Found problems: 85335
Fill each unshaded cell of the grid with a number that is either $1$, $3$, or $5$. For each
cell, exactly one of the touching cells must contain the same number. Here touching includes
cells that only share a point, i.e. touch diagonally.
[asy]
unitsize(1.2cm);
defaultpen(fontsize(30pt));
for(int i=0; i<8; ++i){
draw((i,0)--(i,7));
draw((0,i)--(7,i));
}
filldraw((0,4)--(1,4)--(1,3)--(0,3)--cycle, gray);
filldraw((1,1)--(1,2)--(2,2)--(2,1)--cycle, gray);
filldraw((4,2)--(4,3)--(5,3)--(5,2)--cycle, gray);
filldraw((3,4)--(4,4)--(4,5)--(3,5)--cycle, gray);
filldraw((6,2)--(6,3)--(7,3)--(7,2)--cycle, gray);
int[][] numbers = {
{0,0,0,0,0,0,0},
{1,0,1,0,3,0,5},
{0,0,0,0,0,0,0},
{0,0,1,1,3,5,0},
{0,0,0,0,0,0,0},
{0,0,0,0,0,0,0},
{5,0,0,5,0,0,0}};
for(int i=0; i<7; ++i){
for(int j=0; j<7; ++j){
if(numbers[i][j]>0){label(string(numbers[i][j]),(j+0.5,6.5-i));}
}
}
[/asy]
There is a unique solution, but you do not need to prove that your answer is the only
one possible. You merely need to find an answer that satisfies the conditions of the problem.
(Note: In any other USAMTS problem, you need to provide a full proof. Only in this
problem is an answer without justification acceptable.)
In graph theory, a [i]triangle[/i] is a set of three vertices, every two of which are connected by an edge. For an integer $n \geq 3$, if a graph on $n$ vertices does not contain two triangles that do not share any vertices, let $f(n)$ be the maximum number of triangles it can contain. Compute $f(3) + f(4) + \cdots + f(100).$
[i]Proposed by Edward Wan[/i]
In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements.
[i]Proposed by Jorge Tipe, Peru[/i]
Let $I, O$ be the incenter and the circumcenter of triangle $ABC$, and $D,E,F$ be the circumcenters of triangle $ BIC, CIA, AIB$. Let $ P, Q, R$ be the midpoints of segments $ DI, EI, FI $. Prove that the circumcenter of triangle $PQR $, $M$, is the midpoint of segment $IO$.
Let $f(x) = 15x - 2016$. If $f(f(f(f(f(x))))) = f(x)$, find the sum of all possible values of $x$.
The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length
$\text{(A)}\ 22 \qquad \text{(B)}\ 58 \qquad \text{(C)}\ 81 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 361$
Let $ABCD$ be a square with side length $6$. Let $E$ be a point on the perimeter of $ABCD$ such that the area of $\triangle{AEB}$ is $\frac{1}{6}$ the area of $ABCD$. Find the maximum possible value of $CE^2$.
[i]Proposed by Anthony Yang[/i]
There are $100$ students who praticipate at exam.Also there are $25$ members of jury.Each student is checked by one jury.Known that every student likes $10$ jury
$a)$ Prove that we can select $7$ jury such that any student likes at least one jury.
$b)$ Prove that we can make this every student will be checked by the jury that he likes and every jury will check at most $10$ students.
Let $a$ and $b$ be positive real numbers with $a + b =1$. Show that $$\left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \ge \frac{25}{2}.$$
Find all continuous functions $f : R \to R$ such that
$$f(x + f(y)) = f(x + y) + y,$$
for all $x, y \in R$. No proof is required for this problem.
Let $p$ be a prime number that is greater than $3$. Show that there exist some integers $a_{1}, a_{2}, \cdots a_{k}$ that satisfy: \[-\frac{p}{2}< a_{1}< a_{2}< \cdots <a_{k}< \frac{p}{2}\] making the product: \[\frac{p-a_{1}}{|a_{1}|}\cdot \frac{p-a_{2}}{|a_{2}|}\cdots \frac{p-a_{k}}{|a_{k}|}\] equals to $3^{m}$ where $m$ is a positive integer.
All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$.
[b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length;
[b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.
Let $f:[0,1]\to\mathbb{R}$ be a continuous function. Prove that \[\lim_{n\to\infty}\int_0^1 f(x^n) \ dx=f(0).\]Furthermore, if $f(0)=0$ and $f$ is right-differentiable in $0{}$, prove that the limits \[\lim_{\varepsilon\to0}\int_\varepsilon^1\frac{f(x)}{x} \ dx\quad\text{and}\quad\lim_{n\to\infty}\left(n\int_0^1f(x^n) \ dx\right)\]exist, are finite and are equal.
