Found problems: 85335
2010 Indonesia TST, 4
Given $3n$ cards, each of them will be written with a number from the following sequence:
$$2, 3, ..., n, n + 1, n + 3, n + 4, ..., 2n + 1, 2n + 2, 2n + 4, ..., 3n + 3$$
with each number used exactly once. Then every card is arranged from left to right in random order. Determine the probability such that for every $i$ with $1\le i \le 3n$, the number written on the $i$-th card, counted from the left, is greater than or equal to $i$.
2023 Turkey Olympic Revenge, 2
Let $ABC$ be a triangle. A point $D$ lies on line $BC$ and points $E,F$ are taken on $AC,AB$ such that $DE \parallel AB$ and $DF\parallel AC$. Let $G = (AEF) \cap (ABC) \neq A$ and $I = (DEF) \cap BC\neq D$. Let $H$ and $O$ denote the orthocenter and the circumcenter of triangle $DEF$. Prove that $A,O,I$ are collinear if and only if $G,H,I$ are collinear.
[i]Proposed by Kaan Bilge[/i]
1981 IMO Shortlist, 8
Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]
1956 AMC 12/AHSME, 28
Mr. J left his entire estate to his wife, his daughter, his son, and the cook. His daughter and son got half the estate, sharing in the ratio of $ 4$ to $ 3$. His wife got twice as much as the son. If the cook received a bequest of $ \$500$, then the entire estate was:
$ \textbf{(A)}\ \$3500 \qquad\textbf{(B)}\ \$5500 \qquad\textbf{(C)}\ \$6500 \qquad\textbf{(D)}\ \$7000 \qquad\textbf{(E)}\ \$7500$
2003 Iran MO (3rd Round), 19
An integer $ n$ is called a good number if and only if $ |n|$ is not square of another intger. Find all integers $ m$ such that they can be written in infinitely many ways as sum of three different good numbers and product of these three numbers is square of an odd number.
2013 China Team Selection Test, 2
Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.
2017 Korea - Final Round, 2
For a positive integer $n$, $(a_0, a_1, \cdots , a_n)$ is a $n+1$-tuple with integer entries.
For all $k=0, 1, \cdots , n$, we denote $b_k$ as the number of $k$s in $(a_0, a_1, \cdots ,a_n)$.
For all $k = 0,1, \cdots , n$, we denote $c_k$ as the number of $k$s in $(b_0, b_1, \cdots ,b_n)$.
Find all $(a_0, a_1, \cdots ,a_n)$ which satisfies $a_0 = c_0$, $a_1=c_1$, $\cdots$, $a_n=c_n$.
2016 NIMO Summer Contest, 1
What is the value of \[\left(9+\dfrac{9}{9}\right)^{9-9/9} - \dfrac{9}{9}?\]
[i]Proposed by David Altizio[/i]
2017 Costa Rica - Final Round, 1
Let the regular hexagon $ABCDEF$ be inscribed in a circle with center $O$, $N$ be such a point Let $E-N-C$, $M$ a point such that $A- M-C$ and $R$ a point on the circumference, such that $D-N- R$. If $\angle EFR = 90^o$, $\frac{AM}{AC}=\frac{CN}{EC}$ and $AC=\sqrt3$, calculate $AM$.
Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.
1977 AMC 12/AHSME, 2
Which one of the following statements is false? All equilateral triangles are
$\textbf{(A)} \ \text{ equiangular} \qquad
\textbf{(B)} \ \text{isosceles} \qquad
\textbf{(C)} \ \text{regular polygons } \qquad
\textbf{(D)} \ \text{congruent to each other} \qquad
\textbf{(E)} \ \text{similar to each other} $
2007 China Team Selection Test, 1
Let $ ABC$ be a triangle. Circle $ \omega$ passes through points $ B$ and $ C.$ Circle $ \omega_{1}$ is tangent internally to $ \omega$ and also to sides $ AB$ and $ AC$ at $ T,\, P,$ and $ Q,$ respectively. Let $ M$ be midpoint of arc $ BC\, ($containing $ T)$ of $ \omega.$ Prove that lines $ PQ,\,BC,$ and $ MT$ are concurrent.
1992 IMO Shortlist, 17
Let $ \alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $ n.$ Prove that:
(a) the inequality $ \alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds;
(b) the above inequality is an equality for infinitely many positive integers, and
(c) there exists a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i }$
goes to zero as $ i$ goes to $ \infty.$
[i]Alternative problem:[/i] Prove that there exists a sequence a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i )}$
(d) $ \infty;$
(e) an arbitrary real number $ \gamma \in (0,1)$;
(f) an arbitrary real number $ \gamma \geq 0$;
as $ i$ goes to $ \infty.$
2001 Greece JBMO TST, 3
$4$ men stand at the entrance of a dark tunnel. Man $A$ needs $10$ minutes to pass through the tunnel, man $B$ needs $5$ minutes, man $C$ needs $2$ minutes and man $D$ needs $1$ minute. There is only one torch, that may be used from anyone that passes through the tunnel. Additionaly, at most $2$ men can pass through at the same time using the existing torch.
