Found problems: 329
2008 Bosnia Herzegovina Team Selection Test, 1
$ 8$ students took part in exam that contains $ 8$ questions. If it is known that each question was solved by at least $ 5$ students, prove that we can always find $ 2$ students such that each of questions was solved by at least one of them.
1996 Canadian Open Math Challenge, 10
Determine the sum of angles $A,B,$ where $0^\circ \leq A,B, \leq 180^\circ$ and
\[ \sin A + \sin B = \sqrt{\frac{3}{2}}, \cos A + \cos B = \sqrt{\frac{1}{2}} \]
2010 India IMO Training Camp, 10
Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$
2014 Cezar Ivănescu, 2
While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$.
[hide="Clarifications"]
[list]
[*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect.
[*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide]
[i]Ray Li[/i]
1993 India National Olympiad, 3
If $a,b,c,d \in \mathbb{R}_{+}$ and $a+b +c +d =1$, show that \[ ab +bc +cd \leq \dfrac{1}{4}. \]
1999 AMC 12/AHSME, 6
What is the sum of the digits of the decimal form of the product $ 2^{1999}\cdot 5^{2001}$?
$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 10$
1997 Turkey Team Selection Test, 3
In a football league, whenever a player is transferred from a team $X$ with $x$ players to a team $Y$ with $y$ players, the federation is paid $y-x$ billions liras by $Y$ if $y \geq x$, while the federation pays $x-y$ billions liras to $X$ if $x > y$. A player is allowed to change as many teams as he wishes during a season. Suppose that a season started with $18$ teams of $20$ players each. At the end of the season, $12$ of the teams turn out to have again $20$ players, while the remaining $6$ teams end up with $16,16, 21, 22, 22, 23$ players, respectively. What is the maximal amount the federation may have won during the season?
1990 AIME Problems, 11
Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.
1978 AMC 12/AHSME, 1
If $1-\frac{4}{x}+\frac{4}{x^2}=0$, then $\frac{2}{x}$ equals
$\textbf{(A) }-1\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }-1\text{ or }2\qquad \textbf{(E) }-1\text{ or }-2$
2002 AMC 12/AHSME, 13
Two different positive numbers $ a$ and $ b$ each differ from their reciprocals by 1. What is $ a \plus{} b$?
\[ \textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } \sqrt {5} \qquad \textbf{(D) } \sqrt {6} \qquad \textbf{(E) } 3
\]
1989 APMO, 2
Prove that the equation \[ 6(6a^2 + 3b^2 + c^2) = 5n^2 \] has no solutions in integers except $a = b = c = n = 0$.
2015 AMC 10, 9
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata?
[asy]
import cse5;pathpen=black;pointpen=black;
size(1.5inch);
D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W));
path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle;
draw(P,linewidth(2));
fill(P,gray);
[/asy]
$\textbf{(A) } \dfrac{4\pi}{5}
\qquad\textbf{(B) } \dfrac{9\pi}{8}
\qquad\textbf{(C) } \dfrac{4\pi}{3}
\qquad\textbf{(D) } \dfrac{7\pi}{5}
\qquad\textbf{(E) } \dfrac{3\pi}{2}
$
2014 Iran Team Selection Test, 3
we named a $n*n$ table $selfish$ if we number the row and column with $0,1,2,3,...,n-1$.(from left to right an from up to down)
for every {$ i,j\in{0,1,2,...,n-1}$} the number of cell $(i,j)$ is equal to the number of number $i$ in the row $j$.
for example we have such table for $n=5$
1 0 3 3 4
1 3 2 1 1
0 1 0 1 0
2 1 0 0 0
1 0 0 0 0
prove that for $n>5$ there is no $selfish$ table
2010 AMC 10, 20
A fly trapped inside a cubical box with side length $ 1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?
$ \textbf{(A)}\ 4 \plus{} 4\sqrt2 \qquad \textbf{(B)}\ 2 \plus{} 4\sqrt2 \plus{} 2\sqrt3 \qquad \textbf{(C)}\ 2 \plus{} 3\sqrt2 \plus{} 3\sqrt3 \qquad \textbf{(D)}\ 4\sqrt2 \plus{} 4\sqrt3 \\ \textbf{(E)}\ 3\sqrt2 \plus{} 5\sqrt3$
2005 Nordic, 4
The circle $\zeta_{1}$ is inside the circle $\zeta_{2}$, and the circles touch each other at $A$. A line through $A$ intersects $\zeta_{1}$ also at $B$, and $\zeta_{2}$ also at $C$. The tangent to $\zeta_{1}$ at $B$ intersects $\zeta_{2}$ at $D$ and $E$. The tangents of $\zeta_{1}$ passing thorugh $C$ touch $\zeta_{2}$ at $F$ and $G$. Prove that $D$, $E$, $F$ and $G$ are concyclic.
