Found problems: 85335
1992 Baltic Way, 10
Find all fourth degree polynomial $ p(x)$ such that the following four conditions are satisfied:
(i) $ p(x)\equal{}p(\minus{}x)$ for all $ x$,
(ii) $ p(x)\ge0$ for all $ x$,
(iii) $ p(0)\equal{}1$
(iv) $ p(x)$ has exactly two local minimum points $ x_1$ and $ x_2$ such that $ |x_1\minus{}x_2|\equal{}2$.
2007 Romania Team Selection Test, 3
Let $ABCDE$ be a convex pentagon, such that $AB=BC$, $CD=DE$, $\angle B+\angle D=180^{\circ}$, and it's area is $\sqrt2$.
a) If $\angle B=135^{\circ}$, find the length of $[BD]$.
b) Find the minimum of the length of $[BD]$.
2021 ABMC., 2021 Oct
[b]p1.[/b] How many perfect squares are in the set: $\{1, 2, 4, 9, 10, 16, 17, 25, 36, 49\}$?
[b]p2.[/b] If $a \spadesuit b = a^b - ab - 5$, what is the value of $2 \spadesuit 11$?
[b]p3.[/b] Joe can catch $20$ fish in $5$ hours. Jill can catch $35$ fish in $7$ hours. If they work together, and the number of days it takes them to catch $900$ fish is represented by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, what is $m + n$? Assume that they work at a constant rate without taking breaks and that there are an infinite number of fish to catch.
[b]p4.[/b] What is the units digit of $187^{10}$?
[b]p5.[/b] What is the largest number of regions we can create by drawing $4$ lines in a plane?
[b]p6.[/b] A regular hexagon is inscribed in a circle. If the area of the circle is $2025\pi$, given that the area of the hexagon can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$ where $gcd(a, c) = 1$ and $b$ is not divisible by the square of any number other than $1$, find $a + b + c$.
[b]p7.[/b] Find the number of trailing zeroes in the product $3! \cdot 5! \cdot 719!$.
[b]p8.[/b] How many ordered triples $(x, y, z)$ of odd positive integers satisfy $x + y + z = 37$?
[b]p9.[/b] Let $N$ be a number with $2021$ digits that has a remainder of $1$ when divided by $9$. $S(N)$ is the sum of the digits of $N$. What is the value of $S(S(S(S(N))))$?
[b]p10.[/b] Ayana rolls a standard die $10$ times. If the probability that the sum of the $10$ die is divisible by $6$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$, what is $m + n$?
[b]p11.[/b] In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. The inscribed circle touches the side $BC$ at point $D$. The line $AI$ intersects side $BC$ at point $K$ given that $I$ is the incenter of triangle $ABC$. What is the area of the triangle $KID$?
[b]p12.[/b] Given the cubic equation $2x^3+8x^2-42x-188$, with roots $a, b, c$, evaluate $|a^2b+a^2c+ab^2+b^2c+c^2a+bc^2|$.
[b]p13.[/b] In tetrahedron $ABCD$, $AB=6$, $BC=8$, $CA=10$, and $DA$, $DB$, $DC=20$. If the volume of $ABCD$ is $a\sqrt{b}$ where $a$, $b$ are positive integers and in simplified radical form, what is $a + b$?
[b]p14.[/b] A $2021$-digit number starts with the four digits $2021$ and the rest of the digits are randomly chosen from the set $0$,$1$,$2$,$3$,$4$,$5$,$6$. If the probability that the number is divisible by $14$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. what is $m + n$?
[b]p15.[/b] Let $ABCD$ be a cyclic quadrilateral with circumcenter $O_1$ and circumradius $20$, Let the intersection of $AC$ and $BD$ be $E$. Let the circumcenter of $\vartriangle EDC$ be $O_2$. Given that the circumradius of 4EDC is $13$; $O_1O_2 = 11$, $BE = 11 \sqrt2$, find $O_1E^2$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1995 Belarus National Olympiad, Problem 4
Given a triangle $ABC$, let $K$ be the midpoint of $AB$ and $L$ be the point on the side $AC$ such that $AL = LC + CB$. Show that if $\angle KLB = 90^o$ then $AC = 3 CB$ and conversely, if $AC = 3 CB$ then $\angle KLB = 90^o$.
