Found problems: 85335
Prove that you can arrange arrows on the edges of a convex polyhedron such that each vertex contains at most three arrows.
In a country, there are $28$ cities and between some cities there are two-way flights. In every city there is exactly one airport and this airport is either small or medium or big. For every route which contains more than two cities, doesn't contain a city twice and ends where it begins; has all types of airports. What is the maximum number of flights in this country?
Let $ABC$ be an equilateral triangle. For an arbitrary line $\ell$. through $B$, the orthogonal projections of $A$ and $C$ on $\ell$ are denoted by $D$ and $E$ respectively. If $D\ne E$, equilateral triangles $DEP$ and $DET$ are constructed on different sides of $\ell$. Find the loci of $P$ and $T$.
Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first me at the point A again, then the number of times they meet, excluding the start and finish, is
$ \textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 25 \qquad
\textbf{(C)}\ 44 \qquad
\textbf{(D)}\ \text{infinity} \qquad
\textbf{(E)}\ \text{none of these}$
Let $ a_1, $ $ a_2, $ $ \ldots, $ $ a_ {99} $ be positive real numbers such that $ ia_j + ja_i \ge i + j $ for all $ 1 \le i <j \le 99. $
Prove , that $ (a_1 + 1) (a_2 + 2) \ldots (a_ {99} +99) \ge 100!$ .
For an integer $ n \geq 2 $, let $ s (n) $ be the sum of positive integers not exceeding $ n $ and not relatively prime to $ n $.
a) Prove that $ s (n) = \dfrac {n} {2} \left (n + 1- \varphi (n) \right) $, where $ \varphi (n) $ is the number of integers positive cannot exceed $ n $ and are relatively prime to $ n $.
b) Prove that there is no integer $ n \geq 2 $ such that $ s (n) = s (n + 2021) $
Set
\[A_n=\sum_{k=1}^n \frac{k^6}{2^k}.\]
Find $\lim_{n\to\infty} A_n.$
Cyclic quadrilateral $ABCD$ with circumcircle $\omega$ is given. Let $E$ be a fixed point on segment $AC$. $M$ is an arbitrary point on $\omega$, lines $AM$ and $BD$ meet at a point $P$. $EP$ meets $AB$ and $AD$ at points $R$ and $Q$, respectively, $S$ is the intersection of $BQ,DR$ and lines $MS$ and $AC$ meet at a point $T$. Prove that as $M$ varies the circumcircle of triangle $\bigtriangleup CMT$ passes through a fixed point other than $C$.
[i]Proposed by Chunlai Jin - China[/i]
Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$.
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
Let $n\geq 3$ be an integer. Find the largest real number $M$ such that for any positive real numbers $x_1,x_2,\cdots,x_n$, there exists an arrangement $y_1,y_2,\cdots,y_n$ of real numbers satisfying
\[\sum_{i=1}^n \frac{y_i^2}{y_{i+1}^2-y_{i+1}y_{i+2}+y_{i+2}^2}\geq M,\]
where $y_{n+1}=y_1,y_{n+2}=y_2$.
We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24+k$?
[i]2020 CCA Math Bonanza Lightning Round #2.1[/i]
Find the $ \mathcal{C}^1 $ class functions $ f:[0,1]\longrightarrow\mathbb{R} $ satisfying the following three clauses:
$ \text{(i) } f(0)=0 $
$ \text{(ii) } \text{Im} f'\subset (0,1] $
$ \text{(iii) }F(1)-\frac{\left( f(1) \right)^3}{3} =F(0)=0, $ where $ F $ is a primitive of $ f. $
Determine two last digits of number $Q = 2^{2017} + 2017^2$
Let $N$ be the number of ordered triples of 3 positive integers $(a,b,c)$ such that $6a$, $10b$, and $15c$ are all perfect squares and $abc = 210^{210}$. Find the number of divisors of $N$.
[i]Proposed by Andy Xu[/i]
Let $ABC$ be a non-right triangle and let $M$ be the midpoint of $BC$. Let $D$ be a point on $AM$ (D≠A, D≠M). Let ω1 be a circle through $D$ that intersects $BC$ at $B$ and let ω2 be a circle through $D$ that intersects $BC$ at $C$. Let $AB$ intersect ω1 at $B$ and $E$, and let $AC$ intersect ω2 at $C$ and $F$.
Prove, that the tangent on ω1 at $E$ and the tangent on ω2 at $F$ intersect on $AM$.
Given two rational numbers $ a,b, $ find the functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify
$$ f(x+a+f(y))=f(x+b)+y, $$
for any rational $ x,y. $
[i]Vasile Pop[/i]
Describe all groups $ G $ which have the property that:
$$ (\forall H\le G)(\forall x,y\in G)(xy\in H\implies (x,y\in H\vee xy=1)) $$
Prove that there is a positive integer $m$ such that the number $5^{2021}m$ has no even digits (in its decimal representation).
Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$.
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
Let $x,y,z$ be positive real numbers such that $x+y+z \ge x^2+y^2+z^2$. Show that;
$$\dfrac{x^2+3}{x^3+1}+\dfrac{y^2+3}{y^3+1}+\dfrac{z^2+3}{z^3+1}\ge6$$
Given a set of integers $S$ satisfies that: for any $a,b,c\in S$ ($a,b,c$ can be the same), $ab+c\in S$\\
Find all pairs of integers $(x,y)$ such that if $x,y\in S$, then $S=\mathbb{Z}$.
a) Do there exist 5 circles in the plane such that every circle passes through centers
of exactly 3 circles?
b) Do there exist 6 circles in the plane such that every circle passes through centers
of exactly 3 circles?
$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\]
(We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.)
[i]Proposed by the4seasons.[/i]
How many different combinations of $5$ bills and $2$ bills can be used to make a total of $17$? Order does not matter in this problem.
$ \text{(A)}\ 2\qquad\text{(B)}\ 3\qquad\text{(C)}\ 4\qquad\text{(D)}\ 5\qquad\text{(E)}\ 6 $