Found problems: 85335
2009 Harvard-MIT Mathematics Tournament, 3
Let $T$ be a right triangle with sides having lengths $3$, $4$, and $5$. A point $P$ is called [i]awesome[/i] if P is the center of a parallelogram whose vertices all lie on the boundary of $T$. What is the area of the set of awesome points?
2012 Princeton University Math Competition, A3
Six ants are placed on the vertices of a regular hexagon with an area of $12$. At each point in time, each ant looks at the next ant in the hexagon (in counterclockwise order), and measures the distance, $s$, to the next ant. Each ant then proceeds towards the next ant at a speed of $\frac{s}{100}$ units per year. After T years, the ants’ new positions are the vertices of a new hexagon with an area of $4$. T is of the form $a \ln b$, where $b$ is square-free. Find $a + b$.
2022 IMAR Test, 2
Let $n, k$ be natural numbers, $1 \leq k < n$. In each vertex of a regular polygon with $n$ sides is written $1$ or $-1$. At each step we choose $k$ consecutive vertices and change their signs. Is it possible that, starting from a certain configuration and by doing the operation a few times to obtain any other configuration?
2011 China Team Selection Test, 2
Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.
1983 All Soviet Union Mathematical Olympiad, 364
The kindergarten group is standing in the column of pairs. The number of boys equals the number of girls in each of the two columns. The number of mixed (boy and girl) pairs equals to the number of the rest pairs. Prove that the total number of children in the group is divisible by eight.
2020 Middle European Mathematical Olympiad, 4#
Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \dots, x_n$ such that
$$ \frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.$$
2011 Postal Coaching, 1
Let $ABCD$ be a quadrilateral with an inscribed circle, centre $O$. Let
\[AO = 5, BO =6, CO = 7, DO = 8.\]
If $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, determine $\frac{OM}{ON}$ .
1974 Spain Mathematical Olympiad, 8
The sides of a convex regular polygon of $L + M + N$ sides are to be given draw in three colors: $L$ of them with a red stroke, $M$ with a yellow stroke, and $N$ with a blue. Express, through inequalities, the necessary and sufficient conditions so that there is a solution (several, in general) to the problem of doing it without leaving two adjacent sides drawn with the same color.
2005 France Team Selection Test, 2
Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle).
Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.
2015 Iran MO (2nd Round), 3
Consider a triangle $ABC$ . The points $D,E$ are on sides $AB,AC$ such that $BDEC$ is a cyclic quadrilateral. Let $P$ be the intersection of $BE$ and $CD$. $H$ is a point on $AC$ such that $\angle PHA = 90^{\circ}$. Let $M,N$ be the midpoints of $AP,BC$. Prove that: $ ACD \sim MNH $.
2021 CHKMO, 4
Let $a,b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that
\[\dfrac{1}{a^3+2b^2+2b+4}+\dfrac{1}{b^3+2c^2+2c+4}+\dfrac{1}{c^3+2a^2+2a+4}\leq \dfrac13.\]
2018 ASDAN Math Tournament, 3
Compute $ax^{2018}+by^{2018}$, given that there exist real $a$, $b$, $x$, and $y$ which satisfy the following four equations:
\begin{align*}
ax^{2014}+by^{2014}&=6\\
ax^{2015}+by^{2015}&=7\\
ax^{2016}+by^{2016}&=3\\
ax^{2017}+by^{2017}&=50.
\end{align*}
2023 All-Russian Olympiad, 2
Initially, a word of $250$ letters with $125$ letters $A$ and $125$ letters $B$ is written on a blackboard. In each operation, we may choose a contiguous string of any length with equal number of letters $A$ and equal number of letters $B$, reverse those letters and then swap each $B$ with $A$ and each $A$ with $B$ (Example: $ABABBA$ after the operation becomes $BAABAB$). Decide if it possible to choose initial word, so that after some operations, it will become the same as the first word, but in reverse order.
2019 International Zhautykov OIympiad, 1
Prove that there exist at least $100!$ ways to write $100!$ as sum of elements of set {$1!,2!,3!...99!$}
(each number in sum can be two or more times)
2015 Purple Comet Problems, 20
For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$. Find the minimum possible value for $|p(15)|$.
2022 AIME Problems, 15
Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon.