Several farmers have 128 sheep. If one of them has at least half of all sheep, the rest conspire and dispossess him: everyone takes as many sheep as he already has : If two people have 64 sheep, then one of them is dispossessed. There were 7 dispossessions. Prove that all the sheep were gathered from one peasant.
Find the minimum value of the expression $f(a, b, c) = (a + b)^4 + (b + c)^4 + (c + a)^4 - \frac47 (a^4 + b^4 + c^4)$,
as $a, b, c$ varies over the set of all real numbers
What is the smallest positive integer that when raised to the $6^{th}$ power, it can be represented by a sum of the $6^{th}$ powers of distinct smaller positive integers?
The circles $k_1$, $k_2$, $k_3$ are externally tangent: $k_1$ to $k_2$ at point $A$, $k_2$ to $k_3$ at point $B$, $k_3$ to $k_4$ at point $C$, $k_4$ to $k_1$ at point $D$. The lines $AB$ and $CD$ intersect at the point $S$. A line $ p $ is drawn through point $ S $, tangent to $ k_4 $ at point $ F $. Prove that $ |SE|=|SF| $.
Given three natural numbers $a$, $b$ and $c$. It turned out that they are coprime together. And their least common multiple and their product are perfect squares. Prove that $a$, $b$ and $c$ are perfect squares.
There are three boxes, one blue, one white and one red, and $8$ balls. Each of the balls has a number from $1$ to $8$ written on it, without repetitions. The $8$ balls are distributed in the boxes, so that there are at least two balls in each box. Then, in each box, add up all the numbers written on the balls it contains. The three outcomes are called the blue sum, the white sum, and the red sum, depending on the color of the corresponding box. Find all possible distributions of the balls such that the red sum equals twice the blue sum, and the red sum minus the white sum equals the white sum minus the blue sum.
Let $ABC$ be a triangle with $AB < AC < BC$. On the side $BC$ we consider points $D$
and $E$ such that $BA = BD$ and $CE = CA$. Let $K$ be the circumcenter of triangle $ADE$ and
let $F$, $G$ be the points of intersection of the lines $AD$, $KC$ and $AE$, $KB$ respectively. Let $\omega_1$ be
the circumcircle of triangle $KDE$, $\omega_2$ the circle with center $F$ and radius $FE$, and $\omega_3$ the circle
with center $G$ and radius $GD$.
Prove that $\omega_1$, $\omega_2$, and $\omega_3$ pass through the same point and that this point of intersection lies on the line $AK$.
Let $M$ and $N$ be the midpoints of segments $BC$ and $AC$ of a triangle $ABC$, respectively. Let $Q$ be a point on the line through $N$ parallel to $BC$ such that $Q$ and $C$ are on opposite sides of $AB$ and $\vert QN\vert \cdot \vert BC\vert=\vert AB\vert \cdot \vert AC\vert$.
Suppose that the circumcircle of triangle $AQN$ intersects the segment $MN$ a second time in a point $T \ne N$.
Prove that there is a circle through points $T$ and $N$ touching both the side $BC$ and the incircle of triangle $ABC$.
An integer $n\geq 3$ is [i]poli-pythagorean[/i] if there exist $n$ positive integers pairwise distinct such that we can order these numbers in the vertices of a regular $n$-gon such that the sum of the squares of consecutive vertices is also a perfect square. For instance, $3$ is poli-pythagorean, because if we write $44,117,240$ in the vertices of a triangle we notice:
$$44^2+117^2=125^2, 117^2+240^2=267^2, 240^2+44^2=244^2$$
Determine all poli-pythagorean integers.
In rectangle $ ABCD$, $ AB\equal{}5$ and $ BC\equal{}3$. Points $ F$ and $ G$ are on $ \overline{CD}$ so that $ DF\equal{}1$ and $ GC\equal{}2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \triangle{AEB}$.
[asy]unitsize(6mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3);
pair E=extension(A,F,B,G);
draw(A--B--C--D--A--E--B);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$E$",E,N);
label("$F$",F,SE);
label("$G$",G,SW);
label("$B$",B,SE);
label("1",midpoint(D--F),N);
label("2",midpoint(G--C),N);
label("3",midpoint(B--C),E);
label("3",midpoint(A--D),W);
label("5",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 10 \qquad
\textbf{(B)}\ \frac{21}{2} \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ \frac{25}{2} \qquad
\textbf{(E)}\ 15$
Find all pairs of positive integers $(a,b)$ satisfying the following conditions:
[list]
[*] $a$ divides $b^4+1$,
[*] $b$ divides $a^4+1$,
[*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$.
[/list]
[i]Yang Liu[/i]
Let $x, y, z$ be positive numbers such that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$. Prove that
$$\sqrt{x + yz} +\sqrt{y + zx} +\sqrt{z + xy} \ge\sqrt{xyz}+\sqrt{x }+\sqrt{y} +\sqrt{z}$$