Found problems: 85335
Prove that if all angles of a convex $n$-gon are equal, then there are at least two of its sides that are not longer than their adjacent sides.
[i]A. Berzin’sh, O. Musin[/i]
Let a set $S$ of $n$ points be called [i]cool[/i] if:
[list]
[*] All points lie in a plane
[*] No three points are collinear
[*] There exists a triangle with three distinct vertices in $S$ such that the triangle contains another point in $S$ strictly inside it
[/list]
Define $g(S)$ for a cool set $S$ to be the sum of the number of points strictly inside each triangle with three distinct vertices in $S$. Let $f(n)$ be the minimal possible value of $g(S)$ across all cool sets of size $n$. Find
\[ f(4) + \dots + f(2020) \pmod{1000}\]
[i]Proposed by Timothy Qian[/i]
A rectangle is divided into $9$ small rectangles if by parallel lines to its sides, as shown in the figure.
[img]https://cdn.artofproblemsolving.com/attachments/e/d/1fd545862a3c7950249ec54a631c74e59fb9ed.png[/img]
The four numbers written indicate the areas of the four corresponding rectangles.
Prove that the total area of the rectangle is greater than or equal to $90$.
Benny the Bear has $100$ rabbits in his rabbit farm. He observes that $53$ rabbits are spotted, and $73$ rabbits are blue-eyed. Compute the minimum number of rabbits that are both spotted and blue-eyed.
Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that
\[
a b \leqslant-\frac{1}{2019}.
\]
In a certain population the ratio of the number of women to the number of men is 11 to 10. If the average (arithmetic mean) age of the women is 34 and the average age of the men is 32, then the average age of the population is
$ \textbf{(A)}\ 32\frac{9}{10}\qquad\textbf{(B)}\ 32\frac{20}{21}\qquad\textbf{(C)}\ 33\qquad\textbf{(D)}\ 33\frac{1}{21}\qquad\textbf{(E)}\ 33\frac{1}{10} $
Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$
Let $ABCD$ be a square with side 1. On the sides $BC$ and $CD$ are chosen points $P$ and $Q$ where $AP$ and $AQ$ intersect the diagonal $BD$ in points $M$ and $N$ respectively. If $DQ\neq BP$ and the line through $A$ and the intersection point of $MQ$ and $NP$ is perpendicular to $PQ$, prove that $\angle MAN=45^\circ$.
Given a triangle $ABC$, let $P$ and $Q$ be points on segments $\overline{AB}$ and $\overline{AC}$, respectively, such that $AP=AQ$. Let $S$ and $R$ be distinct points on segment $\overline{BC}$ such that $S$ lies between $B$ and $R$, $\angle BPS=\angle PRS$, and $\angle CQR=\angle QSR$. Prove that $P,Q,R,S$ are concyclic (in other words, these four points lie on a circle).
For a positive integer $n$ with prime factorization $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ let's define $\lambda(n) = (-1)^{\alpha_1 + \alpha_2 + \dots + \alpha_k}$.
Define $L(n)$ as sum of $\lambda(x)$ over all integers from $1$ to $n$.
Define $K(n)$ as sum of $\lambda(x)$ over all [b]composite[/b] integers from $1$ to $n$.
For some $N>1$, we know, that for every $2\le n \le N$, $L(n)\le 0$.
Prove that for this $N$, for every $2\le n \le N$, $K(n)\ge 0$.
[i]Mykhailo Shtandenko[/i]
If $ x<\minus{}2$ then $ |1\minus{}|1\plus{}x|$ $ |$ equals
$ \textbf{(A)}\ 2\plus{}x \qquad
\textbf{(B)}\ \minus{}2\minus{}x \qquad
\textbf{(C)}\ x \qquad
\textbf{(D)}\ \minus{}x \qquad
\textbf{(E)}\ \minus{}2$
Given a positive integer $n$, let $D$ be the set of all positive divisors of $n$. The subsets $A,B$ of $D$ satisfies that for any $a \in A$ and $b \in B$, it holds that $a \nmid b$ and $b \nmid a$. Show that
\[ \sqrt{|A|}+\sqrt{|B|} \le \sqrt{|D|}. \]
Determine all pairs of integers $(a, b)$ that solve the equation $a^3 + b^3 + 3ab = 1$.