[asy]
import geometry;
size(10cm);
point O1=(0,0),O2=(15,0),B=9*dir(30);
circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B);
point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0];
filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black);
draw(w1);
draw(w2);
draw(O1--O2,dashed);
draw(o);
dot(O1);
dot(O2);
dot(A);
dot(D);
dot(C);
dot(B);
label("$\omega_1$",8*dir(110),SW);
label("$\omega_2$",5*dir(70)+(15,0),SE);
label("$O_1$",O1,W);
label("$O_2$",O2,E);
label("$B$",B,N+1/2*E);
label("$A$",A,N+1/2*W);
label("$C$",C,S+1/4*W);
label("$D$",D,S+1/4*E);
label("$15$",midpoint(O1--O2),N);
label("$16$",midpoint(C--D),N);
label("$2$",midpoint(A--B),S);
label("$\Omega$",o.C+(o.r-1)*dir(270));
[/asy]
2023 VN Math Olympiad For High School Students, Problem 5
Given a polynomial$$P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0\in \mathbb{Z}[x]$$
with degree $n\ge 2$ and $a_o\ne 0.$
Prove that if $|a_{n-1}|>1+|a_{n-2}|+...+|a_1|+|a_0|$, then $P(x)$ is irreducible in $\mathbb{Z}[x].$
2018 AMC 12/AHSME, 25
Circles $\omega_1$, $\omega_2$, and $\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\omega_1$, $\omega_2$, and $\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_iP_{i+1}$ is tangent to $\omega_i$ for each $i=1,2,3$, where $P_4 = P_1$. See the figure below. The area of $\triangle P_1P_2P_3$ can be written in the form $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b$. What is $a+b$?
[asy]
unitsize(12);
pair A = (0, 8/sqrt(3)), B = rotate(-120)*A, C = rotate(120)*A;
real theta = 41.5;
pair P1 = rotate(theta)*(2+2*sqrt(7/3), 0), P2 = rotate(-120)*P1, P3 = rotate(120)*P1;
filldraw(P1--P2--P3--cycle, gray(0.9));
draw(Circle(A, 4));
draw(Circle(B, 4));
draw(Circle(C, 4));
dot(P1);
dot(P2);
dot(P3);
defaultpen(fontsize(10pt));
label("$P_1$", P1, E*1.5);
label("$P_2$", P2, SW*1.5);
label("$P_3$", P3, N);
label("$\omega_1$", A, W*17);
label("$\omega_2$", B, E*17);
label("$\omega_3$", C, W*17);
[/asy]
$\textbf{(A) }546\qquad\textbf{(B) }548\qquad\textbf{(C) }550\qquad\textbf{(D) }552\qquad\textbf{(E) }554$
1972 All Soviet Union Mathematical Olympiad, 166
Each of the $9$ straight lines divides the given square onto two quadrangles with the areas ratio as $2:3$. Prove that there exist three of them intersecting in one point
2019 District Olympiad, 3
Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $$2(a_1+a_2+…+a_n)=na_{n+1}~\forall~n \ge 1.$$
$\textbf{a)}$ Prove that the given sequence is an arithmetic progression.
$\textbf{b)}$ If $\lfloor a_1 \rfloor + \lfloor a_2 \rfloor +…+ \lfloor a_n \rfloor = \lfloor a_1+a_2+…+a_n \rfloor~\forall~ n \in \mathbb{N},$ prove that every term of the sequence is an integer.
May Olympiad L1 - geometry, 2002.2
A rectangular sheet of paper (white on one side and gray on the other) was folded three times, as shown in the figure:
Rectangle $1$, which was white after the first fold, has $20$ cm more perimeter than rectangle $2$, which was white after the second fold, and this in turn has $16$ cm more perimeter than rectangle $3$, which was white after the third fold. Determine the area of the sheet.
[img]https://cdn.artofproblemsolving.com/attachments/d/f/8e363b40654ad0d8e100eac38319ee3784a7a7.png[/img]
2022 Rioplatense Mathematical Olympiad, 3
On the table there are $N$ cards. Each card has an integer number written on it.
Beto performs the following operation several times: he chooses two cards from the table, calculates the difference between the numbers written on them, writes the result on his notebook and removes those two cards from the table. He can perform this operation as many times as he wants, as long as there are at least two cards on the table.
After this, Beto multiplies all the numbers that he wrote on his notebook. Beto's goal is that the result of this multiplication is a multiple of $7^{100}$.
Find the minimum value of $N$ such that Beto can always achieve his goal, no matter what the numbers on the cards are.
1996 Moldova Team Selection Test, 8
Let $X$ be set with $n{}$ elements, $n\in\mathbb{N}$. Find the greatest integer $m$ $(m\geq2)$ for which there exist $m$ subsets of $X$ such that each two of them are not disjoint.
2010 AMC 12/AHSME, 13
For how many integer values of $ k$ do the graphs of $ x^2 \plus{} y^2 \equal{} k^2$ and $ xy \equal{} k$ [u]not[/u] intersect?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 8$
1988 Spain Mathematical Olympiad, 3
Prove that if one of the numbers $25x+31y, 3x+7y$ (where $x,y \in Z$) is a multiple of $41$, then so is the other.