Found problems: 85335
$X$ is a set with $100$ members. What is the smallest number of subsets of $X$ such that every pair of elements belongs to at least one subset and no subset has more than $50$ members? What is the smallest number if we also require that the union of any two subsets has at most $80$ members?
Let $P$ be a polynomial with real coefficients. Prove that if for some integer $k$ $P(k)$ isn't integral, then there exist infinitely many integers $m$, for which $P(m)$ isn't integral.
Given is a triangle $ABC$ and $M$ is the midpoint of the minor arc $BC$. Let $M_1$ be the reflection of $M$ with respect to side $BC$. Prove that the nine-point circle bisects $AM_1$.
Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]
Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.
Let $ABCD$ be a tangential quadrilateral. Let $AB$ meet $CD$ at $E, AD$ intersect $BC$ at $F$. Two arbitrary lines through $E$ meet $AD,BC$ at $M,N, P,Q$ respectively ($M,N \in AD$, $P,Q \in BC$). Another arbitrary pair of lines through $F$ intersect $AB,CD$ at $X, Y,Z, T$ respectively ($X, Y \in AB$,$Z, T \in CD$). Suppose that $d_1, d_2$ are the second tangents from $E$ to the incircles of triangles $FXY, FZT,d_3, d_4$ are the second tangents from $F$ to the incircles of triangles $EMN,EPQ$. Prove that the four lines $d_1, d_2, d_3, d_4$ meet each other at four points and these intersections make a tangential quadrilateral.
Nguyễn Văn Linh
Consider a sequence of words , consisting of the letters $A$ and $B$ .
The first word in the sequence is "$A$" . The k-th word i s obtained from the $(k-1)$-th by means of the following transformation : each $A$ is substituted by $AAB$ , and each $B$ is substituted by $A$. It is easily seen that every word is an initial part of the next word. The initial parts of these words coincide to give a sequence of letters $AABAABAAA BAABAAB...$
(a) In which place of this sequence is the $1000$-th letter $A$?
(b ) Prove that this sequence is not periodic.
(V . Galperin , Moscows)
Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.
Determine the minimum value of $\dfrac{m^m}{1\cdot 3\cdot 5\cdot \ldots \cdot(2m-1)}$ for positive integers $m$.
Prove the inequality
\[(n + 1)\cos\frac{\pi}{n + 1}- n\cos\frac{\pi}{n}> 1\]
for all natural numbers $n \ge 2.$
Given cyclic quadrilateral $ABCD$. Four circles each touching its diagonals and the circumcircle internally are equal. Is $ABCD$ a square?
(C.Pohoata, A.Zaslavsky)
Let $ABC$ be a triangle with $\angle C=90^\circ$, and $D$ be a point outside $ABC$, such that $\angle ADC=\angle BAC$. The segments $CD$ and $AB$ meet at point $E$. It is known that the distance from $E$ to $AC$ is equal to the circumradius of triangle $ADE$. Find the angles of triangle $ABC$.
Find all positive integers $ n $ with the following property: It is possible to fill a $ n \times n $ chessboard with one of arrows $ \uparrow, \downarrow, \leftarrow, \rightarrow $ such that
1. Start from any grid, if we follows the arrows, then we will eventually go back to the start point.
2. For every row, except the first and the last, the number of $ \uparrow $ and the number of $ \downarrow $ are the same.
3. For every column, except the first and the last, the number of $ \leftarrow $ and the number of $ \rightarrow $ are the same.
Let $f$ be a continuous real-valued function on $\mathbb R^3$. Suppose that for every sphere $S$ of radius $1$, the integral of $f(x,y,z)$ over the surface of $S$ equals zero. Must $f(x,y,z)$ be identically zero?
Find all real $x$ such that
$$\lfloor x \lceil x \rceil \rfloor = 2022.$$
Express your answer in interval notation.
On the perimeter of a rectangle, point $ M $ is chosen. Find the shortest path whose beginning and end are point $ M $ and which has a point in common with each side of the rectangle.
A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.
There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does?
[i]Mikhail Svyatlovskiy[/i]
$ \frac{2}{1\minus{}\frac{2}{3}}\equal{}$
\[ \textbf{(A)}\ \minus{}3 \qquad
\textbf{(B)}\ \minus{}\frac{4}{3} \qquad
\textbf{(C)}\ \frac{2}{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 6
\]
Let $ABCD$ be a parallelogram with $AB=CD=16$ and $BC=AD=24.$ Suppose the angle bisectors of $\angle A$ and $\angle D$ intersect $BC$ at $E$ and $F,$ respectively. Moreover, suppose $AE$ and $DF$ intersect at $P.$ Given that the sum of the areas of quadrilaterals $ABFP$ and $DCEP$ is $100,$ compute the area of the parallelogram.
Let $ABC$ be an arbitrary triangle. Let the excircle tangent to side $a$ be tangent to lines $AB,BC$ and $CA$ at points $C_a,A_a,$ and $B_a,$ respectively. Similarly, let the excircle tangent to side $b$ be tangent to lines $AB,BC,$ and $CA$ at points $C_b,A_b,$ and $B_b,$ respectively. Finally, let the excircle tangent to side $c$ be tangent to lines $AB,BC,$ and $CA$ at points $C_c,A_c,$ and $B_c,$ respectively. Let $A'$ be the intersection of lines $A_bC_b$ and $A_cB_c.$ Similarly, let $B'$ be the intersection of lines $B_aC_a$ and $A_cB_c,$ and let $C$ be the intersection of lines $B_aC_a$ and $A_bC_b.$ Finally, let the incircle be tangent to sides $a,b,$ and $c$ at points $T_a,T_b,$ and $T_c,$ respectively.
a) Prove that lines $A'A_a,B'B_b,$ and $C'C_c$ are concurrent.
b) Prove that lines $A'T_a, B'T_b,$ and $C'T_c$ are also concurrent, and their point of intersection is on the line defined by the orthocentre and the incentre of triangle $ABC.$
[i]Proposed by Viktor Csaplár, Bátorkeszi and Dániel Hegedűs, Gyöngyös[/i]
Let $a$, $b$, $c$, $d$ be positive numbers. Prove that
$$(a^2 + b^2 + c^2 + d^2)^2 \ge (a+b)(b+c)(c+d)(d+a)$$
When does equality hold?
(Georg Anegg)
We say that the set of step lengths $D\subset \mathbb{Z}_+=\{1,2,\ldots\}$ is [i]excellent[/i] if it has the following property: If we split the set of integers into two subsets $A$ and $\mathbb{Z}\setminus{A}$, at least other set contains element $a-d,a,a+d$ (i.e. $\{a-d,a,a+d\} \subset A$ or $\{a-d,a,a+d\}\in \mathbb{Z}\setminus A$ from some integer $a\in \mathbb{Z},d\in D$.) For example the set of one element $\{1\}$ is not excellent as the set of integer can be split into even and odd numbers, and neither of these contains three consecutive integer. Show that the set $\{1,2,3,4\}$ is excellent but it has no proper subset which is excellent.
Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.
Let $n \geq 1$ be an integer. A [b]path[/b] from $(0,0)$ to $(n,n)$ in the $xy$ plane is a chain of consecutive unit moves either to the right (move denoted by $E$) or upwards (move denoted by $N$), all the moves being made inside the half-plane $x \geq y$. A [b]step[/b] in a path is the occurence of two consecutive moves of the form $EN$. Show that the number of paths from $(0,0)$ to $(n,n)$ that contain exactly $s$ steps $(n \geq s \geq 1)$ is
\[\frac{1}{s} \binom{n-1}{s-1} \binom{n}{s-1}.\]