Found problems: 85335
Let $ABCD$ and $A'B'C'D'$ be two arbitrary squares in the plane that are oriented in the same direction. Prove that the quadrilateral formed by the midpoints of $AA',BB',CC',DD'$ is a square.
There are $2n-1$ twoelement subsets of set $1,2,...,n$. Prove that one can choose $n$ out of these such that their union contains no more than $\frac{2}{3}n+1$ elements.
A list of \(n\) positive integers \(a_1, a_2,a_3,\ldots,a_n\) is said to be [i]good[/i] if it checks simultaneously:
\(\bullet a_1<a_2<a_3<\cdots<a_n,\)
\(\bullet a_1+a_2^2+a_3^3+\cdots+a_n^n\le 2023.\)
For each \(n\ge 1\), determine how many [i]good[/i] lists of \(n\) numbers exist.
a) Let $a,n,m$ be natural numbers, $a > 1$. Prove that if $(a^m + 1)$ is divisible by $(a^n + 1)$ than $m$ is divisible by $n$.
b) Let $a,b,n,m$ be natural numbers, $a>1, a$ and $b$ are relatively prime. Prove that if $(a^m+b^m)$ is divisible by $(a^n+b^n)$ than $m$ is divisible by $n$.
Alex and Kevin are radish watching. The probability that they will see a radish within the next hour is $\frac{1}{17}$. If the probability that they will see a radish within the next $15$ minutes is $p$, determine $\lfloor 1000p \rfloor$. Assume that the probability of seeing a radish at any given moment is uniform for the entire hour.
[i]Proposed by Ephram Chun[/i]
$$\int_1^2\left(x^5+6x^4+14x^3+16x^2+9x+2\right)dx$$
[i]Proposed by Connor Gordon[/i]
Let $ABC$ be a triangle. Point $E,F$ moves on the opposite ray of $BA,CA$ such that $BF=CE$. Let $M,N$ be the midpoint of $BE,CF$. $BF$ cuts $CE$ at $D$
a) Suppost that $I$ is the circumcenter of $(DBE)$ and $J$ is the circumcenter of $(DCF)$, Prove that $MN \parallel IJ$
b) Let $K$ be the midpoint of $MN$ and $H$ be the orthocenter of triangle $AEF$. Prove that when $E$ varies on the opposite ray of $BA$, $HK$ go through a fixed point
Prove that equation $a^2 - b^2=ab - 1$ has infinitely many solutions, if $a,b$ are positive integers
Show that ${2n \choose n} \; \vert \; \text{lcm}(1,2, \cdots, 2n)$ for all positive integers $n$.
Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$
Given a triangle $ABC$ and three circles $x$, $y$ and $z$ such that $A \in y \cap z$, $B \in z \cap x$ and $C \in x \cap y$.
The circle $x$ intersects the line $AC$ at the points $X_b$ and $C$, and intersects the line $AB$ at the points $X_c$ and $B$.
The circle $y$ intersects the line $BA$ at the points $Y_c$ and $A$, and intersects the line $BC$ at the points $Y_a$ and $C$.
The circle $z$ intersects the line $CB$ at the points $Z_a$ and $B$, and intersects the line $CA$ at the points $Z_b$ and $A$.
(Yes, these definitions have the symmetries you would expect.)
Prove that the perpendicular bisectors of the segments $Y_a Z_a$, $Z_b X_b$ and $X_c Y_c$ concur.
Consider the set of all equations $ x^3 \plus{} a_2x^2 \plus{} a_1x \plus{} a_0 \equal{} 0$, where $ a_2$, $ a_1$, $ a_0$ are real constants and $ |a_i| < 2$ for $ i \equal{} 0,1,2$. Let $ r$ be the largest positive real number which satisfies at least one of these equations. Then
$ \textbf{(A)}\ 1 < r < \frac{3}{2}\qquad
\textbf{(B)}\ \frac{3}{2} < r < 2\qquad
\textbf{(C)}\ 2 < r < \frac{5}{2}\qquad
\textbf{(D)}\ \frac{5}{2} < r < 3\qquad \\
\textbf{(E)}\ 3 < r < \frac{7}{2}$
An equilateral triangle is cut as shown in figure 1 and the parts are used to form figure 2. What is the shape of figure 2?
An isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has $AB=13$, $CD=17$, and height $3$. Let $E$ be the intersection of $AC$ and $BD$. Circles $\Omega$ and $\omega$ are circumscribed about triangles $ABE$ and $CDE$. Compute the sum of the radii of $\Omega$ and $\omega$.
Two circles $\Gamma_1$ and $\Gamma_2$ meet at two distinct points $A$ and $B$. A line passing through $A$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ respectively, such that $A$ lies between $C$ and $D$. The tangent at $A$ to $\Gamma_2$ meets $\Gamma_1$ again at $E$. Let $F$ be a point on $\Gamma_2$ such that $F$ and $A$ lie on different sides of $BD$, and $2\angle AFC=\angle ABC$. Prove that the tangent at $F$ to $\Gamma_2$, and lines $BD$ and $CE$ are concurrent.
Suppose that $\triangle ABC$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP = 1$, $BP = \sqrt{3}$, and $CP = 2$. What is $s?$
$\textbf{(A) } 1 + \sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5 + \sqrt{5}} \qquad \textbf{(E) } 2\sqrt{2}$
positive real $C$ is $n-vengeful$ if it is possible to color the cells of an $n \times n$ chessboard such that:
i) There is an equal number of cells of each color.
ii) In each row or column, at least $Cn$ cells have the same color.
a) Show that $\frac{3}{4}$ is $n-vengeful$ for infinitely many values of $n$.
b) Show that it does not exist $n$ such that $\frac{4}{5}$ is $n-vengeful$.
For a positive integer n the number $P(n)$ is the product of the positive divisors of $n$. For example, $P(20) = 8000$, as the positive divisors of $20$ are $1, 2, 4, 5, 10$ and $20$, whose product is $1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20 = 8000$.
(a) Find all positive integers $n$ satisfying $P(n) = 15n$.
(b) Show that there exists no positive integer $n$ such that $P(n) = 15n^2$.
Perpendiculars at their centers of gravity (points of intersection of medians) are restored to the faces of the tetrahedron. Prove that the projections of the three perpendiculars to the fourth face intersect at one point.
For two positive integers $n$ and $d$, let $S_n(d)$ be the set of all ordered $d$-tuples $(x_1,x_2,\dots ,x_d)$ that satisfy all of the following conditions:
i. $x_i\in \{1,2,\dots ,n\}$ for every $i\in\{1,2,\dots ,d\}$;
ii. $x_i\ne x_{i+1}$ for every $i\in\{1,2,\dots ,d-1\}$;
iii. There does not exist $i,j,k,l\in\{1,2,\dots ,d\}$ such that $i<j<k<l$ and $x_i=x_k,\, x_j=x_l$;
a. Compute $|S_3(5)|$
b. Prove that $|S_n(d)|>0$ if and only if $d\leq 2n-1$.
Compute the smallest positive integer $N$ for which $N \cdot 2^{2024}$ is a multiple of $2024$.
Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.
Find the smallest positive integer $n$ with the property that in the set $\{70, 71, 72,... 70 + n\}$ you can choose two different numbers whose product is the square of an integer.
Let $\mathbb{N} = \{1, 2, 3, \dots \}$ be the set of positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for all positive integers $n$ and prime numbers $p$:
$$p \mid f(n)f(p-1)!+n^{f(p)}.$$
[i]Proposed by Dorlir Ahmeti[/i]
Along a circular railroad, $n$ trains circulate in the same direction at equal distances between them. Stations $A, B$ and $C$ on this railroad (denoted as the trains pass them) form an equilateral triangle. Ira enters station $A$ at the same time as Alex enters station $B$ in order to take the nearest train. It is knows that if they enter the stations at the same time as the driver Roma passes a forest, then Ira takes her train earlier than Alex; otherwise Alex takes the train earlier than or simultaneously with Ira. What part of the railroad goes through the forest (between which stations)?