Found problems: 85335
1989 Putnam, A2
Evaluate $\int^{a}_{0}{\int^{b}_{0}{e^{max(b^{2}x^{2},a^{2}y^{2})}dy dx}}$
1999 Gauss, 5
Which one of the following gives an odd integer?
$\textbf{(A)}\ 6^2 \qquad \textbf{(B)}\ 23-17 \qquad \textbf{(C)}\ 9\times24 \qquad \textbf{(D)}\ 96\div8 \qquad \textbf{(E)}\ 9\times41$
May Olympiad L1 - geometry, 1995.5
A tortoise walks $60$ meters per hour and a lizard walks at $240$ meters per hour. There is a rectangle $ABCD$ where $AB =60$ and $AD =120$. Both start from the vertex $A$ and in the same direction ($A \to B \to D \to A$), crossing the edge of the rectangle. The lizard has the habit of advancing two consecutive sides of the rectangle, turning to go back one, turning to go forward two, turning to go back one and so on. How many times and in what places do the tortoise and the lizard meet when the tortoise completes its third turn?
2018 Bosnia and Herzegovina Junior BMO TST, 1
Students are in classroom with $n$ rows. In each row there are $m$ tables. It's given that $m,n \geq 3$. At each table there is exactly one student. We call neighbours of the student students sitting one place right, left to him, in front of him and behind him. Each student shook hands with his neighbours. In the end there were $252$ handshakes. How many students were in the classroom?
2022 CMIMC, 2.1
Alice and Bob live on the same road. At time $t$, they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they didn't realize passing each other. Alice arrived at Bob's house at $3:19\text{pm}$, and Bob arrived at Alice's house at $3:29\text{pm}$. Charlie, who was driving by, noted that Alice and Bob passed each other at $3:11\text{pm}$. Find the difference in minutes between the time Alice and Bob left their own houses and noon on that day.
[i]Proposed by Kevin You[/i]
2023 China Northern MO, 3
Find all solutions of the equation
$$sin\pi \sqrt x+cos\pi \sqrt x=(-1)^{\lfloor \sqrt x \rfloor }$$
2003 BAMO, 3
A lattice point is a point $(x, y)$ with both $x$ and $y$ integers. Find, with proof, the smallest $n$ such that every set of $n$ lattice points contains three points that are the vertices of a triangle with integer area. (The triangle may be degenerate, in other words, the three points may lie on a straight line and hence form a triangle with area zero.)
1970 AMC 12/AHSME, 9
Points $P$ and $Q$ are on line segment $AB$, and both points are on the same side of the midpoint of $AB$. Point $P$ divides $AB$ in the ratio $2:3$ and $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of segment $AB$ is
$\textbf{(A) }12\qquad\textbf{(B) }28\qquad\textbf{(C) }70\qquad\textbf{(D) }75\qquad \textbf{(E) }105$
2021 USMCA, 20
Let $\tau(n)$ be the number of positive divisors of $n$, let $f(n) = \sum_{d \mid n} \tau(d)$, and let $g(n) = \sum_{d \mid n} f(d)$. Let $P_n$ be the product of the first $n$ prime numbers, and let $M = P_1 P_2 \cdots P_{2021}$. Then $\sum_{d \mid M} \frac{1}{g(d)} = \frac{a}{b}$, where $a, b$ are relatively prime positive integers. What is the remainder when $\tau(ab)$ is divided by $2017$? (Here, $\sum_{d \mid n}$ means a sum over the positive divisors of $n$.)
2021 Romanian Master of Mathematics Shortlist, N1
Given a positive integer $N$, determine all positive integers $n$, satisfying the following condition: for any list $d_1,d_2,\ldots,d_k$ of (not necessarily distinct) divisors of $n$ such that $\frac{1}{d_1} + \frac{1}{d_2} + \ldots + \frac{1}{d_k} > N$, some of the fractions $\frac{1}{d_1}, \frac{1}{d_2}, \ldots, \frac{1}{d_k}$ add up to exactly $N$.
2010 Princeton University Math Competition, 6
In regular hexagon $ABCDEF$, $AC$, $CE$ are two diagonals. Points $M$, $N$ are on $AC$, $CE$ respectively and satisfy $AC: AM = CE: CN = r$. Suppose $B, M, N$ are collinear, find $100r^2$.
[asy]
size(120); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
pair A=dir(0), B=dir(60), C=dir(120), D=dir(180), E=dir(240), F=dir(300), N=(4*E+C)/5,M=intersectionpoints(A--C,B--N)[0];
draw(A--B--C--D--E--F--cycle); draw(A--C--E); draw(B--N);
label("$A$",D2(A),plain.E);
label("$B$",D2(B),NE);
label("$C$",D2(C),NW);
label("$D$",D2(D),W);
label("$E$",D2(E),SW);
label("$F$",D2(F),SE);
label("$M$",D2(M),(0,-1.5));
label("$N$",D2(N),SE);
[/asy]
2015 Taiwan TST Round 3, 1
A plane has several seats on it, each with its own price, as shown below(attachment). $2n-2$ passengers wish to take this plane, but none of them wants to sit with any other passenger in the same column or row. The captain realize that, no matter how he arranges the passengers, the total money he can collect is the same. Proof this fact, and compute how much money the captain can collect.
