Found problems: 85335
2021 CMIMC, 2.2
Suppose $a,b$ are positive real numbers such that $a+a^2 = 1$ and $b^2+b^4=1$. Compute $a^2+b^2$.
[i]Proposed by Thomas Lam[/i]
2016 PUMaC Number Theory A, 8
Let $n = 2^8 \cdot 3^9 \cdot 5^{10} \cdot 7^{11}$.
For $k$ a positive integer, let $f(k)$ be the number of integers $0 \le x < n$ such that $x^2 \equiv k^2$ (mod $n$).
Compute the number of positive integers k such that $k | f(k)$.
1993 AMC 8, 10
This line graph represents the price of a trading card during the first $6$ months of $1993$.
[asy]
unitsize(18);
for (int a = 0; a <= 6; ++a)
{
draw((4*a,0)--(4*a,10));
}
for (int a = 0; a <= 5; ++a)
{
draw((0,2*a)--(24,2*a));
}
draw((0,5)--(4,4)--(8,8)--(12,3)--(16,9)--(20,6)--(24,2),linewidth(1.5));
label("$Jan$",(2,0),S);
label("$Feb$",(6,0),S);
label("$Mar$",(10,0),S);
label("$Apr$",(14,0),S);
label("$May$",(18,0),S);
label("$Jun$",(22,0),S);
label("$\textbf{1993 PRICES FOR A TRADING CARD}$",(12,10),N);
label("$\begin{tabular}{c}\textbf{P} \\ \textbf{R} \\ \textbf{I} \\ \textbf{C} \\ \textbf{E} \end{tabular}$",(-2,5),W);
label("$1$",(0,2),W);
label("$2$",(0,4),W);
label("$3$",(0,6),W);
label("$4$",(0,8),W);
label("$5$",(0,10),W);
[/asy]
The greatest monthly drop in price occurred during
$\text{(A)}\ \text{January} \qquad \text{(B)}\ \text{March} \qquad \text{(C)}\ \text{April} \qquad \text{(D)}\ \text{May} \qquad \text{(E)}\ \text{June}$
2013 Princeton University Math Competition, 3
Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at Frist, located $2$ jumps east and $4$ jumps north of Fine Hall. How many ways can he get to McCosh without going through Frist?
2018 PUMaC Individual Finals B, 1
Let a positive integer $n$ have at least four positive divisors. Let the least four positive divisors be $1=d_1<d_2<d_3<d_4$. Find, with proof, all solutions to $n^2=d_1^3+d_2^3+d_4^3$.
2013 Tournament of Towns, 5
A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing exactly two nodes inside. Prove that the straight line connecting these nodes either passes through a vertex or is parallel to a side of the triangle.
2007 Pre-Preparation Course Examination, 18
Prove that the equation $x^3+y^3+z^3=t^4$ has infinitely many solutions in positive integers such that $\gcd(x,y,z,t)=1$.
[i]Mihai Pitticari & Sorin Rǎdulescu[/i]
2014 ASDAN Math Tournament, 2
Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$.
1950 Moscow Mathematical Olympiad, 180
Solve the equation $\sqrt {x + 3 - 4 \sqrt{x -1}} +\sqrt{x + 8 - 6 \sqrt{x - 1}}= 1$.
2023 Stanford Mathematics Tournament, 2
$f(x)$ is a nonconstant polynomial. Given that $f(f(x)) + f(x) = f(x)^2$, compute $f(3)$.
2005 Flanders Junior Olympiad, 1
It is the year 2005 now. According to a legend there is a monster that awakes every now and then to swallow everyone who is solving this problem, and then falls back asleep for as many years as the sum of the digits of that year. The monster first hit AoPS in the year +234. Prove you're safe this year, as well as for the coming 10 years.
Russian TST 2019, P1
Suppose that $A$, $B$, $C$, and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient
\[\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}\]
is a rational number.
2021 Thailand TST, 1
For a positive integer $n$, consider a square cake which is divided into $n \times n$ pieces with at most one strawberry on each piece. We say that such a cake is [i]delicious[/i] if both diagonals are fully occupied, and each row and each column has an odd number of strawberries.
Find all positive integers $n$ such that there is an $n \times n$ delicious cake with exactly $\left\lceil\frac{n^2}{2}\right\rceil$ strawberries on it.
1993 Poland - Second Round, 1
If $ x,y,u,v$ are positiv real numbers, prove the inequality :
\[ \frac {xu \plus{} xv \plus{} yu \plus{} yv}{x \plus{} y \plus{} u \plus{} v} \geq \frac {xy}{x \plus{} y} \plus{} \frac {uv}{u \plus{} v}
\]
2016 Brazil Team Selection Test, 1
For each positive integer $n$, determine the digits of units and hundreds of the decimal representation of the number $$\frac{1 + 5^{2n+1}}{6}$$
2014 Flanders Math Olympiad, 3
Let $PQRS$ be a quadrilateral with $| P Q | = | QR | = | RS |$, $\angle Q= 110^o$ and $\angle R = 130^o$ . Determine $\angle P$ and $\angle S$ .
