Found problems: 85335
Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.
Say a number is [i]tico[/i] if the sum of it's digits is a multiple of $2003$.
$\text{(i)}$ Show that there exists a positive integer $N$ such that the first $2003$ multiples, $N,2N,3N,\ldots 2003N$ are all tico.
$\text{(ii)}$ Does there exist a positive integer $N$ such that all it's multiples are tico?
Let $ABC$ be an equilateral triangle . Let point $D$ lie on side $AB,E$ lie on side $AC, D_1$ and $E_1$ lie on side BC such that $AB=DB+BD_1$ and $AC=CE+CE_1$. Calculate the smallest angle between the lines $DE_1$ and $ED_1$.
Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side $AB$ and is tangent to the sides $AC$ and $BC$. The circles $\Gamma_1$ and $ \Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.
Determine the number of arrangements $ a_1,a_2,...,a_{10}$ of the numbers $ 1,2,...,10$ such that $ a_i>a_{2i}$ for $ 1 \le i \le 5$ and $ a_i>a_{2i\plus{}1}$ for $ 1 \le i \le 4$.
For each positive $ x \in \mathbb{R}$, define
$ E(x)=\{[nx]: n\in \mathbb{N}\}$
Find all irrational $ \alpha >1$ with the following property:
If a positive real $ \beta$ satisfies $ E(\beta) \subset E(\alpha)$. then $ \frac{\beta}{\alpha}$ is a natural number.
We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?
If $ a@b\equal{}\frac{a\plus{}b}{a\minus{}b}$, find $ n$ such that $ 3@n\equal{}3$.
Given hexagon $ \mathcal{P}$ inscribed in a unit square, such that each vertex is on the side of the square. It’s known that all interior angles of the hexagon are equal. Find the maximum possible value of the smallest side length of $\mathcal{P}$.
In $\vartriangle ABC, \angle A = 66^o$ and $| AB | <| AC |$. The outer bisector in $A$ intersects $BC$ in $D$ and $| BD | = | AB | + | AC |$. Determine the angles of $\vartriangle ABC$.
Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$, and the lines $BC$, $DE$, and $FA$ determine a triangle with area $324\sqrt{3}$. The perimeter of hexagon $ABCDEF$ can be expressed as $m + n\sqrt{p}$, where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$?
$\textbf{(A)}~47\qquad\textbf{(B)}~52\qquad\textbf{(C)}~55\qquad\textbf{(D)}~58\qquad\textbf{(E)}~63$
Source: 1976 Euclid Part A Problem 3
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The minimum value of the function $2x^2+6x+7$ is
$\textbf{(A) } 7 \qquad \textbf{(B) } \frac{5}{2} \qquad \textbf{(C) } \frac{9}{4} \qquad \textbf{(D) } -\frac{9}{2} \qquad \textbf{(E) } \frac{5}{4}$
A circle which is tangent to the sides $ [AB]$ and $ [BC]$ of $ \triangle ABC$ is also tangent to its circumcircle at the point $ T$. If $ I$ is the incenter of $ \triangle ABC$ , show that $ \widehat{ATI}\equal{}\widehat{CTI}$
Prove that a harmonic function that is not identically zero in the plane cannot vanish on a two-dimensional positive-measure set.
Let $ABC$ be a right-angled triangle with hypotenuse $AB$. Denote by $D$ the foot of the altitude from $C$. Let $Q, R$, and $P$ be the midpoints of the segments $AD, BD$, and $CD$, respectively. Prove that $\angle AP B + \angle QCR = 180^o$.
Czech Republic
A minus sign is placed on one square of a $5 \times 5$ board and plus signs are placed on the remaining squares. A move is to select a $2 \times 2, 3 \times 3, 4 \times 4$ or $5 \times 5$ square and change all the signs in it. Which initial positions allow a series of moves to change all the signs to plus?
To celebrate the $20$th LMT, the LHSMath Team bakes a cake. Each of the $n$ bakers places $20$ candles on the cake. When they count, they realize that there are $(n -1)!$ total candles on the cake. Find $n$.
[i]Proposed by Christopher Cheng[/i]
We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?
We have a $6 \times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?
A number $n$ is called multiplicatively perfect if the product of all the positive divisors of $n$ is $n^2$. Determine the number of positive multiplicatively perfect numbers less than $100$.
Find the smallest positive integer $k$ which is representable in the form $k=19^n-5^m$ for some positive integers $m$ and $n$.
Let $k$ be a positive integer and let $m=6k-1$. Let
$$S(m)=\sum_{j=1}^{2k-1}(-1)^{j+1}\binom m{3j-1}.$$Prove that $S(m)$ is never zero.
$(BUL 4)$ Let $M$ be the point inside the right-angled triangle $ABC (\angle C = 90^{\circ})$ such that $\angle MAB = \angle MBC = \angle MCA =\phi.$ Let $\Psi$ be the acute angle between the medians of $AC$ and $BC.$ Prove that $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.
The circumcircle of triangle $ABC$ has centre $O$. $P$ is the midpoint of $\widehat{BAC}$ and $QP$ is the diameter. Let $I$ be the incentre of $\triangle ABC$ and let $D$ be the intersection of $PI$ and $BC$. The circumcircle of $\triangle AID$ and the extension of $PA$ meet at $F$. The point $E$ lies on the line segment $PD$ such that $DE=DQ$. Let $R,r$ be the radius of the inscribed circle and circumcircle of $\triangle ABC$, respectively.
Show that if $\angle AEF=\angle APE$, then $\sin^2\angle BAC=\dfrac{2r}R$