Found problems: 85335
2008 Postal Coaching, 2
Let $ABC$ be a triangle, $AD$ be the altitude from $A$ on to $BC$. Draw perpendiculars $DD_1$ and $DD_2$ from $D$ on to $AB$ and $AC$ respectively and let $p(A)$ be the length of the segment $D_1D_2$. Similarly define $p(B)$ and $p(C)$. Prove that $\frac{p(A)p(B)p(C)}{s^3}\le \frac18$ , where s is the semi-perimeter of the triangle $ABC$.
2013 Israel National Olympiad, 4
Determine the number of positive integers $n$ satisfying:
[list]
[*] $n<10^6$
[*] $n$ is divisible by 7
[*] $n$ does not contain any of the digits 2,3,4,5,6,7,8.
[/list]
LMT Team Rounds 2021+, 4
Jeff has a deck of $12$ cards: $4$ $L$s, $4$ $M$s, and $4$ $T$s. Armaan randomly draws three cards without replacement. The probability that he takes $3$ $L$s can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m +n$.
CNCM Online Round 1, 1
Pooki Sooki has $8$ hoodies, and he may wear any of them throughout a 7 day week. He changes his hoodie exactly $2$ times during the week, and will only do so at one of the $6$ midnights. Once he changes out of a hoodie, he never wears it for the rest of the week. The number of ways he can wear his hoodies throughout the week can be expressed as $\frac{8!}{2^k}$. Find $k$.
Proposed by Minseok Eli Park (wolfpack)
LMT Guts Rounds, 5
Big Welk writes the letters of the alphabet in order, and starts again at $A$ each time he gets to $Z.$ What is the $4^3$-rd letter that he writes down?
1996 AIME Problems, 10
Find the smallest positive integer solution to $\tan 19x^\circ=\frac{\cos 96^\circ+\sin 96^\circ}{\cos 96^\circ-\sin 96^\circ}.$
2016 Vietnam National Olympiad, 1
Find all $a\in\mathbb{R}$ such that there is function $f:\mathbb{R}\to\mathbb{R}$
i) $f(1)=2016$
ii) $f(x+y+f(y))=f(x)+ay\quad\forall x,y\in\mathbb{R}$
1980 All Soviet Union Mathematical Olympiad, 287
The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.
2014 NIMO Problems, 8
Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is [i]monotonically bounded[/i] if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$. We say that a term $a_k$ in the sequence with $2\leq k\leq 2^{16}-1$ is a [i]mountain[/i] if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$. Evan writes out all possible monotonically bounded sequences. Let $N$ be the total number of mountain terms over all such sequences he writes. Find the remainder when $N$ is divided by $p$.
[i]Proposed by Michael Ren[/i]
2019 CIIM, Problem 2
Consider the set
\[\{0, 1\}^n = \{X = (x_1, x_2,\dots , x_n) : x_i \in \{0, 1\}, 1 \leq i \leq n\}.\]
We say that $X > Y$ if $X \neq Y$ and the following $n$ inequalities are satisfy
\[x_1 \geq y_1, x_1 + x_2 \geq y_1 + y_2,\dots , x_1 + x_2 + \cdots + x_n \geq y_1 + y_2 + \cdots + y_n.\]
We define a chain of length $k$ as a subset ${Z_1,\dots , Z_k} \subseteq \{0, 1\}^n$ of distinct elements such that $Z_1 > Z_2 > \cdots > Z_k.$
Determine the lenght of longest chain in $\{0,1\}^n$.
2003 Mexico National Olympiad, 4
The quadrilateral $ABCD$ has $AB$ parallel to $CD$. $P$ is on the side $AB$ and $Q$ on the side $CD$ such that $\frac{AP}{PB}= \frac{DQ}{CQ}$. M is the intersection of $AQ$ and $DP$, and $N$ is the intersection of $PC$ and $QB$. Find $MN$ in terms of $AB$ and $CD$.
2016 MMPC, 5
Consider four real numbers $x$, $y$, $a$, and $b$, satisfying $x + y = a + b$ and $x^2 + y^2 = a^2 + b^2$. Prove that $x^n + y^n = a^n + b^n$, for all $n \in \mathbb{N}$.
2008 SDMO (Middle School), 4
Find the number of ordered triples of positive integers $\left(a,b,c\right)$ such that $a\times b\times c=2008^2$.
2017 USA TSTST, 5
Let $ABC$ be a triangle with incenter $I$. Let $D$ be a point on side $BC$ and let $\omega_B$ and $\omega_C$ be the incircles of $\triangle ABD$ and $\triangle ACD$, respectively. Suppose that $\omega_B$ and $\omega_C$ are tangent to segment $BC$ at points $E$ and $F$, respectively. Let $P$ be the intersection of segment $AD$ with the line joining the centers of $\omega_B$ and $\omega_C$. Let $X$ be the intersection point of lines $BI$ and $CP$ and let $Y$ be the intersection point of lines $CI$ and $BP$. Prove that lines $EX$ and $FY$ meet on the incircle of $\triangle ABC$.
[i]Proposed by Ray Li[/i]
2010 AMC 12/AHSME, 23
Monic quadratic polynomials $ P(x)$ and $ Q(x)$ have the property that $ P(Q(x))$ has zeroes at $ x\equal{}\minus{}23,\minus{}21,\minus{}17, \text{and} \minus{}15$, and $ Q(P(x))$ has zeroes at $ x\equal{}\minus{}59, \minus{}57, \minus{}51, \text{and} \minus{}49$. What is the sum of the minimum values of $ P(x)$ and $ Q(x)$?
