Found problems: 85335
2020 CMIMC Combinatorics & Computer Science, 6
The nation of CMIMCland consists of 8 islands, none of which are connected. Each citizen wants to visit the other islands, so the government will build bridges between the islands. However, each island has a volcano that could erupt at any time, destroying that island and any bridges connected to it. The government wants to guarantee that after any eruption, a citizen from any of the remaining $7$ islands can go on a tour, visiting each of the remaining islands exactly once and returning to their home island (only at the end of the tour). What is the minimum number of bridges needed?
2004 National High School Mathematics League, 4
$O$ is a point inside $\triangle ABC$, and $\overrightarrow{OA}+2\overrightarrow{OB}+3\overrightarrow{OC}=\overrightarrow{0}$, then the ratio of the area of $\triangle ABC$ to $\triangle AOC$ is
$\text{(A)}2\qquad\text{(B)}\frac{3}{2}\qquad\text{(C)}3\qquad\text{(D)}\frac{5}{3}$
2017 Taiwan TST Round 2, 1
There is a $2n\times 2n$ rectangular grid and a chair in each cell of the grid. Now, there are $2n^2$ pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating at $(3,3)$ and $(2,5)$ respectively, then the distance between them is $|3-2|+|3-5|=3$. Moreover, define the total distance to be the sum of the distance in each pair. Find the maximal total distance among all possibilities.
2010 BAMO, 1
We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.
1989 Putnam, B4
Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?
2017 MMATHS, 1
For any integer $n > 4$, prove that $2^n > n^2$.
1983 IMO, 3
Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that
\[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0.
\]
Determine when equality occurs.
1999 Miklós Schweitzer, 5
Let $\alpha>-2$ , $n\in \mathbb{N}$ and $y_1,\cdots,y_n$ be the solutions to the system of equations:
$\sum_{j=1}^n \frac{y_j}{j+k+\alpha}= \frac{1}{n+1+k+\alpha}$ , $k=1,\cdots,n$
Prove that $y_{j-1}y_{j+1}\leq y_j^2 \,\forall 1<j<n$
2017 AMC 12/AHSME, 5
At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
$\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$
2015 Harvard-MIT Mathematics Tournament, 7
Let $ABCD$ be a square pyramid of height $\frac{1}{2}$ with square base $ABCD$ of side length $AB=12$ (so $E$ is the vertex of the pyramid, and the foot of the altitude from $E$ to $ABCD$ is the center of square $ABCD$). The faces $ADE$ and $CDE$ meet at an acute angle of measure $\alpha$ (so that $0^{\circ}<\alpha<90^{\circ}$). Find $\tan \alpha$.
2015 BMT Spring, 19
Two sequences $(x_n)_{n\in N}$ and $(y_n)_{n\in N}$ are defined recursively as follows:
$x_0 = 2015$ and $x_{n+1} =\left \lfloor x_n \cdot \frac{y_{n+1}}{y_{n-1}} \right \rfloor$ for all $n \ge 0$,
$y_0 = 307$ and $y_{n+1} = y_n + 1$ for all $n \ge 0$.
Compute $\lim_{n\to \infty} \frac{x_n}{(y_n)^2}$.
1998 AMC 12/AHSME, 6
If 1998 is written as a product of two positive integers whose difference is as small as possible, then the difference is
$\text{(A)} \ 8 \qquad \text{(B)} \ 15 \qquad \text{(C)} \ 17 \qquad \text{(D)} \ 47 \qquad \text{(E)} \ 93$
2013 IFYM, Sozopol, 7
Let $O$ be the center of the inscribed circle of $\Delta ABC$ and point $D$ be the middle point of $AB$.
If $\angle AOD=90^\circ$, prove that $AB+BC=3AC$.
2014 JHMMC 7 Contest, 6
Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write (and submit) before he has written half of the total problems?
2019 Junior Balkan Team Selection Tests - Moldova, 5
Find all triplets of positive integers $(a, b, c)$ that verify $\left(\frac{1}{a}+1\right)\left(\frac{1}{b}+1\right)\left(\frac{1}{c}+1\right)=2$.
1991 AMC 8, 18
The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for $5$ years or more?
