Found problems: 85335
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?
1994 China Team Selection Test, 3
Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.
1990 IMO Longlists, 54
Let $M = \{1, 2, \ldots, n\}$ and $\phi : M \to M$ be a bijection.
(i) Prove that there exist bijections $\phi_1, \phi_2 : M \to M$ such that $\phi_1 \cdot \phi_2 = \phi , \phi_1^2 =\phi_2^2=E$, where $E$ is the identity mapping.
(ii) Prove that the conclusion in (i) is also true if $M$ is the set of all positive integers.
PEN P Problems, 6
Show that every integer greater than $1$ can be written as a sum of two square-free integers.
1994 China Team Selection Test, 1
Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i
t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.
2013 National Chemistry Olympiad, 37
Three metals, $A, B $and $C$, with solutions of their respective cations are tested in a voltaic cell with the following results:
$A$ and $B$: $A$ is the cathode
$B$ and $C$: $C$ is the cathode
$A$ and $C$: $A$ is the anode
What is the order of the reduction potentials from highest to lowest for the cations of these metals?
$ \textbf{(A)}\ A>B>C \qquad\textbf{(B)}\ B>C>A\qquad$
${\textbf{(C)}\ C>A>B\qquad\textbf{(D}}\ B>A>C\qquad$
2022 China Northern MO, 2
(1) Find the smallest positive integer $a$ such that $221|3^a -2^a$,
(2) Let $A=\{n\in N^*: 211|1+2^n+3^n+4^n\}$.
Are there infinitely many numbers $n$ such that both $n$ and $n+1$ belong to set $A$?
2021 Austrian MO National Competition, 4
On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game.
Alice starts and they alternately play the following moves:
$\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$
$\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$.
Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps.
Prove that Alice has a winning strategy.
(Daniel Holmes)
2000 Mexico National Olympiad, 1
Circles $A,B,C,D$ are given on the plane such that circles $A$ and $B$ are externally tangent at $P, B$ and $C$ at $Q, C$ and $D$ at $R$, and $D$ and $A$ at $S$. Circles $A$ and $C$ do not meet, and so do not $B$ and $D$.
(a) Prove that the points $P,Q,R,S$ lie on a circle.
(b) Suppose that $A$ and $C$ have radius $2, B$ and $D$ have radius $3$, and the distance between the centers of $A$ and $C$ is $6$. Compute the area of the quadrilateral $PQRS$.