$ \textbf{(A)}\ \text{\minus{}100} \qquad \textbf{(B)}\ \text{\minus{}82} \qquad \textbf{(C)}\ \text{\minus{}73} \qquad \textbf{(D)}\ \text{\minus{}64} \qquad \textbf{(E)}\ 0$
2006 Federal Math Competition of S&M, Problem 3
Show that for an arbitrary tetrahedron there are two planes such that the ratio of the areas of the projections of the tetrahedron onto the two planes is not less than $\sqrt2$.
1984 Vietnam National Olympiad, 3
Consider a trihedral angle $Sxyz$ with $\angle xSz = \alpha , \angle xSy = \beta$ and $\angle ySz =\gamma$. Let $A,B,C$ denote the dihedral angles at edges $y, z, x$ respectively.
$(a)$ Prove that $\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C}$
$(b)$ Show that $\alpha + \beta = 180^{\circ}$ if and only if $\angle A + \angle B = 180^{\circ}.$
$(c)$ Assume that $\alpha=\beta =\gamma = 90^{\circ}$. Let $O \in Sz$ be a fixed point such that $SO = a$ and let $M,N$ be variable points on $x, y$ respectively. Prove that $\angle SOM +\angle SON +\angle MON$ is constant and find the locus of the incenter of $OSMN$.
2024 Korea Junior Math Olympiad (First Round), 8.
Find the number of 4 digit positive integers '$n$' that follow these.
1) the number of digit $ \le $ 6
2) $ 3 \mid n$, but $ 6 \nmid n $
2018 Taiwan TST Round 3, 2
A [i]calendar[/i] is a (finite) rectangular grid. A calendar is [i]valid[/i] if it satisfies the following conditions:
(i) Each square of the calendar is colored white or red, and there are exactly 10 red squares.
(ii) Suppose that there are $N$ columns of squares in the calendar. Then if we fill in the numbers $1,2,\ldots$ from the top row to the bottom row, and within each row from left to right, there do not exist $N$ consecutive numbers such that the squares they are in are all white.
(iii) Suppose that there are $M$ rows of squares in the calendar. Then if we fill in the numbers $1,2,\ldots$ from the left-most column to the right-most column, and within each column from bottom to top, there do not exist $M$ consecutive numbers such that the squares they are in are all white. In other words, if we rotate the calendar clockwise by $90^{\circ}$, the resulting calendar still satisfies (ii).
How many different kinds of valid calendars are there?
(Remark: During the actual exam, the contestants were confused about what counts as different calendars. So although this was not in the actual exam, I would like to specify that two calendars are considered different if they have different side lengths or if the $10$ red squares are at different locations.)
1975 Bulgaria National Olympiad, Problem 2
Let $F$ be a polygon the boundary of which is a broken line with vertices in the knots (units) of a given in advance regular square network. If $k$ is the count of knots of the network situated over the boundary of $F$, and $\ell$ is the count of the knots of the network lying inside $F$, prove that if the surface of every square from the network is $1$, then the surface $S$ of $F$ is calculated with the formulae:
$$S=\frac k2+\ell-1$$
[i]V. Chukanov[/i]
2019 Balkan MO Shortlist, N2
Let $S \subset \{ 1, \dots, n \}$ be a nonempty set, where $n$ is a positive integer. We denote by $s$ the greatest common divisor of the elements of the set $S$. We assume that $s \not= 1$ and let $d$ be its smallest divisor greater than $1$. Let $T \subset \{ 1, \dots, n \}$ be a set such that $S \subset T$ and $|T| \ge 1 + \left[ \frac{n}{d} \right]$. Prove that the greatest common divisor of the elements in $T$ is $1$.
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[Second Version]
Let $n(n \ge 1)$ be a positive integer and $U = \{ 1, \dots, n \}$. Let $S$ be a nonempty subset of $U$ and let $d (d \not= 1)$ be the smallest common divisor of all elements of the set $S$. Find the smallest positive integer $k$ such that for any subset $T$ of $U$, consisting of $k$ elements, with $S \subset T$, the greatest common divisor of all elements of $T$ is equal to $1$.
2012 AMC 10, 11
Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3$, respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$. What is $BC$?
$ \textbf{(A)}\ 4
\qquad\textbf{(B)}\ 4.8
\qquad\textbf{(C)}\ 10.2
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ 14.4
$
1991 Arnold's Trivium, 7
How many normals to an ellipse can be drawn from a given point in plane? Find the region in which the number of normals is maximal.
2021 Azerbaijan IMO TST, 2
Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other.
$\emph{Slovakia}$
2022 JHMT HS, 5
Let $P(x)$ be a quadratic polynomial satisfying the following conditions:
[list]
[*] $P(x)$ has leading coefficient $1$.
[*] $P(x)$ has nonnegative integer roots that are at most $2022$.
[*] the set of the roots of $P(x)$ is a subset of the set of the roots of $P(P(x))$.
[/list]
Let $S$ be the set of all such possible $P(x)$, and let $Q(x)$ be the polynomial obtained upon summing all the elements of $S$. Find the sum of the roots of $Q(x)$.