[asy]
for(int a=1; a<11; ++a)
{
draw((a,0)--(a,-.5));
}
draw((0,10.5)--(0,0)--(10.5,0));
label("$1$",(1,-.5),S); label("$2$",(2,-.5),S); label("$3$",(3,-.5),S); label("$4$",(4,-.5),S);
label("$5$",(5,-.5),S); label("$6$",(6,-.5),S); label("$7$",(7,-.5),S); label("$8$",(8,-.5),S);
label("$9$",(9,-.5),S); label("$10$",(10,-.5),S); label("Number of years with company",(5.5,-2),S);
label("X",(1,0),N); label("X",(1,1),N); label("X",(1,2),N); label("X",(1,3),N); label("X",(1,4),N);
label("X",(2,0),N); label("X",(2,1),N); label("X",(2,2),N); label("X",(2,3),N); label("X",(2,4),N);
label("X",(3,0),N); label("X",(3,1),N); label("X",(3,2),N); label("X",(3,3),N);
label("X",(3,4),N); label("X",(3,5),N); label("X",(3,6),N); label("X",(3,7),N);
label("X",(4,0),N); label("X",(4,1),N); label("X",(4,2),N); label("X",(5,0),N); label("X",(5,1),N);
label("X",(6,0),N); label("X",(6,1),N); label("X",(7,0),N); label("X",(7,1),N);
label("X",(8,0),N); label("X",(9,0),N); label("X",(10,0),N);
label("Gauss Company",(5.5,10),N);
[/asy]
$\text{(A)}\ 9\% \qquad \text{(B)}\ 23\frac{1}{3}\% \qquad \text{(C)}\ 30\% \qquad \text{(D)}\ 42\frac{6}{7}\% \qquad \text{(E)}\ 50\% $
2022 LMT Spring, 7
Kevin has a square piece of paper with creases drawn to split the paper in half in both directions, and then each of the four small formed squares diagonal creases drawn, as shown below.
[img]https://cdn.artofproblemsolving.com/attachments/2/2/70d6c54e86856af3a977265a8054fd9b0444b0.png[/img]
Find the sum of the corresponding numerical values of figures below that Kevin can create by folding the above piece
of paper along the creases. (The figures are to scale.) Kevin cannot cut the paper or rip it in any way.
[img]https://cdn.artofproblemsolving.com/attachments/a/c/e0e62a743c00d35b9e6e2f702106016b9e7872.png[/img]
2020 CCA Math Bonanza, L4.2
Let $a_0,a_1,\ldots$ be a sequence of positive integers such that $a_0=1$, and for all positive integers $n$, $a_n$ is the smallest composite number relatively prime to all of $a_0,a_1,\ldots,a_{n-1}$. Compute $a_{10}$.
[i]2020 CCA Math Bonanza Lightning Round #4.2[/i]
2004 Harvard-MIT Mathematics Tournament, 2
Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$.
2013 National Chemistry Olympiad, 38
In which pair of substances do the nitrogen atoms have the same oxidation state?
$ \textbf{(A)}\ \ce{HNO3} \text{ and } \ce{ N2O5} \qquad\textbf{(B)}\ \ce{NO} \text{ and } \ce{HNO2} \qquad$
${\textbf{(C)}\ \ce{N2} \text{ and } \ce{N2O} \qquad\textbf{(D)}}\ \ce{HNO2} \text{ and } \ce{HNO3} \qquad $
2001 Singapore MO Open, 2
Let $n$ be a positive integer, and let $a_1,a_2,...,a_n$ be $n$ positive real numbers such that $a_1+a_2+...+a_n = 1$. Is it true that $\frac{a_1^4}{a_1^2+a_2^2}+\frac{a_2^4}{a_2^2+a_3^2}+\frac{a_3^4}{a_3^2+a_4^2}+...+\frac{a_{n-1}^4}{a_{n-1}^2+a_n^2}+\frac{a_n^4}{a_n^2+a_1^2}\ge \frac{1}{2n}$ ?
Justify your answer.
2009 Turkey Team Selection Test, 1
For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?
2010 Romania Team Selection Test, 3
Let $\mathcal{L}$ be a finite collection of lines in the plane in general position (no two lines in $\mathcal{L}$ are parallel and no three are concurrent). Consider the open circular discs inscribed in the triangles enclosed by each triple of lines in $\mathcal{L}$. Determine the number of such discs intersected by no line in $\mathcal{L}$, in terms of $|\mathcal{L}|$.
[i]B. Aronov et al.[/i]
1992 Bulgaria National Olympiad, Problem 3
Let $m$ and $n$ are fixed natural numbers and $Oxy$ is a coordinate system in the plane. Find the total count of all possible situations of $n+m-1$ points $P_1(x_1,y_1),P_2(x_2,y_2),\ldots,P_{n+m-1}(x_{n+m-1},y_{n+m-1})$ in the plane for which the following conditions are satisfied:
(i) The numbers $x_i$ and $y_i~(i=1,2,\ldots,n+m-1)$ are integers and $1\le x_i\le n,1\le y_i\le m$.
(ii) Every one of the numbers $1,2,\ldots,n$ can be found in the sequence $x_1,x_2,\ldots,x_{n+m-1}$ and every one of the numbers $1,2,\ldots,m$ can be found in the sequence $y_1,y_2,\ldots,y_{n+m-1}$.
(iii) For every $i=1,2,\ldots,n+m-2$ the line $P_iP_{i+1}$ is parallel to one of the coordinate axes. [i](Ivan Gochev, Hristo Minchev)[/i]
2023/2024 Tournament of Towns, 1
1. Every square of a $8 \times 8$ board is filled with a positive integer, such that the following condition holds: if a chess knight can move from some square to another then the ratio of numbers from these two squares is a prime number. Is it possible that some square is filled with 5 , and another one with 6 ?
Egor Bakaev