Found problems: 85335
1987 Tournament Of Towns, (157) 1
From vertex $A$ in square $ABCD$ (of side length $1$ ) two lines are drawn , one intersecting side $BC$ and the other intersecting side $CD$. The angle between these lines is $\theta$. From vertices $B$ and $D$ we construct perpendiculars to each of these lines . Find the area of the quadrilateral whose vertices are the four feet of these perpendiculars.
2013 Stanford Mathematics Tournament, 6
Compute the largest root of $x^4-x^3-5x^2+2x+6$.
1979 Austrian-Polish Competition, 5
The circumcenter and incenter of a given tetrahedron coincide. Prove that all its faces are congruent.
1993 Korea - Final Round, 4
An integer which is the area of a right-angled triangle with integer sides is called [i]Pythagorean[/i]. Prove that for every positive integer $n > 12$ there exists a Pythagorean number between $n$ and $2n.$
2010 Mathcenter Contest, 2
Let $k$ and $d$ be integers such that $k>1$ and $0\leq d<9$. Prove that there exists some integer $n$ such that the $k$th digit from the right of $2^n$ is $d$.
[i](tatari/nightmare)[/i]
2008 Princeton University Math Competition, B1
Find all pairs of positive real numbers $(a,b)$ such that $\frac{n-2}{a} \leq \left\lfloor bn \right\rfloor < \frac{n-1}{a}$ for all positive integes $n$.
2014 PUMaC Combinatorics A, 4
Amy has a $2 \times 10$ puzzle grid which she can use $1 \times 1$ and $1 \times 2$ (1 vertical, 2 horizontal) tiles to cover. How many ways can she exactly cover the grid without any tiles overlapping and without rotating the tiles?
2019 USAMTS Problems, 3
Call a quadruple of positive integers $(a, b, c, d)$ fruitful if there are infinitely many integers $m$ such that $\text{gcd} (am + b, cm + d) = 2019$. Find all possible values of $|ad-bc|$ over fruitful quadruples $(a, b, c, d)$.
2009 QEDMO 6th, 9
For every natural $n$ let $\phi (n)$ be the number of coprime numbers $k \in \{1,2,...,n\}$. (Example: $\phi (12) = 4$, because among the numbers $1, 2, ..., 12$ there are only the$ 4$ numbers, $1, 5, 7$ and $11$ coprime to$12.$)
If $k$ is a natural number, then one defines $\phi^k (n)=\underbrace{\strut \phi (\phi ...(\phi (n)) ...)}_{(k \, times \phi)}$ (Example: $\phi^3 (n)=\phi (\phi (\phi (n))) $)
For every whole $n> 2$ let $c(n)$ be the smallest natural number $k$ with $\phi^k (n)= 2$.
Prove that $c (ab) = c (a) + c (b)$ for odd integers $a$ and $b$, both of which are greater than $2$, .
2012 Sharygin Geometry Olympiad, 4
Determine all integer $n > 3$ for which a regular $n$-gon can be divided into equal triangles by several (possibly intersecting) diagonals.
(B.Frenkin)
PEN M Problems, 16
Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i\geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$?
1996 Putnam, 6
Let $(a_1,b_1),(a_2,b_2),\ldots ,(a_n,b_n)$ be the vertices of a convex polygon containing the origin in its interior. Prove that there are positive real numbers $x,y$ such that :
\[ (a_1,b_1)x^{a_1}y^{b_1}+(a_2,b_2)x^{a_2}y^{b_2}+\ldots +(a_n,b_n)x^{a_n}y^{b_n}=(0,0) \]
2021 Peru Iberoamerican Team Selection Test, P1
Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$
2020 European Mathematical Cup, 4
Let \(a,b,c\) be positive real numbers such that \(ab+bc+ac = a+b+c\). Prove the following inequality:
\[\sqrt{a+\frac{b}{c}} + \sqrt{b+\frac{c}{a}} + \sqrt{c+\frac{a}{b}} \leq \sqrt{2} \cdot \min \left\{ \frac{a}{b}+\frac{b}{c}+\frac{c}{a},\ \frac{b}{a}+\frac{c}{b}+\frac{a}{c} \right\}.\] \\ \\ [i]Proposed by Dorlir Ahmeti.[/i]
2013 Balkan MO Shortlist, N2
Determine all positive integers $x$, $y$ and $z$ such that $x^5 + 4^y = 2013^z$.
([i]Serbia[/i])
2005 Iran Team Selection Test, 3
Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.
2010 Denmark MO - Mohr Contest, 4
It is stated that $2^{2010}$ is a $606$-digit number that begins with $1$. How many of the numbers $1, 2,2^2,2^3, ..., 2^{2009}$ start with $4$?
2019 SG Originals, Q7
Let $n$ be a natural number. A sequence is $k-$complete if it contains all residues modulo $n^k$. Let $Q(x)$ be a polynomial with integer coefficients. For $k\ge 2$, define $Q^k(x)=Q(Q^{k-1}(x))$, where $Q^1(x)=Q(x)$. Show that if $$0,Q(0),Q^2(0),Q^3(0),\ldots $$is $2018-$complete, then it is $k-$complete for all positive integers $k$.
[i]Proposed by Ma Zhao Yu[/i]
2012 Math Prize for Girls Olympiad, 2
Let $m$ and $n$ be integers greater than 1. Prove that $\left\lfloor \dfrac{mn}{6} \right\rfloor$ non-overlapping 2-by-3 rectangles can be placed in an $m$-by-$n$ rectangle. Note: $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.
2010 Contests, 1
Solve the system equations
\[\left\{\begin{array}{cc}x^{4}-y^{4}=240\\x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.\]
2019 Serbia JBMO TST, 1
Does there exist a positive integer $n$, such that the number of divisors of $n!$ is divisible by $2019$?
2003 China Team Selection Test, 2
Denote by $\left(ABC\right)$ the circumcircle of a triangle $ABC$.
Let $ABC$ be an isosceles right-angled triangle with $AB=AC=1$ and $\measuredangle CAB=90^{\circ}$. Let $D$ be the midpoint of the side $BC$, and let $E$ and $F$ be two points on the side $BC$.
Let $M$ be the point of intersection of the circles $\left(ADE\right)$ and $\left(ABF\right)$ (apart from $A$).
Let $N$ be the point of intersection of the line $AF$ and the circle $\left(ACE\right)$ (apart from $A$).
Let $P$ be the point of intersection of the line $AD$ and the circle $\left(AMN\right)$.
Find the length of $AP$.
2014 AMC 10, 6
Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days?
${ \textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}$
2000 Belarus Team Selection Test, 3.1
In a triangle $ABC$, let $a = BC, b = AC$ and let $m_a,m_b$ be the corresponding medians. Find all real numbers $k$ for which the equality $m_a+ka = m_b +kb$ implies that $a = b$.
2003 National Olympiad First Round, 5
Let $ABC$ be a triangle and $D$ be the foot of the altitude from $C$ to $AB$. If $|CH|=|HD|$ where $H$ is the orthocenter, what is $\tan \widehat {A} \cdot \tan \widehat{B}$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ \sqrt 2
\qquad\textbf{(C)}\ 3/2
\qquad\textbf{(D)}\ \sqrt 3
\qquad\textbf{(E)}\ \text{None of the preceding}
$