Found problems: 85335
Seven positive integers are written on a piece of paper. No matter which five numbers one chooses, each of the remaining two numbers divides the sum of the five chosen numbers. How many distinct numbers can there be among the seven?
Represent $1957$ as the sum of $12$ positive integer summands $a_1, a_2, ... , a_{12}$ for which the number $a_1! \cdot a_2! \cdot a_3! \cdot ... \cdot a_{12}!$ is minimal.
Find all pairs $(m, n)$ of positive integers such that the $m \times n$ grid contains exactly $225$ rectangles whose side lengths are odd and whose edges lie on the lines of the grid.
Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if
(i) $a_n < a_{n+1}$ for all $n\ge 1$,
(ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.
Let $p\geq 5$ be a prime number, and set $M=\{1,2,\cdots,p-1\}.$ Define $$T=\{(n,x_n):p|nx_n-1\ \textup{and}\ n,x_n\in M\}.$$ If $\sum_{(n,x_n)\in T}n\left[\dfrac{nx_n}{p}\right]\equiv k \pmod {p},$ with $0\leq k\leq p-1,$ where $\left[\alpha\right]$ denotes the largest integer that does not exceed $\alpha,$ determine the value of $k.$
In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.
Find all natural numbers $n$ and nonnegative integers $x_{1},x_{2},...,x_{n}$ such that $\sum_{i=1}^{n}x_{i}^{2}=1+\frac{4}{4n+1}(\sum_{i=1}^{n}x_{i})^{2}$.
Let $P(x)$ be a polynomial with integer coefficients that has at least one rational root. Let $n$ be a positive integer.
Alan and Allan are playing a game. First, Alan writes down $n$ integers at $n$ different locations on a board. Then Allan may make moves of the following kind: choose a position that has integer $a$ written, then choose a different position that has integer $b$ written, then at the first position erase $a$ and in its place write $a+P(b)$. After any nonnegative number of moves, Allan may choose to end the game. Once Allan ends the game, his score is the number of times the mode (most common element) of the integers on the board appears.
Find, in terms of $P(x)$ and $n$, the maximum score Allan can guarantee.
[i]Henrick Rabinovitz[/i]
Let $\triangle ABC$ be an arbitary triangle, and construct $P,Q,R$ so that each of the angles marked is $30^\circ$. Prove that $\triangle PQR$ is an equilateral triangle.
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair ext30(pair pt1, pair pt2) {
pair r1 = pt1+rotate(-30)*(pt2-pt1), r2 = pt2+rotate(30)*(pt1-pt2);
draw(anglemark(r1,pt1,pt2,25)); draw(anglemark(pt1,pt2,r2,25));
return intersectionpoints(pt1--r1, pt2--r2)[0];
}
pair A = (0,0), B=(10,0), C=(3,7), P=ext30(B,C), Q=ext30(C,A), R=ext30(A,B);
draw(A--B--C--A--R--B--P--C--Q--A); draw(P--Q--R--cycle, linetype("8 8"));
label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$P$", P, NE); label("$Q$", Q, NW); label("$R$", R, S);[/asy]
In some cells of the table $10\times 10$ arranged several $X$'s and a few $O$'s. It is known that there is no line (row or column) completely filled with identical symbols (crosses or zeros). However, if in any empty If you place any icon in a cell, this condition will be violated. What is the minimum number of icons that can appear in a table?
We are numbering the rows and columns of a $29 \text{x} 29$ chess table with numbers $1, 2, ..., 29$ in order (Top row is numbered with $1$ and first columns is numbered with $1$ as well). We choose some of the squares in this chess table and for every selected square, we know that there exist at most one square having a row number greater than or equal to this selected square's row number and a column number greater than or equal to this selected square's column number. How many squares can we choose at most?
Let $ABC$ be a triangle with incenter $I$. Two distinct points $P$ and $Q$ are chosen on the circumcircle of $ABC$ such that
\[ \angle API = \angle AQI = 45^\circ. \]
Lines $PQ$ and $BC$ meet at $S$. Let $H$ denote the foot of the altitude from $A$ to $BC$. Prove that $\angle AHI = \angle ISH$.
[i]Matsvei Zorka[/i]
Let be a matrix $ A\in\mathcal{M}_2(\mathbb{R}) $ having the property that the numbers $ \det (A+X) ,\det (A^2+X^2) ,\det (A^3+X^3) $ are (in this order) in geometric progression, for any matrix $ X\in\mathcal{M}_2(\mathbb{R}) . $
Prove that $ A=0. $
[i]Marius Ghergu[/i]
Let $A$ be a family of subsets of $\{1,2,\ldots,n\}$ such that no member of $A$ is contained in another. Sperner’s Theorem states that $|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}$. Find all the families for which the equality holds.