Determine the smallest possible time the four men need to reach the exit of the tunnel.
1974 AMC 12/AHSME, 28
Which of the following is satisfied by all numbers $ x$ of the form
\[ x \equal{} \frac {a_1}{3} \plus{} \frac {a_2}{3^2} \plus{} \cdots \plus{} \frac {a_{25}}{3^{25}},\]
where $ a_1$ is $ 0$ or $ 2$, $ a_2$ is $ 0$ or $ 2$,...,$ a_{25}$ is $ 0$ or $ 2$?
$ \textbf{(A)}\ 0 \le x < 1/3 \qquad \textbf{(B)}\ 1/3 \le x < 2/3 \qquad \textbf{(C)}\ 2/3 \le x < 1 \\
\textbf{(D)}\ 0 \le x < 1/3 \text{ or } 2/3 \le x < 1 \qquad \textbf{(E)}\ 1/2 \le x \le 3/4$
2012 Czech-Polish-Slovak Match, 3
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$. Let $I, J$ and $K$ be the incentres of the triangles $ABC, ACD$ and $ABD$ respectively. Let $E$ be the midpoint of the arc $DB$ of circle $\omega$ containing the point $A$. The line $EK$ intersects again the circle $\omega$ at point $F$ $(F \neq E)$. Prove that the points $C, F, I, J$ lie on a circle.
2013 Today's Calculation Of Integral, 870
Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$
(1) Find all points of intersection of $E$ and $H$.
(2) Find the area of the region expressed by the system of inequality
\[\left\{
\begin{array}{ll}
3x^2+y^2\leq 3 &\quad \\
xy\geq \frac 34 , &\quad
\end{array}
\right.\]
1979 Austrian-Polish Competition, 4
Determine all functions $f : N_0 \to R$ satisfying $f (x+y)+ f (x-y)= f (3x)$ for all $x,y$.
2021-IMOC qualification, G2
Given a triangle $ABC$, $D$ is the reflection from the perpendicular foot from $A$ to $BC$ through the midpoint of $BC$. $E$ is the reflection from the perpendicular foot from $B$ to $CA$ through the midpoint of $CA$. $F$ is the reflection from the perpendicular foot from $C$ to $AB$ through the midpoint of $AB$. Prove: $DE \perp AC$ if and only if $DF \perp AB$
1966 AMC 12/AHSME, 21
An "n-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\cdots,k,\cdots,n$, $n\geq 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong the $n$ pairs of sides numbered $k$ and $k+2$ until they meet. (A figure is shown for the case $n=5$)
[img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=704&sid=8da93909c5939e037aa99c429b2d157a[/img]
Let $S$ be the degree-sum of the interior angles at the $n$ points of the star; then $S$ equals:
$\text{(A)} \ 180 \qquad \text{(B)} \ 360 \qquad \text{(C)} \ 180(n+2) \qquad \text{(D)} \ 180(n-2) \qquad \text{(E)} \ 180(n-4)$
2015 India PRMO, 6
$6.$ How many two digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square $?$
1991 All Soviet Union Mathematical Olympiad, 550
a) $r_1, r_2, ... , r_{100}, c_1, c_2, ... , c_{100}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $100 \times 100$ array. The product of the numbers in each column is $1$. Show that the product of the numbers in each row is $-1$.
b) $r_1, r_2, ... , r_{2n}, c_1, c_2, ... , c_{2n}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $2n \times 2n$ array. The product of the numbers in each column is the same. Show that the product of the numbers in each row is also the same.
2013 NIMO Summer Contest, 7
Circle $\omega_1$ and $\omega_2$ have centers $(0,6)$ and $(20,0)$, respectively. Both circles have radius $30$, and intersect at two points $X$ and $Y$. The line through $X$ and $Y$ can be written in the form $y = mx+b$. Compute $100m+b$.
[i]Proposed by Evan Chen[/i]
2022 HMNT, 32
Suppose point $P$ is inside triangle $ABC.$ Let $AP, BP,$ and $CP$ intersect sides $BC, CA,$ and $AB$ at points $D,$ $E,$ and $F,$ respectively. Suppose $\angle APB = \angle BPC = \angle CPA, PD = \tfrac{1}{4}, PE = \tfrac{1}{5},$ and $PF = \tfrac{1}{7}.$ Compute $AP +BP +CP.$
1986 Tournament Of Towns, (111) 5
$20$ football teams take part in a tournament . On the first day all the teams play one match . On the second day all the teams play a further match . Prove that after the second day it is possible to select $10$ teams, so that no two of them have yet played each other.
( S . A . Genkin)
1999 Mongolian Mathematical Olympiad, Problem 2
Any two vertices $A,B$ of a regular $n$-gon are connected by an oriented segment (i.e. either $A\to B$ or $B\to A$). Find the maximum possible number of quadruples $(A,B,C,D)$ of vertices such that $A\to B\to C\to D\to A$.