Found problems: 85335
The numbers $a$, $b$, $c$, $d$ satisfy $0<a \leq b \leq d \leq c$ and
${a+c=b+d}$. Prove that for every internal point $P$ of a segment with
length $a$ this segment is a side of a circumscribed quadrilateral with consecutive sides
$a$, $b$, $c$, $d$, such that its incircle contains~$P$.
Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.
Suppose that $(a_n)$ is a sequence of positive integers such that $\lim\limits_{n\to \infty} \dfrac{n}{a_n}=0$
Prove that there exists $k$ such that there are at least $1990$ perfect squares between $a_1 + a_2 + ... + a_k$ and $a_1 + a_2 + ... + a_{k+1}$.
Caitlin and Donal play a game called [i]Basketball Shoot-Out[/i]. The game consists of $10$ rounds. In each round, Caitlin and Donal both throw a ball simultaneously at each other's basket. If a player's ball falls into the basket, that player scores one point; otherwise, they score zero points. The scoreboard shows the complete sequence of points scored by each player in each of the $10$ rounds of the game.
It turns out that Caitlin has scored at least as many points in total as Donal after every round of the game. Prove the number of possible scoreboards is divisible by $4$ but not by $8$.
Find the number of ways to partition $S = \{1, 2, 3, \dots, 2020\}$ into two disjoint sets $A$ and $B$ with $A \cup B = S$ so that if you choose an element $a$ from $A$ and an element $b$ from $B$, $a+b$ is never a multiple of $20$. $A$ or $B$ can be the empty set, and the order of $A$ and $B$ doesn't matter. In other words, the pair of sets $(A,B)$ is indistinguishable from the pair of sets $(B,A)$.
[i]Proposed by Timothy Qian[/i]
Given $n \geq 3$ pairwise different prime numbers $p_1, p_2, ....,p_n$. Given, that for any $k \in \{ 1,2,....,n \}$ residue by division of $ \prod_{i \neq k} p_i$ by $p_k$ equals one number $r$. Prove, that $r \leq n-2 $.
$n+m-1$ unit squares are arranged in $L$-shape whose one side contains $n$ squares and the other side contains $m$ squares. Ayse and Betul plays a turn based game with following rules: Ayse plays first. At each move, the player captures desired number of adjacent squares in same side of $L$. The one who captures the last square loses the game. If four games are played for pairs $(n,m)=(2003,2003)$, $(2002,2003)$, $(2003,3)$, $(2001,2003)$; how many times can Ayse guarantee to win?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 4
$
Let $ABC$ be a triangle, let the $A$-altitude meet $BC$ at $D$, let the $B$-altitude meet $AC$ at $E$, and let $T\neq A$ be the point on the circumcircle of $ABC$ such that $AT || BC$. Given that $D,E,T$ are collinear, if $BD=3$ and $AD=4$, then the area of $ABC$ can be written as $a+\sqrt{b}$, where $a$ and $b$ are positive integers. What is $a+b$?
[i]2021 CCA Math Bonanza Individual Round #12[/i]
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
Determine all triples $(x, y, z)$ of nonnegative real numbers that verify the following system of equations:
$$x^2 - y = (z -1)^2 $$
$$y^2 - z = (x -1)^2$$
$$z^2 - x = (y - 1)^2$$
We are given fixed circles $\Omega$ and $\omega$ such that there exists a hexagon $ABCDEF$ inscribed in $\Omega$ and circumscribed around $\omega$. (Note that then, by virtue of Poncelet's theorem, there is an infinite family of such hexagons.) Prove that the value of $\dfrac{S_{ABCDEF}}{AD+BE+CF}$ it does not depend on the choice of the hexagon $ABCDEF$.
[i]A. Zaslavsky and Tran Quang Hung[/i]
Given the polynomial \[ f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n \] with integer coefficients $a_1,a_2,\ldots,a_n$, and given also that there exist four distinct integers $a$, $b$, $c$ and $d$ such that \[ f(a)=f(b)=f(c)=f(d)=5, \] show that there is no integer $k$ such that $f(k)=8$.
For the equation \[ (3x^2+y^2-4y-17)^3-(2x^2+2y^2-4y-6)^3=(x^2-y^2-11)^3, \]
determine its solutions $(x, y)$ where both $x$ and $y$ are integers. Prove that your answer lists all the integer solutions.
Create $2024$ by $2024$ grid of integers numbered from $1$ to $2024^2$ such that the integer in row $n$ and column $k$ is $2024(n-1)+k$. Let the sum of the numbers in the grid be $S$. Arthur picks 2024 numbers such that their sum is $\dfrac{S}{2024}$. Prove that in every case, using the remaining numbers, Bob can create 2023 sets of 2024 numbers with equal numbers. For clarification, All numbers can only be used once
Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$
Two equal circles in the same plane cannot have the following number of common tangents:
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{none of these}$
A circle is divided in $100$ equal parts and the points of this division are colored green or yellow, such that when between two points of division $A,B$ there are exactly $4$ division points and the point $A$ is green, then the point $B$ shall be yellow. Which points are more, the green or the yellow ones?
How many of the following statements are false?
a. $2005$ distinct positive integers exist such that the sum of their squares is a cube and the sum of their cubes is a square.
b. There are $2$ integral solutions to $x^2 + y^2 + z^2 = x^2y^2$.
c. If the vertices of a triangle are lattice points in a plane, the diameter of the triangle’s circumcircle will never exceed the product of the triangle’s side lengths.
Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its
sides.
Let $G$ be a simple graph with $11$ vertices labeled as $v_{1} , v_{2} , ... , v_{11}$ such that the degree of $v_1$ equals to $2$ and the degree of other vertices are equal to $3$.If for any set $A$ of these vertices which $|A| \le 4$ , the number of vertices which are adjacent to at least one verex in $A$ and are not in $A$ themselves is at least equal to $|A|$ , then find the maximum possible number for the diameter of $G$. (The distance between two vertices of graph is the number of edges of the shortest path between them and the diameter of a graph , is the largest distance between arbitrary pairs in $V(G)$. )
[i]Proposed by Alireza Haqi[/i]
Construct a set $S$ of polynomials inductively by the rules:
(i) $x\in S$;
(ii) if $f(x)\in S$, then $xf(x)\in S$ and $x+(1-x)f(x)\in S$.
Prove that there are no two distinct polynomials in $S$ whose graphs intersect within the region $\{0 < x < 1\}$.
Consider the radii of normal curvature of a surface at one of its points $ P_0$ in two conjugate direction (with respect to the Dupin indicatrix). Show that their sum does not depend on the choice of the conjugate directions. (We exclude the choice of asymptotic directions in the case of a hyperbolic point.)
Is it true that every positive integer greater than 30 is a sum of 4 positive integers such that each two of them have a common divisor greater than 1?
Given a triangular pyramid $ABCD$. Sphere $S_1$ passing through points $A$, $B$, $C$, intersects edges $AD$, $BD$, $CD$ at points $K$, $L$, $M$, respectively; sphere $S_2$ passing through points $A$, $B$, $D$ intersects the edges $AC$, $BC$, $DC$ at points $P$, $Q$, $M$ respectively. It turned out that $KL \parallel PQ$. Prove that the bisectors of plane angles $KMQ$ and $LMP$ are the same.
How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{infinitely many}$