Found problems: 85335
2007 Sharygin Geometry Olympiad, 2
Points $A', B', C'$ are the feet of the altitudes $AA', BB'$ and $CC'$ of an acute triangle $ABC$. A circle with center $B$ and radius $BB'$ meets line $A'C'$ at points $K$ and $L$ (points $K$ and $A$ are on the same side of line $BB'$). Prove that the intersection point of lines $AK$ and $CL$ belongs to line $BO$ ($O$ is the circumcenter of triangle $ABC$).
2010 Contests, 2
If $a @ b = \frac{a\times b}{a+b}$, for $a,b$ positive integers, then what is $5 @10$?
$\textbf{(A)}\ \frac{3}{10} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{10}{3} \qquad\textbf{(E)}\ 50$
2018 CCA Math Bonanza, L5.1
Estimate the number of five-card combinations from a standard $52$-card deck that contain a pair (two cards with the same number).
An estimate of $E$ earns $2e^{-\frac{\left|A-E\right|}{20000}}$ points, where $A$ is the actual answer.
[i]2018 CCA Math Bonanza Lightning Round #5.1[/i]
2023 Regional Olympiad of Mexico West, 4
Prove that you can pick $15$ distinct positive integers between $1$ and $2023$, such that each one of them and the sum between some of them is never a perfect square, nor a perfect cube or any other greater perfect power.
2016 AMC 12/AHSME, 9
Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?
$\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$
1952 AMC 12/AHSME, 12
The sum to infinity of the terms of an infinite geometric progression is $ 6$. The sum of the first two terms is $ 4\frac {1}{2}$. The first term of the progression is:
$ \textbf{(A)}\ 3 \text{ or } 1\frac {1}{2} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2\frac {1}{2} \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 9 \text{ or } 3$
V Soros Olympiad 1998 - 99 (Russia), 9.6
On the coordinate plane, draw all points $M(x, y)$, whose coordinates satisfy the equation:
$$ |x-y| + |1-x| + |y|=1 $$
2018 Iran MO (2nd Round), 4
Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that:
$$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$
for all reals $x, y $.
2016 Postal Coaching, 5
Is it possible to define an operation $\star$ on $\mathbb Z$ such that[list=a][*] for any $a, b, c$ in $\mathbb Z, (a \star b) \star c = a \star (b \star c)$ holds;
[*] for any $x, y$ in $\mathbb Z, x \star x \star y = y \star x \star x=y$?[/list]
2022 HMNT, 19
Define the [i]annoyingness[/i] of a permutation of the first $n$ integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence $1,2 \ldots ,n$ appears. For instance, the annoyingness of $3,2,1$ is $3,$ and the annoyingness of $1,3,4,2$ is $2.$
A random permutation of $1,2, \ldots, 2022$ is selected. Compute the expected value of the annoyingness of this permutation.
2021 BMT, 9
Druv has a $33 \times 33$ grid of unit squares, and he wants to color each unit square with exactly one of three distinct colors such that he uses all three colors and the number of unit squares with each color is the same. However, he realizes that there are internal sides, or unit line segments that have exactly one unit square on each side, with these two unit squares having different colors. What is the minimum possible number of such internal sides?
2007 Regional Olympiad of Mexico Center Zone, 6
Certain tickets are numbered as follows: $1, 2, 3, \dots, N $. Exactly half of the tickets have the digit $ 1$ on them. If $N$ is a three-digit number, determine all possible values of $N $.
1997 Czech And Slovak Olympiad IIIA, 3
A tetrahedron $ABCD$ is divided into five polyhedra so that each face of the tetrahedron is a face of (exactly) one polyhedron, and that the intersection of any two of the polyhedra is either a common vertex, a common edge, or a common face. What is the smallest possible sum of the numbers of faces of the five polyhedra?
2022 Kazakhstan National Olympiad, 1
Given a triangle $ABC$ draw the altitudes $AD$, $BE$, $CF$. Take points $P$ and $Q$ on $AB$ and $AC$, respectively such that $PQ \parallel BC$. Draw the circles with diameters $BQ$ and $CP$ and let them intersect at points $R$
and $T$ where $R$ is closer to $A$ than $T$. Draw the altitudes $BN$ and $CM$ in the triangle $BCR$. Prove that $FM$, $EN$ and $AD$ are concurrent.\\
2003 Iran MO (2nd round), 2
$\angle{A}$ is the least angle in $\Delta{ABC}$. Point $D$ is on the arc $BC$ from the circumcircle of $\Delta{ABC}$. The perpendicular bisectors of the segments $AB,AC$ intersect the line $AD$ at $M,N$, respectively. Point $T$ is the meet point of $BM,CN$. Suppose that $R$ is the radius of the circumcircle of $\Delta{ABC}$. Prove that:
\[ BT+CT\leq{2R}. \]
1986 China Team Selection Test, 3
Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that:
[b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$.
[b]II.[/b] Find such $A_k$ for $19^{86}.$
2012 International Zhautykov Olympiad, 1
Do there exist integers $m, n$ and a function $f\colon \mathbb R \to \mathbb R$ satisfying simultaneously the following two conditions?
$\bullet$ i) $f(f(x))=2f(x)-x-2$ for any $x \in \mathbb R$;
$\bullet$ ii) $m \leq n$ and $f(m)=n$.
1969 IMO Shortlist, 63
$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.
PEN F Problems, 8
Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.
2002 China Team Selection Test, 3
Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n \minus{} x^m) \equal{} (ax^m \minus{} 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.
1988 China Team Selection Test, 2
Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying
(i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$.
(ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.
2021 USA TSTST, 8
Let $ABC$ be a scalene triangle. Points $A_1,B_1$ and $C_1$ are chosen on segments $BC,CA$ and $AB$, respectively, such that $\triangle A_1B_1C_1$ and $\triangle ABC$ are similar. Let $A_2$ be the unique point on line $B_1C_1$ such that $AA_2=A_1A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that $\triangle A_2B_2C_2$ and $\triangle ABC$ are similar.
[i]Fedir Yudin [/i]
1983 IMO Shortlist, 2
Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$
Prove that there exists an infinity of [i]superabundant[/i] numbers.
1950 AMC 12/AHSME, 25
The value of $ \log_5 \frac {(125)(625)}{25}$ is equal to:
$\textbf{(A)}\ 725 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 3125 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ \text{None of these}$
2019 LIMIT Category B, Problem 2
Let $\mathbb C$ denote the set of all complex numbers. Define
$$A=\{(z,w)|z,w\in\mathbb C\text{ and }|z|=|w|\}$$$$B=\{(z,w)|z,w\in\mathbb C\text{ and }z^2=w^2\}$$$\textbf{(A)}~A=B$
$\textbf{(B)}~A\subset B\text{ and }A\ne B$
$\textbf{(C)}~B\subset A\text{ and }B\ne A$
$\textbf{(D)}~\text{None of the above}$