Found problems: 85335
1999 National High School Mathematics League, 12
The bottom surface of triangular pyramid $S-ABC$ is a regular triangle. Projection of $A$ on plane $SBC$ is $H$, which is the orthocenter of $\triangle SBC$. If $H-AB-C=30^{\circ},SA=2\sqrt3$, then the volume of $S-ABC$ is________.
2021 Brazil National Olympiad, 2
Let \(n\) be a positive integer. On a \(2 \times 3 n\) board, we mark some squares, so that any square (marked or not) is adjacent to at most two other distinct marked squares (two squares are adjacent when they are distinct and have at least one vertex in common, i.e. they are horizontal, vertical or diagonal neighbors; a square is not adjacent to itself).
(a) What is the greatest possible number of marked square?
(b) For this maximum number, in how many ways can we mark the squares? configurations that can be achieved through rotation or reflection are considered distinct.
2023 India National Olympiad, 5
Euler marks $n$ different points in the Euclidean plane. For each pair of marked points, Gauss writes down the number $\lfloor \log_2 d \rfloor$ where $d$ is the distance between the two points. Prove that Gauss writes down less than $2n$ distinct values.
[i]Note:[/i] For any $d>0$, $\lfloor \log_2 d\rfloor$ is the unique integer $k$ such that $2^k\le d<2^{k+1}$.
[i]Proposed by Pranjal Srivastava[/i]
2006 AMC 10, 24
Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?
$ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 16 \qquad \textbf{(C) } \frac 14 \qquad \textbf{(D) } \frac 13 \qquad \textbf{(E) } \frac 12$
1951 AMC 12/AHSME, 20
When simplified and expressed with negative exponents, the expression $ (x \plus{} y)^{ \minus{} 1}(x^{ \minus{} 1} \plus{} y^{ \minus{} 1})$ is equal to:
$ \textbf{(A)}\ x^{ \minus{} 2} \plus{} 2x^{ \minus{} 1}y^{ \minus{} 1} \plus{} y^{ \minus{} 2} \qquad\textbf{(B)}\ x^{ \minus{} 2} \plus{} 2^{ \minus{} 1}x^{ \minus{} 1}y^{ \minus{} 1} \plus{} y^{ \minus{} 2} \qquad\textbf{(C)}\ x^{ \minus{} 1}y^{ \minus{} 1}$
$ \textbf{(D)}\ x^{ \minus{} 2} \plus{} y^{ \minus{} 2} \qquad\textbf{(E)}\ \frac {1}{x^{ \minus{} 1}y^{ \minus{} 1}}$
2010 Moldova Team Selection Test, 1
Find all $ 3$-digit numbers such that placing to the right side of the number its successor we get a $ 6$-digit number which is a perfect square.
Geometry Mathley 2011-12, 11.3
Let $ABC$ be a triangle such that $AB = AC$ and let $M$ be a point interior to the triangle. If $BM$ meets $AC$ at $D$. show that $\frac{DM}{DA}=\frac{AM}{AB}$ if and only if $\angle AMB = 2\angle ABC$.
Michel Bataille
2007 Balkan MO Shortlist, A1
Find the minimum and maximum value of the function
\begin{align*} f(x,y)=ax^2+cy^2 \end{align*}
Under the condition $ax^2-bxy+cy^2=d$, where $a,b,c,d$ are positive real numbers such that $b^2 -4ac <0$
2022 Serbia JBMO TST, 4
Initially in every cell of a $5\times 5$ board is the number $0$. In one move you may take any cell of this board and add $1$ to it and all of its adjacent cells (two cells are adjacent if they share an edge). After a finite number of moves, number $n$ is written in all cells. Find all possible values of $n$.
1952 Moscow Mathematical Olympiad, 227
$99$ straight lines divide a plane into $n$ parts. Find all possible values of $n$ less than $199$.
PEN O Problems, 6
Let $S$ be a set of integers such that [list][*] there exist $a, b \in S$ with $\gcd(a, b)=\gcd(a-2,b-2)=1$, [*] if $x,y\in S$, then $x^2 -y\in S$.[/list] Prove that $S=\mathbb{Z}$.
2013 AMC 12/AHSME, 8
Line $\ell_1$ has equation $3x-2y=1$ and goes through $A=(-1,-2)$. Line $\ell_2$ has equation $y=1$ and meets line $\ell_1$ at point $B$. Line $\ell_3$ has positive slope, goes through point $A$, and meets $\ell_2$ at point $C$. The area of $\triangle ABC$ is $3$. What is the slope of $\ell_3$?