Juan makes a list of $2018$ numbers. The first is $ 1$. Then each number is obtained by adding to the previous number, one of the numbers $ 1$, $2$, $3$, $4$, $5$, $6$, $7$, $ 8$ or $9$. Knowing that none of the numbers in the list ends in $0$, what is the largest value you can have the last number on the list?
Let $a, b, c$ be positive real numbers. Prove that $$8(a+b+c) \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \right) \le 9 \left(1+\frac{a}{b} \right)\left(1+\frac{b}{c} \right)\left(1+\frac{c}{a} \right)$$
Prove that $\sqrt{x}+\sqrt{y}+\sqrt{xy}$ is equal to$ \sqrt{x}+\sqrt{y+xy+2y\sqrt{x}}$
and compare the numbers $\sqrt{3}+\sqrt{10+2\sqrt{3}}$ and $\sqrt{5+\sqrt{22}}+\sqrt{8-
\sqrt{22}+2\sqrt{15-3\sqrt{22}}}$.
How many ordered pairs of integers $ (x, y)$ are there such that
\[ 0 < \left\vert xy \right\vert < 36?\]
Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and
\[a^3 + b^3 = p^n\mbox{.}\]
[i](2nd Benelux Mathematical Olympiad 2010, Problem 4)[/i]
The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.
[i]Proposed by Christopher Cheng[/i]
[hide=Solution][i]Solution. [/i] $\boxed{6}$
Note that $(20,23)$ is the intersection of the lines $\ell_1$ and $\ell_2$. Thus, we only care about lattice points on the the two lines that are an integer distance away from $(20,23)$. Notice that $7$ and $24$ are part of the Pythagorean triple $(7,24,25)$ and $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$. Thus, points on $\ell_1$ only satisfy the conditions when $n$ is divisible by $25$ and points on $\ell_2$ only satisfy the conditions when $n$ is divisible by $13$. Therefore, $a$ is just the number of positive integers less than $2023$ that are divisible by both $25$ and $13$. The LCM of $25$ and $13$ is $325$, so the answer is $\boxed{6}$.[/hide]
Let $ABCD$ be a rectangle. Construct equilateral triangles $BCX$ and $DCY$, in such a way that both of these triangles share some of their interior points with some interior points of the rectangle. Line $AX$ intersects line $CD$ on $P$, and line $AY$ intersects line $BC$ on $Q$. Prove that triangle $APQ$ is equilateral.
Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square.
[i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]
Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.
In a small town, there are $n \times n$ houses indexed by $(i, j)$ for $1 \leq i, j \leq n$ with $(1, 1)$ being the house at the top left corner, where $i$ and $j$ are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by $(1, c)$, where $c \leq \frac{n}{2}$. During each subsequent time interval $[t, t+1]$, the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended [i]neighbors[/i] of each house which was on fire at time t. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters?
A house indexed by $(i, j)$ is a [i]neighbor[/i] of a house indexed by $(k, l)$ if $|i - k| + |j - l|=1$.
Let $ABC$ be a triangle such that $AB = 7$, $BC = 8$, and $CA = 9$. There exists a unique point $X$ such that $XB = XC$ and $XA$ is tangent to the circumcircle of $ABC$. If $XA = \tfrac ab$, where $a$ and $b$ are coprime positive integers, find $a + b$.
[i]Proposed by Alexander Wang[/i]
Call a polynomial $f$ with positive integer coefficients \textit{triangle-compatible} if any three coefficients of $f$ satisfy the triangle inequality. For instance, $3x^3 + 4x^2 + 6x + 5$ is triangle-compatible, but $3x^3 + 3x^2 + 6x + 5$ is not. Given that $f$ is a degree $20$ triangle-compatible polynomial with $-20$ as a root, what is the least possible value of $f(1)$?
[i] Note: this problem is also Premier #3 [/i]