Found problems: 85335
2004 Thailand Mathematical Olympiad, 1
Given that $\cos 4A =\frac13$ and $-\frac{\pi}{4} \le A \le \frac{\pi}{4}$ , find the value of $\cos^8 A - \sin^8 A$.
2010 District Olympiad, 4
Determine all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that
\[ f(n)\plus{}f(n\plus{}1)\plus{}f(f(n))\equal{}3n\plus{}1, \quad \forall n\in \mathbb{N}.\]
2007 Greece Junior Math Olympiad, 2
If $n$ is is an integer such that $4n+3$ is divisible by $11,$ find the form of $n$ and the remainder of $n^{4}$ upon division by $11$.
1953 AMC 12/AHSME, 27
The radius of the first circle is $ 1$ inch, that of the second $ \frac{1}{2}$ inch, that of the third $ \frac{1}{4}$ inch and so on indefinitely. The sum of the areas of the circles is:
$ \textbf{(A)}\ \frac{3\pi}{4} \qquad\textbf{(B)}\ 1.3\pi \qquad\textbf{(C)}\ 2\pi \qquad\textbf{(D)}\ \frac{4\pi}{3} \qquad\textbf{(E)}\ \text{none of these}$
1957 AMC 12/AHSME, 21
Start with the theorem "If two angles of a triangle are equal, the triangle is isosceles," and the following four statements:
1. If two angles of a triangle are not equal, the triangle is not isosceles.
2. The base angles of an isosceles triangle are equal.
3. If a triangle is not isosceles, then two of its angles are not equal.
4. A necessary condition that two angles of a triangle be equal is that the triangle be isosceles.
Which combination of statements contains only those which are logically equivalent to the given theorem?
$ \textbf{(A)}\ 1,\,2,\,3,\,4 \qquad
\textbf{(B)}\ 1,\,2,\,3\qquad
\textbf{(C)}\ 2,\,3,\,4\qquad
\textbf{(D)}\ 1,\,2\qquad
\textbf{(E)}\ 3,\,4$
2024 India IMOTC, 14
Let $ABCD$ be a convex cyclic quadrilateral with circumcircle $\omega$. Let $BA$ produced beyond $A$ meet $CD$ produced beyond $D$, at $L$. Let $\ell$ be a line through $L$ meeting $AD$ and $BC$ at $M$ and $N$ respectively, so that $M,D$ (respectively $N,C$) are on opposite sides of $A$ (resp. $B$). Suppose $K$ and $J$ are points on the arc $AB$ of $\omega$ not containing $C,D$ so that $MK, NJ$ are tangent to $\omega$. Prove that $K,J,L$ are collinear.
[i]Proposed by Rijul Saini[/i]
2006 Finnish National High School Mathematics Competition, 2
Show that the inequality \[3(1 + a^2 + a^4)\geq (1 + a + a^2)^2\]
holds for all real numbers $a.$
1996 Iran MO (2nd round), 4
Let $n$ blue points $A_i$ and $n$ red points $B_i \ (i = 1, 2, \ldots , n)$ be situated on a line. Prove that
\[\sum_{i,j} A_i B_j \geq \sum_{i<j} A_iA_j + \sum_{i<j} B_iB_j.\]
2014 PUMaC Team, 1
The evilest number $666^{666}$ has $1881$ digits. Let $a$ be the sum of digits of $66^{666}$ and let $b$ be the sum of digits of $a$ and let $c$ be the sum of digits of $b$. Find $c$.
2008 China Team Selection Test, 3
Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.
2005 Taiwan National Olympiad, 1
Let $a,b,c$ be three positive real numbers such that $abc=1$. Prove that: \[ 1+\frac{3}{a+b+c}\ge{\frac{6}{ab+bc+ca}} . \]
1966 IMO Shortlist, 27
Given a point $P$ lying on a line $g,$ and given a circle $K.$ Construct a circle passing through the point $P$ and touching the circle $K$ and the line $g.$
2016 NIMO Problems, 6
Let $ABC$ be a triangle with $AB=20$, $AC=34$, and $BC=42$. Let $\omega_1$ and $\omega_2$ be the semicircles with diameters $\overline{AB}$ and $\overline{AC}$ erected outwards of $\triangle ABC$ and denote by $\ell$ the common external tangent to $\omega_1$ and $\omega_2$. The line through $A$ perpendicular to $\overline{BC}$ intersects $\ell$ at $X$ and $BC$ at $Y$. The length of $\overline{XY}$ can be written in the form $m+\sqrt n$ where $m$ and $n$ are positive integers. Find $100m+n$.
