Found problems: 85335
A trapezoid has bases with lengths equal to $5$ and $15$ and legs with lengths equal to $13$ and $13.$ Determine the area of the trapezoid.
Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[f(\alpha f(x)+f(y))=\beta x+f(y)\]
holds for all real $x$ and $y$.
[i](Walther Janous)[/i]
Fix a positive integer $n\ge 2$. For any cyclic $2n$-gon $P_1 P_2\cdots P_{2n}$ in this order, define its score as the maximal possible value of $$\angle P_iXP_{i+1} + \angle P_{i+n}XP_{i+n+1}$$ across all $1\le i\le n$ (indices modulo $n$), and over all points $X$ inside the $2n$-gon including its boundary.
Prove that there exist a real number $r$ such that a cyclic $2n$-gon is regular if and only if it has score $r$.
[i]Proposed by Wong Jer Ren[/i]
Given a simple, connected graph with $n$ vertices and $m$ edges. Prove that one can find at least $m$ ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.
Let $D$ be a point on the side $[AB]$ of the isosceles triangle $ABC$ such that $|AB|=|AC|$. The parallel line to $BC$ passing through $D$ intersects $AC$ at $E$. If $m(\widehat A) = 20^\circ$, $|DE|=1$, $|BC|=a$, and $|BE|=a+1$, then which of the followings is equal to $|AB|$?
$
\textbf{(A)}\ 2a
\qquad\textbf{(B)}\ a^2-a
\qquad\textbf{(C)}\ a^2+1
\qquad\textbf{(D)}\ (a+1)^2
\qquad\textbf{(E)}\ a^2+a
$
Sunaina and Malay play a game on the coordinate plane. Sunaina has two pawns on $(0,0)$ and $(x,0)$, and Malay has a pawn on $(y,w)$, where $x,y,w$ are all positive integers. They take turns alternately, starting with Sunaina. In their turn they can move one of their pawns one step vertically up or down. Sunaina wins if at any point in time all the three pawns are colinear. Find all values of $x,y$ for which Sunaina has a winning strategy irrespective of the value of $w$.
Proposed by NV Tejaswi
The sum of the squares of the first $ n$ positive integers is given by the expression $ \frac{n(n \plus{} c)(2n \plus{} k)}{6}$, if $ c$ and $ k$ are, respectively:
$ \textbf{(A)}\ {1}\text{ and }{2} \qquad
\textbf{(B)}\ {3}\text{ and }{5}\qquad
\textbf{(C)}\ {2}\text{ and }{2}\qquad
\textbf{(D)}\ {1}\text{ and }{1}\qquad
\textbf{(E)}\ {2}\text{ and }{1}$
Find all positive integers $n$ for which all coefficients of polynomial $P(x)$ are divisible by $7,$ where
\[P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.\]
To be able to play the Game of Glory, seven friends have to form four teams which, obviously, may not have the same number of members. In how many different ways can they form these teams? (The order of people within teams is not important, but each person can only be part of one team)
Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Let $x,y,z$ be real numbers such that $x^2+y^2+z^2-2xyz=1$. Prove that
\[ (1+x)(1+y)(1+z)\le 4+4xyz. \]
Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.
Determine the number of odd integers $1 \le n \le 100$ with the property that
\[
\sum_{\substack{1 \le k \le n \\ \gcd(k,n) = 1}} \cos\left(\frac{2\pi k}{n} \right) = 1
\quad\text{and}\quad
\sum_{\substack{1 \le k \le n \\ \gcd(k,n) = 1}} \sin\left(\frac{2\pi k}{n} \right) = 0.
\]
[i]Based on a proposal by Mayank Pandey[/i]
Determine the number of pairs of positive integers \( (p, k) \) such that \( p \) is a prime number and \( p^2 + 2^k \) is a perfect square less than 2023. A number is called a perfect square if it is the square of an integer.
A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$.
Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$
Two integers $n, k$ satisfies $n \ge 2$ and $k \ge \frac{5}{2}n-1$.
Prove that whichever $k$ lattice points with $x$ and $y$ coordinate no less than $1$ and no more than $n$ we pick, there must be a circle passing through at least four of these points.
Three congruent ellipses are mutually tangent. Their major axes are parallel. Two of the ellipses are tangent at the end points of their minor axes as shown. The distance between the centers of these two ellipses is $4$. The distances from those two centers to the center of the third ellipse are both $14$. There are positive integers m and n so that the area between these three ellipses is $\sqrt{n}-m \pi$. Find $m+n$.
[asy]
size(250);
filldraw(ellipse((2.2,0),2,1),grey);
filldraw(ellipse((0,-2),4,2),white);
filldraw(ellipse((0,+2),4,2),white);
filldraw(ellipse((6.94,0),4,2),white);[/asy]
Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Omega$.
Two equal squares, one with red sides, another with blue ones, give an octagon in intersection. Prove that the sum of red octagon sides lengths is equal to the sum of blue octagon sides lengths.
Let $\omega$ be the circumcircle of acute triangle $ABC$. Two tangents of $\omega$ from $B$ and $C$ intersect at $P$, $AP$ and $BC$ intersect at $D$. Point $E$, $F$ are on $AC$ and $AB$ such that $DE \parallel BA$ and $DF \parallel CA$.
(1) Prove that $F,B,C,E$ are concyclic.
(2) Denote $A_{1}$ the centre of the circle passing through $F,B,C,E$. $B_{1}$, $C_{1}$ are difined similarly. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent.
Show that if $x,y,z,w$ are positive reals, then
\[
\frac{3}{2}\sqrt{(x^2+y^2)(w^2+z^2)} + \sqrt{(x^2+w^2)(y^2+z^2) - 3xyzw} \geq (x+z)(y+w)
\]
In the sequence of digits $2,0,2,9,3,...$ any digit it equal to the last digit in the decimal representation of the sum of four previous digits. Do the four numbers $2,0,1,5$ in that order occur in the sequence?
Folklore
There are $ n$ football teams in a round-robin competition where every 2 teams meet once. The winner of each match receives 3 points while the loser receives 0 points. In the case of a draw, both teams receive 1 point each. Let $ k$ be as follows: $ 2 \leq k \leq n \minus{} 1$. At least how many points must a certain team get in the competition so as to ensure that there are at most $ k \minus{} 1$ teams whose scores are not less than that particular team's score?
Interior in alake there are two points $A,B$ from which we can see every other point of the lake. Prove that also from any other point of the segment $AB$, we can see all points of the lake.