Found problems: 85335
2005 Taiwan TST Round 1, 3
$n$ teams take part in a tournament, in which every two teams compete exactly once, and that no draws are possible. It is known that for any two teams, there exists another team which defeated both of the two teams. Find all $n$ for which this is possible.
2008 Baltic Way, 20
Let $ M$ be a point on $ BC$ and $ N$ be a point on $ AB$ such that $ AM$ and $ CN$ are angle bisectors of the triangle $ ABC$. Given that $ \frac {\angle BNM}{\angle MNC} \equal{} \frac {\angle BMN}{\angle NMA}$, prove that the triangle $ ABC$ is isosceles.
2023-24 IOQM India, 1
Let $n$ be a positive integer such that $1 \leq n \leq 1000$. Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$. Let
$$
a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. }
$$
Find $a-b$.
1966 IMO Longlists, 44
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$
1996 All-Russian Olympiad Regional Round, 10.4
In each cell of a square table of size $n \times n$ cells ($n \ge 3$) the number $1$ or $-1$ is written. If you take any two lines, multiply numbers standing above each other in them and add the n resulting products, then the sum will be equal to $0$. Prove that the number $n$ is divisible by $4$.
Kyiv City MO Juniors 2003+ geometry, 2008.8.4
There are two triangles $ABC$ and $BKL$ on the plane so that the segment $AK$ is divided into three equal parts by the point of intersection of the medians $\vartriangle ABC$ and the point of intersection of the bisectors $ \vartriangle BKL $ ($AK $ - median $ \vartriangle ABC$, $KA$ - bisector $\vartriangle BKL $) and quadrilateral $KALC $ is trapezoid. Find the angles of the triangle $BKL$.
(Bogdan Rublev)
2012 Indonesia TST, 4
Given a non-zero integer $y$ and a positive integer $n$. If $x_1, x_2, \ldots, x_n \in \mathbb{Z} - \{0, 1\}$ and $z \in \mathbb{Z}^+$ satisfy $(x_1x_2 \ldots x_n)^2y \le 2^{2(n+1)}$ and $x_1x_2 \ldots x_ny = z + 1$, prove that there is a prime among $x_1, x_2, \ldots, x_n, z$.
[color=blue]It appears that the problem statement is incorrect; suppose $y = 5, n = 2$, then $x_1 = x_2 = -1$ and $z = 4$. They all satisfy the problem's conditions, but none of $x_1, x_2, z$ is a prime. What should the problem be, or did I misinterpret the problem badly?[/color]
2023 MOAA, 13
If real numbers $x$, $y$, and $z$ satisfy $x^2-yz = 1$ and $y^2-xz = 4$ such that $|x+y+z|$ is minimized, then $z^2-xy$ can be expressed in the form $\sqrt{a}-b$ where $a$ and $b$ are positive integers. Find $a+b$.
[i]Proposed by Andy Xu[/i]
1987 IMO Longlists, 10
In a Cartesian coordinate system, the circle $C_1$ has center $O_1(-2, 0)$ and radius $3$. Denote the point $(1, 0)$ by $A$ and the origin by $O$.Prove that there is a constant $c > 0$ such that for every $X$ that is exterior to $C1$,
\[OX- 1 \geq c \min\{AX,AX^2\}.\]
Find the largest possible $c.$
2000 India Regional Mathematical Olympiad, 7
Find all real values of $a$ such that $x^4 - 2ax^2 + x + a^2 -a = 0$ has all its roots real.
1963 Putnam, A3
Find an integral formula for the solution of the differential equation
$$\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\;\, x\geq 1,$$
for $y$ as a function of $f$ satisfying the initial conditions $y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0$, where $f$ is continuous and $\delta$ is the differential operator $ x \frac{d}{dx}.$
2015 Sharygin Geometry Olympiad, P2
Let $O$ and $H$ be the circumcenter and the orthocenter of a triangle $ABC$. The line passing through the midpoint of $OH$ and parallel to $BC$ meets $AB$ and $AC$ at points $D$ and $E$. It is known that $O$ is the incenter of triangle $ADE$. Find the angles of $ABC$.
2003 Switzerland Team Selection Test, 2
In an acute-angled triangle $ABC, E$ and $F$ are the feet of the altitudes from $B$ and $C$, and $G$ and $H$ are the projections of $B$ and $C$ on $EF$, respectively. Prove that $HE = FG$.
2018 Bosnia And Herzegovina - Regional Olympiad, 1
if $a$, $b$ and $c$ are real numbers such that $(a-b)(b-c)(c-a) \neq 0$, prove the equality:
$\frac{b^2c^2}{(a-b)(a-c)}+\frac{c^2a^2}{(b-c)(b-a)}+\frac{a^2b^2}{(c-a)(c-b)}=ab+bc+ca$
2007 Purple Comet Problems, 15
We have some identical paper squares which are black on one side of the sheet and white on the other side. We can join nine squares together to make a $3$ by $3$ sheet of squares by placing each of the nine squares either white side up or black side up. Two of these $3$ by $3$ sheets are distinguishable if neither can be made to look like the other by rotating the sheet or by turning it over. How many distinguishable $3$ by $3$ squares can we form?
2021 BMT, 3
Compute $\log_2 6 \cdot \log_3 72 - \log_2 9 - \log_3 8$.
2023 USAJMO, 2
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.
[i]Proposed by Holden Mui[/i]
2011 F = Ma, 25
A hollow cylinder with a very thin wall (like a toilet paper tube) and a block are placed at rest at the top of a plane with inclination $\theta$ above the horizontal. The cylinder rolls down the plane without slipping and the block slides down the plane; it is found that both objects reach the bottom of the plane simultaneously. What is the coefficient of kinetic friction between the block and the plane?
(A) $0$
(B) $\frac{1}{3}\tan \theta$
(C) $\frac{1}{2}\tan \theta$
(D) $\frac{2}{3}\tan \theta$
(E) $\tan \theta$
2019 Regional Olympiad of Mexico Southeast, 3
Eight teams are competing in a tournament all against all (every pair of team play exactly one time among them). There are not ties and both results of every game are equally probable. What is the probability that in the tournament every team had lose at least one game and won at least one game?
2012 Hitotsubashi University Entrance Examination, 5
At first a fair dice is placed in such way the spot $1$ is shown on the top face. After that, select a face from the four sides at random, the process that the side would be the top face is repeated $n$ times. Note the sum of the spots of opposite face is 7.
(1) Find the probability such that the spot $1$ would appear on the top face after the $n$-repetition.
(2) Find the expected value of the number of the spot on the top face after the $n$-repetition.
IV Soros Olympiad 1997 - 98 (Russia), 11.3
Draw on the coordinate plane the set of points whose coordinates satisfy the equation $$\sin x \cos^2 y +\sin y \cos^2 x =0$$
2004 Bulgaria Team Selection Test, 3
In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.
2011 Hanoi Open Mathematics Competitions, 4
Among the five statements on real numbers below, how many of them are correct?
"If $a < b < 0$ then $a < b^2$" ,
"If $0 < a < b$ then $a < b^2$",
"If $a^3 < b^3$ then $a < b$",
"If $a^2 < b^2$ then $a < b$",
"If $|a| < |b|$ then $a < b$",
(A) $0$, (B) $1$, (C) $2$, (D) $3$, (E) $4$
2010 IMC, 1
Let $0 < a < b$. Prove that
$\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.
2009 German National Olympiad, 2
Find all positive interger $ n$ so that $ n^3\minus{}5n^2\plus{}9n\minus{}6$ is perfect square number.