$ \textbf{(A)}\ \frac{2}{3}\qquad\textbf{(B)}\ \frac{3}{4}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{4}{3}\qquad\textbf{(E)}\ \frac{3}{2} $
2009 Abels Math Contest (Norwegian MO) Final, 3a
In the triangle $ABC$ the edge $BC$ has length $a$, the edge $AC$ length $b$, and the edge $AB$ length $c$. Extend all the edges at both ends – by the length $a$ from the vertex $A, b$ from $B$, and $c$ from $C$. Show that the six endpoints of the extended edges all lie on a common circle.
[img]https://cdn.artofproblemsolving.com/attachments/8/7/14c8c6a4090d4fade28893729a510d263e7abb.png[/img]
2019 AMC 12/AHSME, 19
Raashan, Sylvia, and Ted play the following game. Each starts with $\$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $\$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $\$1$? (For example, Raashan and Ted may each decide to give $\$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $\$0$, Sylvia would have $\$2$, and Ted would have $\$1$, and and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their $\$1$ to, and and the holdings will be the same as the end of the second [sic] round.
$\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{2}{3}$
2024 Caucasus Mathematical Olympiad, 2
The rhombuses $ABDK$ and $CBEL$ are arranged so that $B$ lies on the segment $AC$ and $E$ lies on the segment $BD$. Point $M$ is the midpoint of $KL$. Prove that $\angle DME=90^{\circ}$.
2008 National Olympiad First Round, 32
At a party with $n\geq 4$ people, if every $3$ people have exactly $1$ common friend, how many different values can $n$ take?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of the above}
$
2010 CHMMC Fall, 8
Rachel writes down a simple inequality: one $2$-digit number is greater than another. Matt is sitting across from Rachel and peeking at her paper. If Matt, reading upside down, sees a valid inequality between two $2$-digit numbers, compute the number of different inequalities that Rachel could have written. Assume that each digit is either a $1, 6, 8$, or $9$.
2016-2017 SDML (Middle School), 10
For how many positive integer values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$\begin{cases} 2x > 3x - 3 \\ 3x - a > -6 \end{cases}$$
$\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$
2009 Today's Calculation Of Integral, 487
Suppose two functions $ f(x)\equal{}x^4\minus{}x,\ g(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ satisfy $ f(1)\equal{}g(1),\ f(\minus{}1)\equal{}g(\minus{}1)$.
Find the values of $ a,\ b,\ c,\ d$ such that $ \int_{\minus{}1}^1 (f(x)\minus{}g(x))^2dx$ is minimal.
2021 AMC 10 Spring, 11
For which of the following integers $b$ is the base-$b$ number $2021_b - 221_b$ not divisible by $3$?
$\textbf{(A) } 3 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
2012 Hanoi Open Mathematics Competitions, 1
[b]Q1.[/b] Assum that $a-b=-(a-b).$ Then:
$(A) \; a=b; \qquad (B) \; a<b; \qquad (C) \; a>b \qquad (D) \; \text{ It is impossible to compare those of a and b.}$
2016 Croatia Team Selection Test, Problem 4
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime.
Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.
III Soros Olympiad 1996 - 97 (Russia), 9.2
How many solutions, depending on the value of the parameter $a$, has the equation $$\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a ?$$
2011 Denmark MO - Mohr Contest, 1
Georg writes the numbers from $1$ to $15$ on different pieces of paper.
He attempts to sort these pieces of paper into two stacks so that none of the stacks contains two numbers whose sum is a square number.Prove that this is impossible.
(The square numbers are the numbers $0 = 0^2$, $1 = 1^2$, $4 = 2^2$, $9 = 3^2$ etc.)
2021 Turkey MO (2nd round), 3
A circle $\Gamma$ is tangent to the side $BC$ of a triangle $ABC$ at $X$ and tangent to the side $AC$ at $Y$. A point $P$ is taken on the side $AB$. Let $XP$ and $YP$ intersect $\Gamma$ at $K$ and $L$ for the second time, $AK$ and $BL$ intersect $\Gamma$ at $R$ and $S$ for the second time. Prove that $XR$ and $YS$ intersect on $AB$.