Found problems: 85335
Cono Sur Shortlist - geometry, 2020.G4
Let $ABC$ be a triangle with circumcircle $\omega$. The bisector of $\angle BAC$ intersects $\omega$ at point $A_1$. Let $A_2$ be a point on the segment $AA_1$, $CA_2$ cuts $AB$ and $\omega$ at points $C_1$ and $C_2$, respectively. Similarly, $BA_2$ cuts $AC$ and $\omega$ at points $B_1$ and $B_2$, respectively. Let $M$ be the intersection point of $B_1C_2$ and $B_2C_1$. Prove that $MA_2$ passes the midpoint of $BC$.
[i]proposed by Jhefferson Lopez, Perú[/i]
1963 AMC 12/AHSME, 40
If $x$ is a number satisfying the equation $\sqrt[3]{x+9}-\sqrt[3]{x-9}=3$, then $x^2$ is between:
$\textbf{(A)}\ 55\text{ and }65 \qquad
\textbf{(B)}\ 65\text{ and }75\qquad
\textbf{(C)}\ 75\text{ and }85 \qquad
\textbf{(D)}\ 85\text{ and }95 \qquad
\textbf{(E)}\ 95\text{ and }105$
2003 Moldova National Olympiad, 12.2
For every natural number $n\geq{2}$ consider the following affirmation $P_n$:
"Consider a polynomial $P(X)$ (of degree $n$) with real coefficients. If its derivative $P'(X)$ has $n-1$ distinct real roots, then there is a real number $C$ such that the equation $P(x)=C$ has $n$ real,distinct roots."
Are $P_4$ and $P_5$ both true? Justify your answer.
2002 Tournament Of Towns, 7
[list]
[*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this?
[*] Previous problem for the grid of $5\times 5$ lattice.[/list]
2009 Croatia Team Selection Test, 4
Prove that there are infinite many positive integers $ n$ such that
$ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.
MathLinks Contest 5th, 6.3
Let $x, y, z$ be three positive numbers such that $(x + y-z) \left( \frac{1}{x}+ \frac{1}{y}- \frac{1}{z} \right)=4$.
Find the minimal value of the expression $$E(x, y, z) = (x^4 + y^4 + z^4) \left( \frac{1}{x^4}+ \frac{1}{y^4}+ \frac{1}{z^4} \right) .$$
2021 Bundeswettbewerb Mathematik, 3
Consider a triangle $ABC$ with $\angle ACB=120^\circ$. Let $A’, B’, C’$ be the points of intersection of the angular bisector through $A$, $B$ and $C$ with the opposite side, respectively.
Determine $\angle A’C’B’$.
2001 Greece National Olympiad, 1
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 $\frac{1}{KB}=\frac{1}{KA}+\frac{1}{KC}.$
2021 AIME Problems, 8
Find the number of integers $c$ such that the equation $$\left||20|x|-x^2|-c\right|=21$$ has $12$ distinct real solutions.
2005 iTest, 5
Find the sum of the answers to all even numbered Short Answer problems, with the exception of #26, rounded to the nearest tenth.
[i](.7 points)[/i]
2025 NEPALTST, 2
Find all integers $n$ such that if
\[
1 = d_1 < d_2 < \cdots < d_{k-1} < d_k = n
\]
are the divisors of $n$, then the sequence
\[
d_2 - d_1,\, d_3 - d_2,\, \ldots,\, d_k - d_{k-1}
\]
forms a permutation of an arithmetic progression.
[i](Kritesh Dhakal, Nepal)[/i]
2022 Harvard-MIT Mathematics Tournament, 4
Suppose $n \ge 3$ is a positive integer. Let $a_1 < a_2 < ... < a_n$ be an increasing sequence of positive real numbers, and let $a_{n+1} = a_1$. Prove that $$\sum_{k=1}^{n}\frac{a_k}{a_{k+1}}>\sum_{k=1}^{n}\frac{a_{k+1}}{a_k}$$
2017 Princeton University Math Competition, 15
How many ordered pairs of positive integers $(x, y)$ satisfy $yx^y = y^{2017}$?
2018 Swedish Mathematical Competition, 6
For which positive integers $n$ can the polynomial $p(x) = 1 + x^n + x^{2n}$ is written as a product of two polynomials with integer coefficients (of degree $\ge 1$)?
2007 Balkan MO Shortlist, A6
Find all real functions $f$ defined on $ \mathbb R$, such that \[f(f(x)+y) = f(f(x)-y)+4f(x)y ,\] for all real numbers $x,y$.
1968 Putnam, A6
Find all polynomials whose coefficients are all $\pm1$ and whose roots are all real.
1975 IMO, 3
In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$.
Prove that
[b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and
[b]b.)[/b] $QR = RP.$
2022 Cyprus TST, 4
Let
\[M=\{1, 2, 3, \ldots, 2022\}\]
Determine the least positive integer $k$, such that for every $k$ subsets of $M$ with the cardinality of each subset equal to $3$, there are two of these subsets with exactly one common element.
2016 PUMaC Geometry B, 8
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through $A$ parallel to $BD$ intersect line $CD$ at $E$ and $\omega$ at $Y \ne A$. If $AB = 10, AD = 24, XA = 17$, and $XB = 21$, then the area of $\vartriangle DEY$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.
2014 China Team Selection Test, 5
Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$, if $|E|\geq c|V|$, then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord.
Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$, $v_iv_{i+1}\in E$ for all $1\leq i\leq n$ $(n\geq 3, v_{n+1}=v_1)$; a cycle containing a chord is the cycle ${v_1,v_2...,v_n}$, such that there exist $i,j, 1< i-j< n-1$, satisfying $v_iv_j\in E$.
2023 UMD Math Competition Part II, 3
Let $p$ be a prime, and $n > p$ be an integer. Prove that
\[ \binom{n+p-1}{p} - \binom{n}{p} \]
is divisible by $n$.
1988 IMO Longlists, 7
Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $ \frac {n}{2}$ zeros?
2015 Princeton University Math Competition, A1
How many integer pairs $(a, b)$ with $1 < a, b \le 2015$ are there such that $\log_a b$ is an integer?
LMT Guts Rounds, 4
The perimeter of a square is equal in value to its area. Determine the length of one of its sides.
2003 Irish Math Olympiad, 1
If $a,b,c$ are the sides of a triangle whose perimeter is equal to 2 then prove that:
a) $abc+\frac{28}{27}\geq ab+bc+ac$;
b) $abc+1<ab+bc+ac$
See also [url]http://www.mathlinks.ro/Forum/viewtopic.php?t=47939&view=next[/url] (problem 1)
:)