Found problems: 85335
Let $P(x)$ be a polynomial with the coefficients being $0$ or $1$ and degree $2023$. If $P(0)=1$, then prove that every real root of this polynomial is less than $\frac{1-\sqrt{5}}{2}$.
Let $ \leq$ be a reflexive, antisymmetric relation on a finite set $ A$. Show that this relation can be extended to an appropriate finite superset $ B$ of $ A$ such that $ \leq$ on $ B$ remains reflexive, antisymmetric, and any two elements of $ B$ have a least upper bound as well as a greatest lower bound. (The relation $ \leq$ is extended to $ B$ if for $ x,y \in A , x \leq y$ holds in $ A$ if and only if it holds in $ B$.)
[i]E. Freid[/i]
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$.
Show that $A,X,Y$ are collinear.
[asy]size(180);
draw((1,0)--(2,0)--(2,10)--(1,10)--cycle);
draw((3,0)--(4,0)--(4,8)--(3,8)--cycle);
draw((5,0)--(6,0)--(6,6)--(5,6)--cycle);
draw((7,0)--(8,0)--(8,6)--(7,6)--cycle);
draw((9,0)--(10,0)--(10,10)--(9,10)--cycle);
draw((0,2)--(-0.5,2));
draw((0,4)--(-0.5,4));
draw((0,6)--(-0.5,6));
draw((0,8)--(-0.5,8));
draw((0,10)--(-0.5,10));
draw((0,10)--(0,0));
draw((0,0)--(10,0));
label("1",(-0.5,2),W);
label("2",(-0.5,4),W);
label("3",(-0.5,6),W);
label("4",(-0.5,8),W);
label("5",(-0.5,10),W);
label("A",(1.5,-0.5),S);
label("B",(3.5,-0.5),S);
label("C",(5.5,-0.5),S);
label("D",(7.5,-0.5),S);
label("F",(9.5,-0.5),S);
label("Grade",(5,-3),S);
label("$\#$ of Students",(-4,5),W);[/asy]
The bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory?
\[ \textbf{(A)}\ \frac{1}{2} \qquad
\textbf{(B)}\ \frac{2}{3} \qquad
\textbf{(C)}\ \frac{3}{4} \qquad
\textbf{(D)}\ \frac{4}{5} \qquad
\textbf{(E)}\ \frac{9}{10}
\]
A segment $AB$ is given. We erect the equilateral triangles $ABC$ and $ADB$ above and below $AB$. We denote the midpoints of $AC$ and $BC$ by $E$ and $F$ respectively. Prove that the straight lines $DE$ and $DF$ divide the segment $AB$ into three parts of equal length .
A cube with edge length $ n $ is divided into $ n^3 $ unit cubes by planes parallel to its faces. How many pairs of such unit cubes exist that have no more than two vertices in common?
Let $\Gamma$ be the circumcircle of a fixed triangle $ABC$, and let $M$ and $N$ be the midpoints of the arcs $BC$ and $CA$, respectively. For any point $X$ on the arc $AB$, let $O_1$ and $O_2$ be the incenters of $\vartriangle XAC$ and $\vartriangle XBC$, and let the circumcircle of $\vartriangle XO_1O_2$ intersect $\Gamma$ at $X$ and $Q$. Prove that triangles $QNO_1$ and $QMO_2$ are similar, and find all possible locations of point $Q$.
Prove that $\tau ((n+1)!) \leq 2 \tau (n!)$ for all positive integers $n$.
Find the smallest positive integer $G$ such that there exist distinct positive integers $a, b, c$ with the following properties:
$\: \bullet \: \gcd(a, b, c) = G$.
$\: \bullet \: \text{lcm}(a, b) = \text{lcm}(a, c) = \text{lcm}(b, c)$.
$\: \bullet \: \frac{1}{a} + \frac{1}{b}, \frac{1}{a} + \frac{1}{c},$ and $\frac{1}{b} + \frac{1}{c}$ are reciprocals of integers.
$\: \bullet \: \gcd(a, b) + \gcd(a, c) + \gcd(b, c) = 16G$.
Let $ABCD$ be a rectangle with $\angle ABD = 15^o, BD = 6$ cm. Compute the area of the rectangle.
