Found problems: 329
2006 Team Selection Test For CSMO, 2
Let $AA_1$ and $BB_1$ be the altitudes of an acute-angled, non-isosceles triangle $ABC$. Also, let $A_0$ and $B_0$ be the midpoints of its sides $BC$ and $CA$, respectively. The line $A_1B_1$ intersects the line $A_0B_0$ at a point $C'$. Prove that the line $CC'$ is perpendicular to the Euler line of the triangle $ABC$ (this is the line that joins the orthocenter and the circumcenter of the triangle $ABC$).
1991 Romania Team Selection Test, 3
Prove the following identity for every $ n\in N$:
$ \sum_{j\plus{}h\equal{}n,j\geq h}\frac{(\minus{}1)^h2^{j\minus{}h}\binom{j}{h}}{j}\equal{}\frac{2}{n}$
2020 Thailand TST, 4
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.
2000 Putnam, 1
Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^{\infty} x_j^2, $ given that $x_0, x_1, \cdots$ are positive numbers for which $\displaystyle\sum_{j=0}^{\infty} x_j = A$?
1994 Canada National Olympiad, 3
$25$ men sit around a circular table. Every hour there is a vote, and each must respond [i]yes [/i]or [i]no[/i]. Each man behaves as follows: on the $n^{\text{th}}$, vote if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)^{\text{th}}$ vote as on the $n^{\text{th}}$ vote; but if his response is different from that of both his neighbours on the $n^{\text{th}}$ vote, then his response on the $(n+1)^{\text{th}}$ vote will be different from his response on the $n^{\text{th}}$ vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.
2013 Online Math Open Problems, 38
Triangle $ABC$ has sides $AB = 25$, $BC = 30$, and $CA=20$. Let $P,Q$ be the points on segments $AB,AC$, respectively, such that $AP=5$ and $AQ=4$. Suppose lines $BQ$ and $CP$ intersect at $R$ and the circumcircles of $\triangle{BPR}$ and $\triangle{CQR}$ intersect at a second point $S\ne R$. If the length of segment $SA$ can be expressed in the form $\frac{m}{\sqrt{n}}$ for positive integers $m,n$, where $n$ is not divisible by the square of any prime, find $m+n$.
[i]Victor Wang[/i]
2006 ITAMO, 2
Solve $p^n+144=m^2$ where $m,n\in \mathbb{N}$ and $p$ is a prime number.
2010 BMO TST, 4
Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number.
[b]a)[/b] Prove the inequality for $ k\equal{}1$.
[b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles.
2007 AIME Problems, 1
How many positive perfect squares less than $10^{6}$ are multiples of $24$?
2007 Germany Team Selection Test, 2
Find all quadruple $ (m,n,p,q) \in \mathbb{Z}^4$ such that \[ p^m q^n \equal{} (p\plus{}q)^2 \plus{} 1.\]
2006 Peru IMO TST, 3
[color=blue][size=150]PERU TST IMO - 2006[/size]
Saturday, may 20.[/color]
[b]Question 03[/b]
In each square of a board drawn into squares of $2^n$ rows and
$n$ columns $(n\geq 1)$ are written a 1 or a -1, in such a way
that the rows of the board constitute all the possible sequences
of length $n$ that they are possible to be formed with numbers 1
and -1.
Next, some of the numbers are replaced by zeros.
Prove that it is possible to choose some of the rows of the board
(It could be a row only) so that in the chosen rows, is fulfilled that the
sum of the numbers in each column is zero.
----
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88511]Spanish version[/url]
$\text{\LaTeX}{}$ed by carlosbr
1951 AMC 12/AHSME, 31
A total of $ 28$ handshakes was exchanged at the conclusion of a party. Assuming that each participant was equally polite toward all the others, the number of people present was:
$ \textbf{(A)}\ 14 \qquad\textbf{(B)}\ 28 \qquad\textbf{(C)}\ 56 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 7$
1957 AMC 12/AHSME, 47
In circle $ O$, the midpoint of radius $ OX$ is $ Q$; at $ Q$, $ \overline{AB} \perp \overline{XY}$. The semi-circle with $ \overline{AB}$ as diameter intersects $ \overline{XY}$ in $ M$. Line $ \overline{AM}$ intersects circle $ O$ in $ C$, and line $ \overline{BM}$ intersects circle $ O$ in $ D$. Line $ \overline{AD}$ is drawn. Then, if the radius of circle $ O$ is $ r$, $ AD$ is:
[asy]defaultpen(linewidth(.8pt));
unitsize(2.5cm);
real m = 0;
real b = 0;
pair O = origin;
pair X = (-1,0);
pair Y = (1,0);
pair Q = midpoint(O--X);
pair A = (Q.x, -1*sqrt(3)/2);
pair B = (Q.x, -1*A.y);
pair M = (Q.x + sqrt(3)/2,0);
m = (B.y - M.y)/(B.x - M.x);
b = (B.y - m*B.x);
pair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));
m = (A.y - M.y)/(A.x - M.x);
b = (A.y - m*A.x);
pair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));
draw(Circle(O,1));
draw(Arc(Q,sqrt(3)/2,-90,90));
draw(A--B);
draw(X--Y);
draw(B--D);
draw(A--C);
draw(A--D);
dot(O);dot(M);
label("$B$",B,NW);
label("$C$",C,NE);
label("$Y$",Y,E);
label("$D$",D,SE);
label("$A$",A,SW);
label("$X$",X,W);
label("$Q$",Q,SW);
label("$O$",O,SW);
label("$M$",M,NE+2N);[/asy]$ \textbf{(A)}\ r\sqrt {2} \qquad \textbf{(B)}\ r\qquad \textbf{(C)}\ \text{not a side of an inscribed regular polygon}\qquad \textbf{(D)}\ \frac {r\sqrt {3}}{2}\qquad \textbf{(E)}\ r\sqrt {3}$
1996 Canada National Olympiad, 5
Let $r_1$, $r_2$, $\ldots$, $r_m$ be a given set of $m$ positive rational numbers such that $\sum_{k=1}^m r_k = 1$. Define the function $f$ by $f(n)= n-\sum_{k=1}^m \: [r_k n]$ for each positive integer $n$. Determine the minimum and maximum values of $f(n)$. Here ${\ [ x ]}$ denotes the greatest integer less than or equal to $x$.