[i]Proposed by David Altizio[/i]
2016 ASDAN Math Tournament, 3
Real numbers $x,y,z$ form an arithmetic sequence satisfying
\begin{align*}
x+y+z&=6\\
xy+yz+zx&=10.
\end{align*}
What is the absolute value of their common difference?
2001 Greece Junior Math Olympiad, 4
Let $ABC$ be a triangle with altitude $AD$ , angle bisectors $AE$ and $BZ$ that intersecting at point $I$. From point $I$ let $IT$ be a perpendicular on $AC$. Also let line $(e)$ be perpendicular on $AC$ at point $A$. Extension of $ET$ intersects line $(e)$ at point $K$. Prove that $AK=AD$.
1996 Baltic Way, 17
Using each of the eight digits $1,3,4,5,6,7,8$ and $9$ exactly once, a three-digit number $A$, two two-digit numbers $B$ and $C$, $B<C$, and a one digit number $D$ are formed. The numbers are such that $A+D=B+C=143$. In how many ways can this be done?
1984 Tournament Of Towns, (065) A3
An infinite (in both directions) sequence of rooms is situated on one side of an infinite hallway. The rooms are numbered by consecutive integers and each contains a grand piano. A finite number of pianists live in these rooms. (There may be more than one of them in some of the rooms.) Every day some two pianists living in adjacent rooms (the Arth and ($k +1$)st) decide that they interfere with each other’s practice, and they move to the ($k - 1$)st and ($k + 2$)nd rooms, respectively. Prove that these moves will cease after a finite number of days.
(VG Ilichev)
2022 Kyiv City MO Round 1, Problem 3
Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$.
[i](Proposed by Oleksii Masalitin)[/i]
2011 Federal Competition For Advanced Students, Part 2, 1
Every brick has $5$ holes in a line. The holes can be filled with bolts (fi tting in one hole) and braces (fi tting into two neighboring holes). No hole may remain free.
One puts $n$ of these bricks in a line to form a pattern from left to right. In this line no two braces and no three bolts may be adjacent.
How many diff erent such patterns can be produced with $n$ bricks?
1951 AMC 12/AHSME, 32
If $ \triangle ABC$ is inscribed in a semicircle whose diameter is $ AB$, then $ AC \plus{} BC$ must be
$ \textbf{(A)}\ \text{equal to }AB \qquad\textbf{(B)}\ \text{equal to }AB\sqrt {2} \qquad\textbf{(C)}\ \geq AB\sqrt {2}$
$ \textbf{(D)}\ \leq AB\sqrt {2} \qquad\textbf{(E)}\ AB^2$
2021 Iran MO (2nd Round), 1
There are two distinct Points $A$ and $B$ on a line. We color a point $P$ on segment $AB$, distinct from $A,B$ and midpoint of segment $AB$ to red. In each move , we can reflect one of the red point wrt $A$ or $B$ and color the midpoint of the resulting point and the point we reflected from ( which is one of $A$ or $B$ ) to red. For example , if we choose $P$ and the reflection of $P$ wrt to $A$ is $P'$ , then midpoint of $AP'$ would be red. Is it possible to make the midpoint of $AB$ red after a finite number of moves?
2011 Math Prize For Girls Problems, 11
The sequence $a_0$, $a_1$, $a_2$, $\ldots\,$ satisfies the recurrence equation
\[
a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}
\]
for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?
2015 Tuymaada Olympiad, 7
In $\triangle ABC$ points $M,O$ are midpoint of $AB$ and circumcenter. It is true, that $OM=R-r$. Bisector of external $\angle A$ intersect $BC$ at $D$ and bisector of external $\angle C$ intersect $AB$ at $E$.
Find possible values of $\angle CED$
[i]D. Shiryaev [/i]
2023 Nordic, P4
Let $ABC$ be a triangle, and $M$ the midpoint of the side $BC$. Let $E$ and $F$ be points on the sides $AC$ and $AB$, respectively, so that $ME=MF$. Let $D$ be the second intersection of the circumcircle of $MEF$ and the side $BC$. Consider the lines $\ell_D$, $\ell_E$ and $\ell_F$ through $D, E$ and $F$, respectively, such that $\ell_D \perp BC$, $\ell_E \perp AC$ and $\ell_F \perp AB$. Show that $\ell_D, \ell_E$ and $\ell_F$ are concurrent.
1967 IMO, 4
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$