A. $9$ cm$^2$ B. $9 \sqrt3$ cm$^2$ C. $18$ cm$^2$ D. $18 \sqrt3$ cm$^2$ E. $24 \sqrt3$ cm$^2$
Determine all triplets $(a,b,c)$ of real numbers such that sets $\{a^2-4c, b^2-2a, c^2-2b \}$ and $\{a-c,b-4c,a+b\}$ are equal and $2a+2b+6=5c$. In every set all elements are pairwise distinct
Given that $x$, $y$ and $z$ are positive integers such that $x^3y^5z^6$ is a perfect 7th power of a positive integer, show that also $x^5y^6z^3$ is a perfect 7th power.
The repeating decimal $2.0151515\ldots$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Let $ S$ be the set of nonnegative integers. Find all functions $ f,g,h: S\rightarrow S$ such that
$ f(m\plus{}n)\equal{}g(m)\plus{}h(n),$ for all $ m,n\in S$, and
$ g(1)\equal{}h(1)\equal{}1$.
Some of the faces of a convex polyhedron $M$ are painted in blue, others are painted in white and there are no two walls with a common edge. Prove that if the sum of surfaces of the blue walls is bigger than half surface of $M$ then it may be inscribed a sphere in the polyhedron given $(M)$.
[i](H. Lesov)[/i]
Find all real polynomials $P(x, y)$ such that $P(x+y, x-y) = 2P(x, y)$, for all $x, y$ in $R$.
Find all complex solutions to the system
\begin{align*}
(a+ic)^3+(ia+b)^3+(-b+ic)^3&=-6,\\
(a+ic)^2+(ia+b)^2+(-b+ic)^2&=6,\\
(1+i)a+2ic&=0.\end{align*}
A triangle $ABC$ is inscribed in a circle of radius $R.$ Let $BD$ and $CE$ be the bisectors of the angles $B$ and $C$ respectively and let the line $DE$ meet the arc $AB$ not containing $C$ at point $K.$ Let $A_1, B_1, C_1$ be the feet of perpendiculars from $K$ to $BC, AC, AB,$ and $x, y$ be the distances from $D$ and $E$ to $BC,$ respectively.
(a) Express the lengths of $KA_1, KB_1, KC_1$ in terms of $x, y$ and the ratio $l = KD/ED.$
(b) Prove that $\frac{1}{KB}=\frac{1}{KA}+\frac{1}{KC}.$
Prove that there exists an infinite sequence of positive integers $a_1,a_2,a_3,\dots$ such that for any positive integer $k$, $a_k^2+a_k+2023$ has at least $k$ distinct positive divisors.
A pack of MIT students are holding an escape room, where students may compete in teams of 4, 5, or 6. There is \$60 dollars worth of prize money in Amazon gift cards for the winning team. If each gift card can contain any whole number of dollars, what is the minimum number of gift cards required so that the prize money can be distributed evenly among any team?
[i]Lightning 4.1[/i]
The decimal representation of all integers from $1$ to an arbitrary integer $n$ are written one after another as such:
$$123... 91011... 99100... (n).$$
Does there exist $n$ such that each of the digits $0,1,2,...,9$ appears the same number of times in the given sequence?
(A Andzans)
You are given a $2 \times 2$ array with a positive integer in each field. If we add the product of the numbers in the first column, the product of the numbers in the second column, the product of the numbers in the first row and the product of the numbers in the second row, we get $2021$.
a) Find possible values for the sum of the four numbers in the table.
b) Find the number of distinct arrays that satisfy the given conditions that contain four pairwise distinct numbers in arrays.
[u]Set 1[/u]
[b]1.1[/b] How many positive integer divisors are there of $2^2 \cdot 3^3 \cdot 5^4$?
[b]1.2[/b] Let $T$ be the answer from the previous problem. For how many integers $n$ between $1$ and $T$ (inclusive) is $\frac{(n)(n - 1)(n - 2)}{12}$ an integer?
[b]1.3[/b] Let $T$ be the answer from the previous problem. Find $\frac{lcm(T, 36)}{gcd(T, 36)}$.
PS. You should use hide for answers.
A sequence of natural numbers $c_1, c_2,\dots$ is called [i]perfect[/i] if every natural
number $m$ with $1\le m \le c_1 +\dots+ c_n$ can be represented as
$m =\frac{c_1}{a_1}+\frac{c_2}{a_2}+\dots+\frac{c_n}{a_n}$
Given $n$, find the maximum possible value of $c_n$ in a perfect sequence $(c_i)$.
The angle bisectors at $A$ and $C$ in a non-isosceles triangle $ABC$ with incenter $I$ intersect its circumcircle $k$ at $A_0$ and $C_0$, respectively. The line through $I$, parallel to $AC$, intersects $A_0C_0$ at $P$. Prove that $PB$ is tangent to $k$.