Found problems: 85335
2004 AMC 8, 1
On a map, a 12-centimeter length represents $72$ kilometers. How many kilometers does a 17-centimeter length represent?
$\textbf{(A)}\ 6\qquad
\textbf{(B)}\ 102\qquad
\textbf{(C)}\ 204\qquad
\textbf{(D)}\ 864\qquad
\textbf{(E)}\ 1224$
2010 Hanoi Open Mathematics Competitions, 6
Find the greatest integer less than $(2 +\sqrt3)^5$ .
(A): $721$ (B): $722$ (C): $723$ (D): $724$ (E) None of the above.
2014 ASDAN Math Tournament, 3
A robot is standing on the bottom left vertex $(0,0)$ of a $5\times5$ grid, and wants to go to $(5,5)$, only moving to the right $(a,b)\mapsto(a+1,b)$ or upward $(a,b)\mapsto(a,b+1)$. However this robot is not programmed perfectly, and sometimes takes the upper-left diagonal path $(a,b)\mapsto(a-1,b+1)$. As the grid is surrounded by walls, the robot cannot go outside the region $0\leq a,b\leq5$. Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.
1995 Poland - First Round, 12
Find out whether there exist two congruent cubes with a common center such that each face of one cube has a common point with each face of the other.
1964 Swedish Mathematical Competition, 5
$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.
2000 May Olympiad, 3
Let $S$ be a circle with radius $2$, let $S_1$ be a circle,with radius $1$ and tangent, internally to $S$ in $B$ and let $S_2$ be a circle, with radius $1$ and tangent to $S_1$ in $A$, but $S_2$ isn't tangent to $S$. If $K$ is the point of intersection of the line $AB$ and the circle $S$, prove that $K$ is in the circle $S_2$.
1969 Kurschak Competition, 2
A triangle has side lengths $a, b, c$ and angles $A, B, C$ as usual (with $b$ opposite $B$ etc). Show that if $$a(1 - 2 \cos A) + b(1 - 2 \cos B) + c(1 - 2 \cos C) = 0$$ then the triangle is equilateral.
Russian TST 2019, P2
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
2008 Stanford Mathematics Tournament, 16
Suppose convex hexagon $ \text{HEXAGN}$ has $ 120^\circ$-rotational symmetry about a point $ P$—that is, if you rotate it $ 120^\circ$ about $ P$, it doesn't change. If $ PX\equal{}1$, find the area of triangle $ \triangle{GHX}$.
2018 CCA Math Bonanza, TB1
What is the maximum number of diagonals of a regular $12$-gon which can be selected such that no two of the chosen diagonals are perpendicular?
Note: sides are not diagonals and diagonals which intersect outside the $12$-gon at right angles are still considered perpendicular.
[i]2018 CCA Math Bonanza Tiebreaker Round #1[/i]
1987 AMC 8, 22
$\text{ABCD}$ is a rectangle, $\text{D}$ is the center of the circle, and $\text{B}$ is on the circle. If $\text{AD}=4$ and $\text{CD}=3$, then the area of the shaded region is between
[asy]
pair A,B,C,D;
A=(0,4); B=(3,4); C=(3,0); D=origin;
draw(circle(D,5));
fill((0,5)..(1.5,4.7697)..B--A--cycle,black);
fill(B..(4,3)..(5,0)--C--cycle,black);
draw((0,5)--D--(5,0));
label("A",A,NW);
label("B",B,NE);
label("C",C,S);
label("D",D,SW);
[/asy]
$\text{(A)}\ 4\text{ and }5 \qquad \text{(B)}\ 5\text{ and }6 \qquad \text{(C)}\ 6\text{ and }7 \qquad \text{(D)}\ 7\text{ and }8 \qquad \text{(E)}\ 8\text{ and }9$
2006 Victor Vâlcovici, 3
Let be a natural number $ n $ and a matrix $ A\in\mathcal{M}_n(\mathbb{R}) $ having the property that sum of the squares of all its elements is strictly less than $ 1. $ Prove that the matrices $ I\pm A $ are invertible.
2018 Korea USCM, 1
Given vector $\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3$ and recursively defined sequence of vectors $\{\mathbf{v}_n\}_{n\geq 0}$
$$\mathbf{v}_0 = (1,2,3),\quad \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1}$$
Evaluate the value of infinite series $\sum_{n=1}^\infty (3,2,1)\cdot \mathbf{v}_{2n}$.