2006 Hong Kong TST., 3
In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B.
2011 IMC, 1
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function. A point $x$ is called a [i]shadow[/i] point if there exists a point $y\in \mathbb{R}$ with $y>x$ such that $f(y)>f(x).$ Let $a<b$ be real numbers and suppose that
$\bullet$ all the points of the open interval $I=(a,b)$ are shadow points;
$\bullet$ $a$ and $b$ are not shadow points.
Prove that
a) $f(x)\leq f(b)$ for all $a<x<b;$
b) $f(a)=f(b).$
[i]Proposed by José Luis Díaz-Barrero, Barcelona[/i]
1956 AMC 12/AHSME, 49
Triangle $ PAB$ is formed by three tangents to circle $ O$ and $ < APB \equal{} 40^{\circ}$; then angle $ AOB$ equals:
$ \textbf{(A)}\ 45^{\circ} \qquad\textbf{(B)}\ 50^{\circ} \qquad\textbf{(C)}\ 55^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 70^{\circ}$
2008 ITest, 88
A six dimensional "cube" (a $6$-cube) has $64$ vertices at the points $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3).$ This $6$-cube has $192\text{ 1-D}$ edges and $240\text{ 2-D}$ edges. This $6$-cube gets cut into $6^6=46656$ smaller congruent "unit" $6$-cubes that are kept together in the tightly packaged form of the original $6$-cube so that the $46656$ smaller $6$-cubes share 2-D square faces with neighbors ($\textit{one}$ 2-D square face shared by $\textit{several}$ unit $6$-cube neighbors). How many 2-D squares are faces of one or more of the unit $6$-cubes?
PEN O Problems, 27
Let $p$ and $q$ be relatively prime positive integers. A subset $S\subseteq \mathbb{N}_0$ is called ideal if $0 \in S$ and, for each element $n \in S$, the integers $n+p$ and $n+q$ belong to $S$. Determine the number of ideal subsets of $\mathbb{N}_0$.
PEN E Problems, 32
Let $n \ge 5$ be an integer. Show that $n$ is prime if and only if $n_{i} n_{j} \neq n_{p} n_{q}$ for every partition of $n$ into $4$ integers, $n=n_{1}+n_{2}+n_{3}+n_{4}$, and for each permutation $(i, j, p, q)$ of $(1, 2, 3, 4)$.
2014 AMC 8, 21
The $7$-digit numbers $\underline{7}$ $ \underline{4}$ $ \underline{A}$ $ \underline{5}$ $ \underline{2}$ $ \underline{B}$ $ \underline{1}$ and $\underline{3}$ $ \underline{2}$ $ \underline{6}$ $ \underline{A}$ $ \underline{B}$ $ \underline{4}$ $ \underline{C}$ are each multiples of $3$. Which of the following could be the value of $C$?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad \textbf{(E) }8$
PEN H Problems, 9
Determine all integers $a$ for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions $x, \;y$.
1951 AMC 12/AHSME, 25
The apothem of a square having its area numerically equal to its perimeter is compared with the apothem of an equilateral triangle having its area numerically equal to its perimeter. The first apothem will be:
$ \textbf{(A)}\ \text{equal to the second} \qquad\textbf{(B)}\ \frac {4}{3} \text{ times the second} \qquad\textbf{(C)}\ \frac {2}{\sqrt {3}} \text{ times the second} \\
\textbf{(D)}\ \frac {\sqrt {2}}{\sqrt {3}} \text{ times the second} \qquad\textbf{(E)}\ \text{indeterminately related to the second}$
[i][Note: The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides.][/i]
2013 India IMO Training Camp, 2
In a triangle $ABC$ with $B = 90^\circ$, $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal. If $\widehat{ADB} = \varphi$ then prove that $\tan^2 (\varphi/2) = \tan (C/2)$.
2005 District Olympiad, 1
Prove that for all $a\in\{0,1,2,\ldots,9\}$ the following sum is divisible by 10:
\[ S_a = \overline{a}^{2005} + \overline{1a}^{2005} + \overline{2a}^{2005} + \cdots + \overline{9a}^{2005}. \]