>6$ Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of $n$.
Let $A_1A_2 ... A_8$ be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each $i = 1, ... , 8$, define $B_i$ as the intersection between segments $A_iA_{i+4}$ and $A_{i-1}A_{i+1}$, where $A_{j+8} = A_j$ and $B_{j+8} = B_j$ for all $j$. Show some number $i$, amongst 1, 2, 3, and 4 satisfies
\[\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}\]
Denote $ S_n $ as being the sum of the squares of the first $ n\in\mathbb{N} $ terms of a given arithmetic sequence of natural numbers.
[b]a)[/b] If $ p\ge 5 $ is a prime, then $ p\big| S_p. $
[b]b)[/b] $ S_5 $ is not a perfect square.
Find the maximum possible value obtainable by inserting a single set of parentheses into the expression $1 + 2 \times 3 + 4 \times 5 + 6$.
Calculate the sum of the coordinates of all pairs of positive integers $(n, k)$ such that $k\equiv 0, 3\pmod 4$, $n > k$, and $\displaystyle\sum^n_{i = k + 1} i^3 = (96^2\cdot3 - 1)\displaystyle\left(\sum^k_{i = 1} i\right)^2 + 48^2$
Let $a \neq 1$ and be a real positive number and $n$ be an integer greater than $1.$ Demonstrate that $n^2 < \frac{(a^n + a^{-n} -2)}{(a + a^{-1} -2)}.$
In this problem, we consider only polynomials with integer coeffients. Call two polynomials $p$ and $q$ [i]really close[/i] if $p(2k + 1) \equiv q(2k + 1)$ (mod $210$) for all $k \in Z^+$. Call a polynomial $p$ [i]partial credit[/i] if no polynomial of lesser degree is [i]really close[/i] to it. What is the maximum possible degree of [i]partial credit[/i]?