Found problems: 85335
Kyiv City MO Juniors 2003+ geometry, 2016.9.5
On the sides $BC$ and $AB$ of the triangle $ABC$ the points ${{A} _ {1}}$ and ${{C} _ {1}} $ are selected accordingly so that the segments $A {{A} _ {1}}$ and $C {{C} _ {1}}$ are equal and perpendicular. Prove that if $\angle ABC = 45 {} ^ \circ$, then $AC = A {{A} _ {1}} $.
(Gogolev Andrew)
2009 National Olympiad First Round, 23
The minimum value of $ x(x \plus{} 4)(x \plus{} 8)(x \plus{} 12)$ in real numbers is ?
$\textbf{(A)}\ \minus{} 240 \qquad\textbf{(B)}\ \minus{} 252 \qquad\textbf{(C)}\ \minus{} 256 \qquad\textbf{(D)}\ \minus{} 260 \qquad\textbf{(E)}\ \minus{} 280$
2022 Harvard-MIT Mathematics Tournament, 2
Rectangle $R_0$ has sides of lengths $3$ and $4$. Rectangles $R_1$, $R_2$, and $R_3$ are formed such that:
$\bullet$ all four rectangles share a common vertex $P$,
$\bullet$ for each $n = 1, 2, 3$, one side of $R_n$ is a diagonal of $R_{n-1}$,
$\bullet$ for each $n = 1, 2, 3$, the opposite side of $R_n$ passes through a vertex of $R_{n-1}$ such that the center of $R_n$ is located counterclockwise of the center of $R_{n-1}$ with respect to $P$.
Compute the total area covered by the union of the four rectangles.
[img]https://cdn.artofproblemsolving.com/attachments/3/1/e9edd39e60e4a4defdb127b93b19ab0d0f443c.png[/img]
2015 Irish Math Olympiad, 4
Two circles $C_1$ and $C_2$, with centres at $D$ and $E$ respectively, touch at $B$. The circle having $DE$ as diameter intersects the circle $C_1$ at $H$ and the circle $C_2$ at $K$. The points $H$ and $K$ both lie on the same side of the line $DE$. $HK$ extended in both directions meets the circle $C_1$ at $L$ and meets the circle $C_2$ at $M$. Prove that
(a) $|LH| = |KM|$
(b) the line through $B$ perpendicular to $DE$ bisects $HK$.
2007 Greece Junior Math Olympiad, 4
Each of the $50$ students in a class sent greeting cards to $25$ of the others. Prove that there exist two students who greeted each other.
IV Soros Olympiad 1997 - 98 (Russia), 9.6
A rhombus is circumscribed around a square with side $1997$. Find its diagonals if it is known that they are equal to different integers.
2018 JBMO TST-Turkey, 8
Let $x, y, z$ be positive real numbers such that
$\sqrt {x}, \sqrt {y}, \sqrt {z}$ are sides of a triangle and $\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5$.
Prove that
$\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0$
2012 Sharygin Geometry Olympiad, 1
The altitudes $AA_1$ and $BB_1$ of an acute-angled triangle ABC meet at point $O$. Let $A_1A_2$ and $B_1B_2$ be the altitudes of triangles $OBA_1$ and $OAB_1$ respectively. Prove that $A_2B_2$ is parallel to $AB$.
(L.Steingarts)
2001 IMC, 2
Let $a_{0}=\sqrt{2}, b_{0}=2,a_{n+1}=\sqrt{2-\sqrt{4-a_{n}^{2}}},b_{n+1}=\frac{2b_{n}}{2+\sqrt{4+b_{n}^{2}}}$.
a) Prove that the sequences $(a_{n})$ and $(b_{n})$ are decreasing and converge to $0$.
b) Prove that the sequence $(2^{n}a_{n})$ is increasing, the sequence $(2^{n}b_{n})$ is decreasing and
both converge to the same limit.
c) Prove that there exists a positive constant $C$ such that for all $n$ the following inequality holds: $0 <b_{n}-a_{n} <\frac{C}{8^{n}}$.
2021 Final Mathematical Cup, 2
Let $ABC$ be an acute triangle, where $AB$ is the smallest side and let $D$ be the midpoint of $AB$. Let $P$ be a point in the interior of the triangle $ABC$ such that $\angle CAP = \angle CBP = \angle ACB$. From the point $P$, we draw perpendicular lines on $BC$ and $AC$ where the intersection point with $BC$ is $M$, and with $AC$ is $N$ . Through the point $M$ we draw a line parallel to $AC$, and through $N$ parallel to $BC$. These lines intercept at the point $K$. Prove that $D$ is the center of the circumscribed circle for the triangle $MNK$.
1998 AMC 8, 21
A $4*4*4$ cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch a side or the bottom of the box?