1999 Mongolian Mathematical Olympiad, Problem 5
The edge lengths of a tetrahedron are a, b, c, d, e, f, the areas of its faces
are S1, S2, S3, S4, and its volume is V .
Prove that
2 [S1 S2 S3 S4](1/2) > 3V [abcdef](1/6)
this problem comes from: http://www.imomath.com/othercomp/jkasfvgkusa/MonMO99.pdf
I was just wondering if someone could write it in LATEX form.
[color=red]_____________________________________
EDIT by moderator: If you type[/color]
[code]The edge lengths of a tetrahedron are $a, b, c, d, e, f,$ the areas of its faces are $S_1, S_2, S_3, S_4,$ and its volume is $V.$ Prove that
$2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$[/code]
[color=red]it shows up as:[/color]
The edge lengths of a tetrahedron are $ a, b, c, d, e, f,$ the areas of its faces are $ S_1, S_2, S_3, S_4,$ and its volume is $ V.$ Prove that
$ 2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$
2002 AMC 8, 6
A birdbath is designed to overflow so that it will be self-cleaning. Water flows in at the rate of 20 milliliters per minute and drains at the rate of 18 milliliters per minute. One of these graphs shows the volume of water in the birdbath during the filling time and continuing into the overflow time. Which one is it?
[asy]
size(450);
defaultpen(linewidth(0.8));
path[] p={origin--(8,8)--(14,8), (0,10)--(4,10)--(14,0), origin--(14,14), (0,14)--(14,14), origin--(7,7)--(14,0)};
int i;
for(i=0; i<5; i=i+1) {
draw(shift(21i,0)*((0,16)--origin--(14,0)));
draw(shift(21i,0)*(p[i]));
label("Time", (7+21i,0), S);
label(rotate(90)*"Volume", (21i,8), W);
}
label("$A$", (0*21 + 7,-5), S);
label("$B$", (1*21 + 7,-5), S);
label("$C$", (2*21 + 7,-5), S);
label("$D$", (3*21 + 7,-5), S);
label("$E$", (4*21 + 7,-5), S);
[/asy]
$\text{(A)}\ \text{A} \qquad \text{(B)}\ \text{B} \qquad \text{(C)}\ \text{C} \qquad \text{(D)}\ \text{D} \qquad \text{(E)}\ \text{E}$
2003 Bulgaria National Olympiad, 2
Let $a,b,c$ be rational numbers such that $a+b+c$ and $a^2+b^2+c^2$ are [b]equal[/b] integers. Prove that the number $abc$ can be written as the ratio of a perfect cube and a perfect square which are relatively prime.
2009 China Team Selection Test, 2
Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$
2004 India Regional Mathematical Olympiad, 5
Let ABCD be a quadrilateral; X and Y be the midpoints of AC and BD respectively and lines through X and Y respectively parallel to BD, AC meet in O. Let P,Q,R,S be the midpoints of AB, BC, CD, DA respectively. Prove that
(A) APOS and APXS have the same area
(B) APOS, BQOP, CROQ, DSOR have the same area.
2014 USAMTS Problems, 5:
Let $a_0,a_1,a_2,\dots$ be a sequence of nonnegative integers such that $a_2=5$, $a_{2014}=2015$, and $a_n=a_{a_{n-1}}$ for all positive integers $n$. Find all possible values of $a_{2015}$.
2009 Indonesia MO, 1
In a drawer, there are at most $ 2009$ balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is $ \frac12$. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?
2004 USAMTS Problems, 4
Find, with proof, all integers $n$ such that there is a solution in nonnegative real numbers $(x,y,z)$ to the system of equations
\[2x^2+3y^2+6z^2=n\text{ and }3x+4y+5z=23.\]
1967 IMO Longlists, 39
Show that the triangle whose angles satisfy the equality \[ \frac{sin^2(A) + sin^2(B) + sin^2(C)}{cos^2(A) + cos^2(B) + cos^2(C)} = 2 \] is a rectangular triangle.
2004 China Team Selection Test, 1
Points $D,E,F$ are on the sides $BC, CA$ and $AB$, respectively which satisfy $EF || BC$, $D_1$ is a point on $BC,$ Make $D_1E_1 || D_E, D_1F_1 || DF$ which intersect $AC$ and $AB$ at $E_1$ and $F_1$, respectively. Make $\bigtriangleup PBC \sim \bigtriangleup DEF$ such that $P$ and $A$ are on the same side of $BC.$ Prove that $E, E_1F_1, PD_1$ are concurrent.
[color=red][Edit by Darij: See my post #4 below for a [b]possible correction[/b] of this problem. However, I am not sure that it is in fact the problem given at the TST... Does anyone have a reliable translation?][/color]