2003 India IMO Training Camp, 6
A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find $z_n$, the maximum number of regions into which $n$ zig-zags can divide the plane. For example, $z_1=2,z_2=12$(see the diagram). Of these $z_n$ regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in $n$ of degree not exceeding $2$.
[asy]
draw((30,0)--(-70,0), Arrow);
draw((30,0)--(-20,-40));
draw((-20,-40)--(80,-40), Arrow);
draw((0,-60)--(-40,20), dashed, Arrow);
draw((0,-60)--(0,15), dashed);
draw((0,15)--(40,-65),dashed, Arrow);
[/asy]
2016 Harvard-MIT Mathematics Tournament, 4
Let $n > 1$ be an odd integer. On an $n \times n$ chessboard the center square and four corners are deleted.
We wish to group the remaining $n^2-5$ squares into $\frac12(n^2-5)$ pairs, such that the two squares in each pair intersect at exactly one point
(i.e.\ they are diagonally adjacent, sharing a single corner).
For which odd integers $n > 1$ is this possible?
2016 Harvard-MIT Mathematics Tournament, 9
Victor has a drawer with two red socks,
two green socks,
two blue socks,
two magenta socks,
two lavender socks,
two neon socks,
two mauve socks,
two wisteria socks,
and $2000$ copper socks,
for a total of $2016$ socks.
He repeatedly draws two socks at a time from the drawer at random, and stops if the socks are of the same color. However, Victor is red-green colorblind, so he also stops if he sees a red and green sock.
What is the probability that Victor stops with two socks of the same color? Assume Victor returns both socks to the drawer at each step.
2023-IMOC, G1
Triangle $ABC$ has circumcenter $O$. $M$ is the midpoint of arc $BC$ not containing $A$. $S$ is a point on $(ABC)$ such that $AS$ and $BC$ intersect on the line passing through $O$ and perpendicular to $AM$. $D$ is a point such that $ABDC$ is a parallelogram. Prove that $D$ lies on the line $SM$.
2014-2015 SDML (High School), 4
A rubber band is wrapped around two pipes as shown. One has radius $3$ inches and the other has radius $9$ inches. The length of the band can be expressed as $a\pi+b\sqrt{c}$ where $a$, $b$, $c$ are integers and $c$ is square free. What is $a+b+c$?
[asy]
size(4cm);
draw(circle((0,0),3));
draw(circle((12,0),9));
draw(3*dir(120)--(12,0)+9*dir(120));
draw(3*dir(240)--(12,0)+9*dir(240));
[/asy]
2013 India National Olympiad, 5
In an acute triangle $ABC,$ let $O,G,H$ be its circumcentre, centroid and orthocenter. Let $D\in BC, E\in CA$ and $OD\perp BC, HE\perp CA.$ Let $F$ be the midpoint of $AB.$ If the triangles $ODC, HEA, GFB$ have the same area, find all the possible values of $\angle C.$
2008 District Olympiad, 4
Determine $ x,y,z>0$ for which $ x^3y\plus{}3<\equal{}4z, y^3z\plus{}3<\equal{}4x,z^3x\plus{}3<\equal{}4y.$
2022 Lusophon Mathematical Olympiad, 1
How many triples $(a,b,c)$ with $a,b,c \in \mathbb{R}$ satisfy the following system?
$$\begin{cases} a^4-b^4=c \\ b^4-c^4=a \\ c^4-a^4=b \end{cases}$$.
2019 Ecuador NMO (OMEC), 1
Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.
2021 Peru EGMO TST, 5
Determine all integers $k$ such that the equation:
$$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{k}{xyz}$$
has an infinite number of integer solutions $(x,y,z)$ with gcd$(k,xyz)=1$.
