Found problems: 85335
2017 AIME Problems, 4
A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
1998 Israel National Olympiad, 2
Show that there is a multiple of $2^{1998}$ whose decimal representation consists only of the digits $1$ and $2$.
1970 IMO Shortlist, 5
Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that
\[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\]
($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).
2003 AMC 12-AHSME, 3
A solid box is $ 15$ cm by $ 10$ cm by $ 8$ cm. A new solid is formed by removing a cube $ 3$ cm on a side from each corner of this box. What percent of the original volume is removed?
$ \textbf{(A)}\ 4.5 \qquad
\textbf{(B)}\ 9 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 18 \qquad
\textbf{(E)}\ 24$
2018 IMO, 2
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$
for $i = 1, 2, \dots, n$.
[i]Proposed by Patrik Bak, Slovakia[/i]
1997 Finnish National High School Mathematics Competition, 3
$12$ knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others.
$5$ knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice?
2010 Princeton University Math Competition, 2
Let $f(n)$ be the sum of the digits of $n$. Find $\displaystyle{\sum_{n=1}^{99}f(n)}$.
1999 Moldova Team Selection Test, 13
Let $N$ be a natural number. Find (with prove) the number of solutions in the segment $[1,N]$ of the equation $x^2-[x^2]=(x-[x])^2$, where $[x]$ means the floor function of $x$.
2014 Danube Mathematical Competition, 2
Let $S$ be a set of positive integers such that $\lfloor \sqrt{x}\rfloor =\lfloor \sqrt{y}\rfloor $ for all $x, y \in S$. Show that the products $xy$, where $x, y \in S$, are pairwise distinct.
2023 AMC 12/AHSME, 3
How many positive perfect squares less than $2023$ are divisible by $5$?
$\textbf{(A) } 8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) } 12$
2021 Iran MO (3rd Round), 3
Given triangle $ABC$ variable points $X$ and $Y$ are chosen on segments $AB$ and $AC$, respectively. Point $Z$ on line $BC$ is chosen such that $ZX=ZY$. The circumcircle of $XYZ$ cuts the line $BC$ for the second time at $T$. Point $P$ is given on line $XY$ such that $\angle PTZ = 90^ \circ$. Point $Q$ is on the same side of line $XY$ with $A$ furthermore $\angle QXY = \angle ACP$ and $\angle QYX = \angle ABP$. Prove that the circumcircle of triangle $QXY$ passes through a fixed point (as $X$ and $Y$ vary).
1974 Chisinau City MO, 79
There are many of the same regular triangles. At the vertices of each of them, the numbers $1, 2, 3$ are written in random order. The triangles were superimposed on one another and found the sum of the numbers that fell into each of the three corners of the stack. Could it be that in each corner the sum is equal to:
a) $25$,
b) $50$?
2014 Singapore Senior Math Olympiad, 16
Evaluate the sum $\frac{3!+4!}{2(1!+2!)}+\frac{4!+5!}{3(2!+3!)}+\cdots+\frac{12!+13!}{11(10!+11!)}$
2019 Ecuador NMO (OMEC), 5
Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$4abc = (a + 3) (b + 3) (c + 3).$$
Find the numerical value of $a + b + c$.
2019 CMIMC, 12
Call a convex quadrilateral [i]angle-Pythagorean[/i] if the degree measures of its angles are integers $w\leq x \leq y \leq z$ satisfying $$w^2+x^2+y^2=z^2.$$ Determine the maximum possible value of $x+y$ for an angle-Pythagorean quadrilateral.
2013 German National Olympiad, 1
Find all positive integers $n$ such that $n^{2}+2^{n}$ is square of an integer.
2024 Harvard-MIT Mathematics Tournament, 1
Compute the sum of all integers $n$ such that $n^2-3000$ is a perfect square.
ICMC 6, 6
Consider the sequence defined by $a_1 = 2022$ and $a_{n+1} = a_n + e^{-a_n}$ for $n \geq 1$. Prove that there exists a positive real number $r$ for which the sequence $$\{ra_1\}, \{ra_{10}\}, \{ra_{100}\}, . . . $$converges.
[i]Note[/i]: $\{x \} = x - \lfloor x \rfloor$ denotes the part of $x$ after the decimal point.
[i]Proposed by Ethan Tan[/i]
1959 AMC 12/AHSME, 50
A club with $x$ members is organized into four committees in accordance with these two rules:
$ \text{(1)}\ \text{Each member belongs to two and only two committees}\qquad$
$\text{(2)}\ \text{Each pair of committees has one and only one member in common}$
Then $x$:
$\textbf{(A)} \ \text{cannont be determined} \qquad$
$\textbf{(B)} \ \text{has a single value between 8 and 16} \qquad$
$\textbf{(C)} \ \text{has two values between 8 and 16} \qquad$
$\textbf{(D)} \ \text{has a single value between 4 and 8} \qquad$
$\textbf{(E)} \ \text{has two values between 4 and 8} \qquad$
Kvant 2020, M1
In a country, the time for presidential elections has approached. There are exactly 20 million voters in the country, of which only one percent supports the current president, Miraflores. Naturally, he wants to be elected again, but on the other hand, he wants the elections to seem democratic.
Miraflores established the following voting process: all the voters are divided into several equal groups, then each of these groups is again divided into a number of equal groups, and so on. In the smallest groups, a representative is chosen. Then, the chosen electors choose representatives in the second-smallest groups, to vote in an even larger group, and so on. Finally, the representatives of the largest groups choose the president.
Miraflores divides voters into groups as he wants and instructs his supporters how to vote. Will he be able to organize the elections in such a way that he will be elected president? (If the votes are equal, the opposition wins.)
[i]From the 32nd Moscow Mathematical Olympiad[/i]
1999 IMO Shortlist, 6
Prove that for every real number $M$ there exists an infinite arithmetic progression such that:
- each term is a positive integer and the common difference is not divisible by 10
- the sum of the digits of each term (in decimal representation) exceeds $M$.
1995 All-Russian Olympiad, 4
Prove that if all angles of a convex $n$-gon are equal, then there are at least two of its sides that are not longer than their adjacent sides.
[i]A. Berzin’sh, O. Musin[/i]
1984 AMC 12/AHSME, 19
A box contains 11 balls, numbered 1,2,3,....,11. If 6 balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd?
A. $\frac{100}{231}$
B. $\frac{115}{231}$
C. $\frac{1}{2}$
D. $\frac{118}{231}$
E. $\frac{6}{11}$
2010 LMT, 28
Two knights placed on distinct square of an $8\times8$ chessboard, whose squares are unit squares, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}.$ In how many ways can two identical knights be placed on distinct squares of an $8\times8$ chessboard such that they do NOT attack each other?
2021 SAFEST Olympiad, 1
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that
[list]
[*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
[*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.
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