Find roots of the equation
$$1 -\frac{x}{1}+ \frac{x(x - 1)}{2!} -... +\frac{ (-1)^nx(x-1)...(x - n + 1)}{n!}= 0$$
A ladder style tournament is held with $2016$ participants. The players begin seeded $1,2,\cdots 2016$. Each round, the lowest remaining seeded player plays the second lowest remaining seeded player, and the loser of the game gets eliminated from the tournament. After $2015$ rounds, one player remains who wins the tournament. If each player has probability of $\tfrac{1}{2}$ to win any game, then the probability that the winner of the tournament began with an even seed can be expressed has $\tfrac{p}{q}$ for coprime positive integers $p$ and $q$. Find the remainder when $p$ is divided by $1000$.
[i]Proposed by Nathan Ramesh
$30$ people sit around a table, some of which are Yale students. Each person is asked if the person to their right is a Yale student. Yale students will always answer correctly, but non-Yale students will answer randomly. Find the smallest possible number of Yale students such that, after hearing everyone’s answers and knowing the number of Yale students, it is possible to identify for certain at least one Yale student.
Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives \begin{eqnarray*} &\ & \binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1+\binom{1000}{2}(0.2)^2+\cdots+\binom{1000}{1000}(0.2)^{1000}\\ &\ & = A_0 + A_1 + A_2 + \cdots + A_{1000}, \end{eqnarray*} where $A_k = \binom{1000}{k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For which $k$ is $A_k$ the largest?
In a triangle $ABC$, the external bisector of $\angle BAC$ intersects the ray $BC$ at $D$. The feet of the perpendiculars from $B$ and $C$ to line $AD$
are $E$ and $F$, respectively and the foot of the perpendicular from $D$ to $AC$ is $G$. Show that $\angle DGE + \angle DGF = 180^{\circ}$.
Let $M$ be a finite set and $P=\{ M_1,M_2,\ldots ,M_l\}$ a partition of $M$ (i.e., $\bigcup_{i=1}^k M_i, M_i\not=\emptyset, M_i\cap M_j =\emptyset$ for all $i,j\in\{1,2, \ldots ,k\} ,i\not= j)$. We define the following elementary operation on $P$:
Choose $i,j\in\{1,2,\ldots ,k\}$, such that $i=j$ and $M_i$ has a elements and $M_j$ has $b$ elements such that $a\ge b$. Then take $b$ elements from $M_i$ and place them into $M_j$, i.e., $M_j$ becomes the union of itself and a $b$-element subset of $M_i$, while the same subset is subtracted from $M_i$ (if $a=b$, $M_i$ is thus removed from the partition).
Let a finite set $M$ be given. Prove that the property “for every partition $P$ of $M$ there exists a sequence $P=P_1,P_2,\ldots ,P_r$ such that $P_{i+1}$ is obtained from $P_i$ by an elementary operation and $P_r=\{M\}$” is equivalent to “the number of elements of $M$ is a power of $2$.”
Let $AH_1$ and $BH_2$ be the altitudes of triangle $ABC$. Let the tangent to the circumcircle of $ABC$ at $A$ meet $BC$ at point $S_1$, and the tangent at $B$ meet $AC$ at point $S_2$. Let $T_1$ and $T_2$ be the midpoints of $AS_1$ and $BS_2$ respectively. Prove that $T_1T_2$, $AB$ and $H_1H_2$ concur.
Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square.
[i]David Yang, Alex Zhu.[/i]
Let $n$ be a natural number. Suppose $A$ and $B$ are two sets, each containing $n$ points in the plane, such that no three points of a set are collinear. Let $T(A)$ be the number of broken lines, each containing $n-1$ segments, and such that it doesn't intersect itself and its vertices are points of $A$. Define $T(B)$ similarly. If the points of $B$ are vertices of a convex $n$-gon (are in [i]convex position[/i]), but the points of $A$ are not, prove that $T(B)<T(A)$.
[i]Proposed by Ali Khezeli[/i]
Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely.
The side of the square $ABCD$ has a length equal to $1$. On the sides $(BC)$ ¸and $(CD)$ take respectively the arbitrary points $M$ and $N$ so that the perimeter of the triangle $MCN$ is equal to $2$.
a) Determine the measure of the angle $\angle MAN$.
b) If the point $P$ is the foot of the perpendicular taken from point $A$ to the line $MN$, determine the locus of the points $P$.