1988 Tournament Of Towns, (170) 3
Find all real solutions of the system of equations
$$\begin{cases} (x_3 + x_4 + x_5)^5 = 3x_1 \\
(x_4 + x_5 + x_1)^5 = 3x_2\\
(x_5 + x _1 + x_2)^5 = 3x_3\\
(x_1 + x_2 + x_3)^5 = 3x_4\\
(x_2 + x_3 + x_4)^5 = 3x_5 \end{cases}$$
(L. Tumescu , Romania)
1969 IMO Shortlist, 67
Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.
Estonia Open Senior - geometry, 2009.2.4
a) An altitude of a triangle is also a tangent to its circumcircle. Prove that some angle of the triangle is larger than $90^o$ but smaller than $135^o$.
b) Some two altitudes of the triangle are both tangents to its circumcircle. Find the angles of the triangle.
1999 German National Olympiad, 6b
Determine all pairs ($m,n$) of natural numbers for which $4^m + 5^n$ is a perfect square.
2007 Balkan MO Shortlist, A3
For $n\in\mathbb{N}$, $n\geq 2$, $a_{i}, b_{i}\in\mathbb{R}$, $1\leq i\leq n$, such that \[\sum_{i=1}^{n}a_{i}^{2}=\sum_{i=1}^{n}b_{i}^{2}=1, \sum_{i=1}^{n}a_{i}b_{i}=0. \] Prove that
\[\left(\sum_{i=1}^{n}a_{i}\right)^{2}+\left(\sum_{i=1}^{n}b_{i}\right)^{2}\leq n. \]
[i]Cezar Lupu & Tudorel Lupu[/i]
2017 AMC 10, 12
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
1960 AMC 12/AHSME, 22
The eqquality $(x+m)^2-(x+n)^2=(m-n)^2$, where $m$ and $n$ are [i]unequal[/i] non-zero constants, is satisfied by $x=am+bn$, where:
$ \textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad$
$\textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values}\qquad$
$\textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value}\qquad$
$\textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values}\qquad$
$\textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value} $
2014 ISI Entrance Examination, 3
Consider $f(x)=x^4+ax^3+bx^2+cx+d\; (a,b,c,d\in\mathbb{R})$. It is known that $f$ intersects X-axis in at least $3$ (distinct) points. Show either $f$ has $4$ $\mathbf{distinct}$ real roots or it has $3$ $\mathbf{distinct}$ real roots and one of them is a point of local maxima or minima.
2022 IFYM, Sozopol, 5
Let $\Delta ABC$ be an acute scalene triangle with $AC<BC$, an orthocenter $H$ and altitudes $AE$, $BF$. The points $E'$ and $F'$ are symmetrical to $E$ and $F$ with respect to $A$ and $B$ respectively. Point $O$ is the center of the circumscribed circle of $ABC$ and $M$ is the midpoint of $AB$. Let $N$ be the midpoint of $OM$. Prove that the tangent through $H$ to the circumscribed circle of $\Delta E'HF'$ is perpendicular to line $CN$.
2016 China National Olympiad, 6
Let $G$ be a complete directed graph with $100$ vertices such that for any two vertices $x,y$ one can find a directed path from $x$ to $y$.
a) Show that for any such $G$, one can find a $m$ such that for any two vertices $x,y$ one can find a directed path of length $m$ from $x$ to $y$ (Vertices can be repeated in the path)
b) For any graph $G$ with the properties above, define $m(G)$ to be smallest possible $m$ as defined in part a). Find the minimim value of $m(G)$ over all such possible $G$'s.
2012 Princeton University Math Competition, A3 / B6
Compute $\Sigma_{n=1}^{\infty}\frac{n + 1}{n^2(n + 2)^2}$ .
Your answer in simplest form can be written as $a/b$, where $a, b$ are relatively-prime positive integers. Find $a + b$.