2018 Philippine MO, 4
Determine all ordered pairs $(x, y)$ of nonnegative integers that satisfy the equation $$3x^2 + 2 \cdot 9^y = x(4^{y+1}-1).$$
2012 Tournament of Towns, 1
The decimal representation of an integer uses only two different digits. The number is at least $10$ digits long, and any two neighbouring digits are distinct. What is the greatest power of two that can divide this number?
2024 Iran MO (3rd Round), 1
For positive real numbers $a,b,c,d$ such that
$$
\dfrac{a^2}{b+c+d} + \dfrac{b^2}{a+c+d} +
\dfrac{c^2}{a+b+d} = \dfrac{3d^2}{a+b+c}
$$
prove that
$$
\dfrac{3}{a}+ \dfrac{3}{b} + \dfrac{3}{c}+ \dfrac{3}{d} \geq \dfrac{16}{a+b+2d} + \dfrac{16}{b+c+2d} +
\dfrac{16}{a+c+2d}.
$$
Proposed by [i]Mojtaba Zare[/i]
2015 Tournament of Towns, 3
[b](a)[/b] A $2 \times n$-table (with $n > 2$) is filled with numbers so that the sums in all the columns are different. Prove that it is possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different.
[i]($2$ points)[/i]
[b](b)[/b] A $100 \times 100$-table is filled with numbers such that the sums in all the columns are different. Is it always possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different?
[i]($6$ points)[/i]
2011 F = Ma, 11
A large metal cylindrical cup floats in a rectangular tub half-filled with water. The tap is placed over the cup and turned on, releasing water at a constant rate. Eventually the cup sinks to the bottom and is completely submerged. Which of the following five graphs could represent the water level in the sink as a function of time?
[asy]
size(450);
picture pic;
draw(pic,(0,0)--(10,0)--(10,7)--(0,7)--cycle);
for (int i=1;i<10;++i) {
draw(pic,(i,0)--(i,7),dashed+linewidth(0.4));
}
for (int j=1;j<7;++j) {
draw(pic,(0,j)--(10,j),dashed+linewidth(0.4));
}
label(pic,scale(1.2)*"time",(5.5,-0.5),S);
label(pic,rotate(90)*scale(1.2)*"water level",(-0.5,2.5),W);
add(pic);
path A=(0,1)--(10,6);
draw(A,linewidth(2));
label("(A)",(4.5,-1.5),1.5*S);
picture pic2=shift(13*right)*pic;
add(pic2);
path B=(0,1)--(4,4)--(10,6);
draw(shift(13*right)*B,linewidth(2));
label("(B)",(17.5,-1.5),1.5*S);
picture pic3=shift(26*right)*pic;
add(pic3);
path C=(0,1)--(4,3)--(4,2)--(10,5);
draw(shift(26*right)*C,linewidth(2));
label("(C)",(30.5,-1.5),1.5*S);
picture pic4=shift(13*down)*pic;
add(pic4);
path D=(0,1)--(4,3)--(4,4)--(10,7);
draw(shift(13*down)*D,linewidth(2));
label("(D)",(4.5,-14.5),1.5*S);
picture pic5=shift(13*down)*shift(13*right)*pic;
add(pic5);
path E=(0,1)--(4,3)--(4,2)--(10,4);
draw(shift(13*down)*shift(13*right)*E,linewidth(2));
label("(E)",(17.5,-14.5),1.5*S);
[/asy]
2002 VJIMC, Problem 2
A ring $R$ (not necessarily commutative) contains at least one non-zero zero divisor and the number of zero divisors is finite. Prove that $R$ is finite.
2016 Miklós Schweitzer, 4
Prove that there exists a sequence $a(1),a(2),\dots,a(n),\dots$ of real numbers such that
\[
a(n+m)\le a(n)+a(m)+\frac{n+m}{\log (n+m)}
\]
for all integers $m,n\ge 1$, and such that the set $\{a(n)/n:n\ge 1\}$ is everywhere dense on the real line.
[i]Remark.[/i] A theorem of de Bruijn and Erdős states that if the inequality above holds with $f(n + m)$ in place of the last term on the right-hand side, where $f(n)\ge 0$ is nondecreasing and $\sum_{n=2}^\infty f(n)/n^2<\infty$, then $a(n)/n$ converges or tends to $(-\infty)$.
Geometry Mathley 2011-12, 7.3
Let $ABCD$ be a tangential quadrilateral. Let $AB$ meet $CD$ at $E, AD$ intersect $BC$ at $F$. Two arbitrary lines through $E$ meet $AD,BC$ at $M,N, P,Q$ respectively ($M,N \in AD$, $P,Q \in BC$). Another arbitrary pair of lines through $F$ intersect $AB,CD$ at $X, Y,Z, T$ respectively ($X, Y \in AB$,$Z, T \in CD$). Suppose that $d_1, d_2$ are the second tangents from $E$ to the incircles of triangles $FXY, FZT,d_3, d_4$ are the second tangents from $F$ to the incircles of triangles $EMN,EPQ$. Prove that the four lines $d_1, d_2, d_3, d_4$ meet each other at four points and these intersections make a tangential quadrilateral.
Nguyễn Văn Linh
2003 AIME Problems, 2
One hundred concentric circles with radii $1, 2, 3, \dots, 100$ are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.