2022/2023 Tournament of Towns, P2
Medians $BK{}$ and $CN{}$ of triangle $ABC$ intersect at $M{}.$ Consider quadrilateral $ANMK$ and find the maximum possible number of its sides having length 1.
[i]Egor Bakaev[/i]
2015 Bundeswettbewerb Mathematik Germany, 4
Let $ABC$ be a triangle, such that its incenter $I$ and circumcenter $U$ are distinct. For all points $X$ in the interior of the triangle let $d(X)$ be the sum of distances from $X$ to the three (possibly extended) sides of the triangle.
Prove: If two distinct points $P,Q$ in the interior of the triangle $ABC$ satisfy $d(P)=d(Q)$, then $PQ$ is perpendicular to $UI$.
2014-2015 SDML (High School), 8
Triangles $ABC$ and $BDC$ are such that $\angle{ABC}=\angle{BDC}=90^{\circ}$ and $\angle{DBC}=\angle{CAB}$. Let $Q$ be a point on $\overline{BD}$ such that $\overline{QC}\perp\overline{AD}$. Suppose that $BD=15$. Then $DQ$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2015 AMC 12/AHSME, 7
Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?
$\textbf{(A) }\text{The second height is 10\% less than the first.}$
$\textbf{(B) }\text{The first height is 10\% more than the second.}$
$\textbf{(C) }\text{The second height is 21\% less than the first.}$
$\textbf{(D) }\text{The first height is 21\% more than the second.}$
$\textbf{(E) }\text{The second height is 80\% of the first.}$
2022/2023 Tournament of Towns, P1
Find the maximum integer $m$ such that $m! \cdot 2022!$ is a factorial of an integer.
2001 Tournament Of Towns, 5
[b](a)[/b] One black and one white pawn are placed on a chessboard. You may move the pawns in turn to the neighbouring empty squares of the chessboard using vertical and horizontal moves. Can you arrange the moves so that every possible position of the two pawns will appear on the chessboard exactly once?
[b](b)[/b] Same question, but you don’t have to move the pawns in turn.
2009 Sharygin Geometry Olympiad, 8
Can the regular octahedron be inscribed into regular dodecahedron in such way that all vertices of octahedron be the vertices of dodecahedron?
(B.Frenkin)
2024 Ecuador NMO (OMEC), 2
Let $s(n)$ the sum of digits of $n$. Find the greatest 3-digits number $m$ such that $3s(m)=s(3m)$.
2000 AMC 12/AHSME, 18
In year $ N$, the $ 300^\text{th}$ day of the year is a Tuesday. In year $ N \plus{} 1$, the $ 200^{\text{th}}$ day of the year is also a Tuesday. On what day of the week did the $ 100^\text{th}$ day of year $ N \minus{} 1$ occur?
$ \textbf{(A)}\ \text{Thursday} \qquad \textbf{(B)}\ \text{Friday} \qquad \textbf{(C)}\ \text{Saturday} \qquad \textbf{(D)}\ \text{Sunday} \qquad \textbf{(E)}\ \text{Monday}$
2009 Brazil Team Selection Test, 1
Let $r$ be a positive real number. Prove that the number of right triangles with prime positive integer sides that have an inradius equal to $r$ are zero or a power of $2$.
[hide=original wording]Seja r um numero real positivo. Prove que o numero de triangulos retangulos com lados inteiros positivos primos entre si que possuem inraio igual a r e zero ou uma potencia de 2.[/hide]
2011 QEDMO 10th, 2
Let $n$ be a positive integer. Let $G (n)$ be the number of $x_1,..., x_n, y_1,...,y_n \in \{0,1\}$, for which the number $x_1y_1 + x_2y_2 +...+ x_ny_n$ is even, and similarly let $U (n)$ be the number for which this sum is odd. Prove that $$\frac{G(n)}{U(n)}= \frac{2^n + 1}{2^n - 1}.$$
2001 AMC 8, 9
Problems 7, 8 and 9 are about these kites.
[asy]
for (int a = 0; a < 7; ++a)
{
for (int b = 0; b < 8; ++b)
{
dot((a,b));
}
}
draw((3,0)--(0,5)--(3,7)--(6,5)--cycle);[/asy]
The large kite is covered with gold foil. The foil is cut from a rectangular piece that just covers the entire grid. How many square inches of waste material are cut off from the four corners?
$ \text{(A)}\ 63\qquad\text{(B)}\ 72\qquad\text{(C)}\ 180\qquad\text{(D)}\ 189\qquad\text{(E)}\ 264 $
LMT Speed Rounds, 2011.19
A positive six-digit integer begins and ends in $8$, and is also the product of three consecutive even numbers. What is the sum of the three even numbers?