Kvant 2022, M2703
Given an infinite sequence of numbers $a_1, a_2,...$, in which there are no two equal members. Segment $a_i, a_{i+1}, ..., a_{i+m-1}$ of this sequence is called a monotone segment of length $m$, if $a_i < a_{i+1} <...<a_{i+m-1}$ or $a_i > a_{i+1} >... > a_{i+m-1}$. It turned out that for each natural $k$ the term $a_k$ is contained in some monotonic segment of length $k + 1$. Prove that there exists a natural $N$ such that the sequence $a_N , a_{N+1} ,...$ monotonic.
2002 JBMO ShortLists, 3
Let $ a,b,c$ be positive real numbers such that $ abc\equal{}\frac{9}{4}$. Prove the inequality:
$ a^3 \plus{} b^3 \plus{} c^3 > a\sqrt {b \plus{} c} \plus{} b\sqrt {c \plus{} a} \plus{} c\sqrt {a \plus{} b}$
Jury's variant:
Prove the same, but with $ abc\equal{}2$
2019 Estonia Team Selection Test, 11
Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply:
(a) the circumcircle of each triangle in the set $T$ is $\omega$;
(b) The interior of any two triangles in the set $T$ has no common point.
Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.
2001 German National Olympiad, 2
Determine the maximum possible number of points you can place in a rectangle with lengths $14$ and $28$ such that any two of those points are more than $10$ apart from each other.
VI Soros Olympiad 1999 - 2000 (Russia), 10.5
For what values of $k\ge2$ can the set of natural numbers be colored in $k$ colors in such a way that it contains no single - color infinite arithmetic progression, but for any two colors there is a progression whose members are each colored in one of these two colors?
2010 Gheorghe Vranceanu, 2
Let be a natural number $ n, $ a nonzero number $ \alpha, \quad n $ numbers $ a_1,a_2,\ldots ,a_n $ and $ n+1 $ functions $ f_0,f_1,f_2,\ldots ,f_n $ such that $ f_0=\alpha $ and the rest are defined recursively as
$$ f_k (x)=a_k+\int_0^x f_{k-1} (x)dx . $$
Prove that if all these functions are everywhere nonnegative, then the sum of all these functions is everywhere nonnegative.
2017 Brazil National Olympiad, 5.
[b]5.[/b] In triangle $ABC$, let $r_A$ be the line that passes through the midpoint of $BC$ and is perpendicular to the internal bisector of $\angle{BAC}$. Define $r_B$ and $r_C$ similarly. Let $H$ and $I$ be the orthocenter and incenter of $ABC$, respectively. Suppose that the three lines $r_A$, $r_B$, $r_C$ define a triangle. Prove that the circumcenter of this triangle is the midpoint of $HI$.
2006 Switzerland - Final Round, 1
Find all functions $f : R \to R$ such that for all $x, y \in R$ holds $$yf(2x) - xf(2y) = 8xy(x^2 - y^2).$$
2013 Bosnia and Herzegovina Junior BMO TST, 3
Let $M$ and $N$ be touching points of incircle with sides $AB$ and $AC$ of triangle $ABC$, and $P$ intersection point of line $MN$ and angle bisector of $\angle ABC$. Prove that $\angle BPC =90 ^{\circ}$
2003 All-Russian Olympiad, 1
Suppose that $M$ is a set of $2003$ numbers such that, for any distinct $a, b, c \in M$, the number $a^2 + bc$ is rational. Prove that there is a positive integer $n$ such that $a\sqrt n$ is rational for all $a \in M.$
2013 India PRMO, 13
To each element of the set $S = \{1,2,... ,1000\}$ a colour is assigned. Suppose that for any two elements $a, b$ of $S$, if $15$ divides $a + b$ then they are both assigned the same colour. What is the maximum possible number of distinct colours used?
1979 IMO Longlists, 67
A circle $C$ with center $O$ on base $BC$ of an isosceles triangle $ABC$ is tangent to the equal sides $AB,AC$. If point $P$ on $AB$ and point $Q$ on $AC$ are selected such that $PB \times CQ = (\frac{BC}{2})^2$, prove that line segment $PQ$ is tangent to circle $C$, and prove the converse.