$ \text{(A)}\ 48\qquad\text{(B)}\ 52\qquad\text{(C)}\ 60\qquad\text{(D)}\ 64\qquad\text{(E)}\ 80 $
2022 Iran MO (2nd round), 1
Let $E$ and $F$ on $AC$ and $AB$ respectively in $\triangle ABC$ such that $DE || BC$ then draw line $l$ through $A$ such that $l || BC$ let $D'$ and $E'$ reflection of $D$ and $E$ to $l$ respectively prove that $D'B, E'C$ and $l$ are congruence.
LMT Guts Rounds, 2020 F29
Find the number of pairs of integers $(a,b)$ with $0 \le a,b \le 2019$ where $ax \equiv b \pmod{2020}$ has exactly $2$ integer solutions $0 \le x \le 2019$.
[i]Proposed by Richard Chen[/i]
1998 Korea Junior Math Olympiad, 6
For positive reals $a \geq b \geq c \geq 0$ prove the following inequality:
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{a+b}{a+c}+\frac{b+c}{b+a}+\frac{c+a}{c+b}$$
2016 Online Math Open Problems, 29
Let $n$ be a positive integer. Yang the Saltant Sanguivorous Shearling is on the side of a very steep mountain that is embedded in the coordinate plane. There is a blood river along the line $y=x$, which Yang may reach but is not permitted to go above (i.e. Yang is allowed to be located at $(2016,2015)$ and $(2016,2016)$, but not at $(2016,2017)$). Yang is currently located at $(0,0)$ and wishes to reach $(n,0)$. Yang is permitted only to make the following moves:
(a) Yang may [i]spring[/i], which consists of going from a point $(x,y)$ to the point $(x,y+1)$.
(b) Yang may [i]stroll[/i], which consists of going from a point $(x,y)$ to the point $(x+1,y)$.
(c) Yang may [i]sink[/i], which consists of going from a point $(x,y)$ to the point $(x,y-1)$.
In addition, whenever Yang does a [i]sink[/i], he breaks his tiny little legs and may no longer do a [i]spring[/i] at any time afterwards. Yang also expends a lot of energy doing a [i]spring[/i] and gets bloodthirsty, so he must visit the blood river at least once afterwards to quench his bloodthirst. (So Yang may still [i]spring[/i] while bloodthirsty, but he may not finish his journey while bloodthirsty.) Let there be $a_n$ different ways for which Yang can reach $(n,0)$, given that Yang is permitted to pass by $(n,0)$ in the middle of his journey. Find the $2016$th smallest positive integer $n$ for which $a_n\equiv 1\pmod 5$.
[i]Proposed by James Lin[/i]
2007 Hanoi Open Mathematics Competitions, 2
Which is largest positive integer n satisfying the following inequality: $n^{2007} > (2007)^n$.
1950 AMC 12/AHSME, 44
The graph of $ y\equal{}\log x$
$\textbf{(A)}\text{Cuts the }y\text{-axis} \qquad\\
\textbf{(B)}\ \text{Cuts all lines perpendicular to the }x\text{-axis} \qquad\\
\textbf{(C)}\ \text{Cuts the }x\text{-axis} \qquad\\
\textbf{(D)}\ \text{Cuts neither axis} \qquad\\
\textbf{(E)}\ \text{Cuts all circles whose center is at the origin}$
2021 LMT Spring, A16
Find the number of ordered pairs $(a,b)$ of positive integers less than or equal to $20$ such that \[\gcd(a,b)>1 \quad \text{and} \quad \frac{1}{\gcd(a,b)}+\frac{a+b}{\text{lcm}(a,b)} \geq 1.\]
[i]Proposed by Zachary Perry[/i]
2023 Austrian MO National Competition, 4
Find all pairs of positive integers $(n, k)$ satisfying the equation $$n!+n=n^k.$$
Kvant 2021, M2653
Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$.
[i]Nikolay Belukhov[/i]
2001 Croatia National Olympiad, Problem 3
Let there be given triples of integers $(r_j,s_j,t_j),~j=1,2,\ldots,N$, such that for each $j$, $r_j,t_j,s_j$ are not all even. Show that one can find integers $a,b,c$ such that $ar_j+bs_j+ct_j$ is odd for at least $\frac{4N}7$ of the indices $j$.
2004 Estonia National Olympiad, 5
The alphabet of language $BAU$ consists of letters $B, A$, and $U$. Independently of the choice of the $BAU$ word of length n from which to start, one can construct all the $BAU$ words with length n using iteratively the following rules:
(1) invert the order of the letters in the word;
(2) replace two consecutive letters: $BA \to UU, AU \to BB, UB \to AA, UU \to BA, BB \to AU$ or $AA \to UB$.
Given that $BBAUABAUUABAUUUABAUUUUABB$ is a $BAU$ word, does $BAU$ have
a) the word $BUABUABUABUABAUBAUBAUBAUB$ ?
b) the word $ABUABUABUABUAUBAUBAUBAUBA$ ?
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.
MathLinks Contest 6th, 2.1
Solve in positive real numbers the following equation $x^{-y} + y^{-x} = 4 - x - y$.
1949-56 Chisinau City MO, 27
The areas of two right-angled triangles have ratio equal to the squares of their hypotenuses. Show that these triangles are similar.