Found problems: 623
Prove that on the coordinate plane it is impossible to draw a closed broken line such that
[i](i)[/i] the coordinates of each vertex are rational;
[i](ii)[/i] the length each of its edges is 1;
[i](iii)[/i] the line has an odd number of vertices.
There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?
Let $K, L, M$ be the feet of the altitudes drawn from the vertices $A, B, C$ of triangle $ABC$, respectively. Prove that $\overrightarrow{AK} + \overrightarrow{BL} + \overrightarrow{CM} = \overrightarrow{O}$ if and only if $ABC$ is equilateral.
Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.
Let $A$ be a $n\times n$ matrix with complex elements and let $A^\star$ be the classical adjoint of $A$. Prove that if there exists a positive integer $m$ such that $(A^\star)^m = 0_n$ then $(A^\star)^2 = 0_n$.
[i]Marian Ionescu, Pitesti[/i]
Distinct points $A, B , C, D, E$ are given in this order on a semicircle with radius $1$. Prove that
\[AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.\]
Introduce a standard scalar product in $\mathbb{R}^4.$ Let $V$ be a partial vector space in $\mathbb{R}^4$ produced by $\left(
\begin{array}{c}
1 \\
-1 \\
-1 \\
1
\end{array}
\right),\left(
\begin{array}{c}
1 \\-1 \\
1 \\
-1
\end{array}
\right).$
Find a pair of base of orthogonal complement $W$ for $V$ in $\mathbb{R}^4.$
Consider the regular $1987$-gon $A_1A_2 . . . A_{1987}$ with center $O$. Show that the sum of vectors belonging to any proper subset of $M = \{OA_j | j = 1, 2, . . . , 1987\}$ is nonzero.
Let $\Gamma$ and $\Gamma'$ be two congruent circles centered at $O$ and $O'$, respectively, and let $A$ be one of their two points of intersection. $B$ is a point on $\Gamma$, $C$ is the second point of intersection of $AB$ and $\Gamma'$, and $D$ is a point on $\Gamma'$ such that $OBDO'$ is a parallelogram. Show that the length of $CD$ does not depend on the position of $B$.
A set $M$ consists of $n$ elements. Find the greatest $k$ for which there is a collection of $k$ subsets of $M$ such that for any subsets $A_{1},...,A_{j}$ from the collection, there is an element belonging to an odd number of them
Participants to an olympiad worked on $n$ problems. Each problem was worth a [color=#FF0000]positive [/color]integer number of points, determined by the jury. A contestant gets $0$ points for a wrong answer, and all points for a correct answer to a problem. It turned out after the olympiad that the jury could impose worths of the problems, so as to obtain any (strict) final ranking of the contestants. Find the greatest possible number of contestants.
Students $ A $ and $ B $ play according to the following rules: student $ A $ selects a vector $ \overrightarrow{a_1} $ of length 1 in the plane, then student $ B $ gives the number $ s_1 $, equal to $ 1 $ or $ - $1; then the student $ A $ chooses a vector $ \overrightarrow{a_1} $ of length $ 1 $, and in turn the student $ B $ gives a number $ s_2 $ equal to $ 1 $ or $ -1 $ etc. $ B $ wins if for a certain $ n $ vector $ \sum_{j=1}^n \varepsilon_j \overrightarrow{a_j} $ has a length greater than the number $ R $ determined before the start of the game. Prove that student $B$ can achieve a win in no more than $R^2 + 1$ steps regardless of partner $A$'s actions.
Given a set of $n>2$ planar vectors. A vector from this set is called [i]long[/i], if its length is not less than the length of the sum of other vectors in this set. Prove that if each vector is long, then the sum of all vectors equals to zero.
[i]N. Agakhanov[/i]
Given that $ABC$ is a triangle, points $A_i, B_i, C_i \hspace{0.15cm} (i \in \{1,2,3\})$ and $O_A, O_B, O_C$ satisfy the following criteria:
a) $ABB_1A_2, BCC_1B_2, CAA_1C_2$ are rectangles not containing any interior points of the triangle $ABC$,
b) $\displaystyle \frac{AB}{BB_1} = \frac{BC}{CC_1} = \frac{CA}{AA_1}$,
c) $AA_1A_3A_2, BB_1B_3B_2, CC_1C_3C_2$ are parallelograms, and
d) $O_A$ is the centroid of rectangle $BCC_1B_2$, $O_B$ is the centroid of rectangle $CAA_1C_2$, and $O_C$ is the centroid of rectangle $ABB_1A_2$.
Prove that $A_3O_A, B_3O_B,$ and $C_3O_C$ concur at a point.
[i]Proposed by Farras Mohammad Hibban Faddila[/i]
Suppose $V= \mathbb{Z}_2^n$ and for a vector $x=(x_1,..x_n)$ in $V$ and permutation $\sigma$.We have $x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})$
Suppose $ n=4k+2,4k+3$ and $f:V \to V$ is injective and if $x$ and $y$ differ in more than $n/2$ places then $f(x)$ and $f(y)$ differ in more than $n/2$ places.
Prove there exist permutaion $\sigma$ and vector $v$ that $f(x)=x_{\sigma}+v$
Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.
Is there an infinite sequence of real numbers $a_1,a_2,a_3,\dots$ such that
\[a_1^m+a_2^m+a_3^m+\cdots=m\]
for every positive integer $m?$
Suppose that $W,W_1$ and $W_2$ are subspaces of a vector space $V$ such that $V=W_1\oplus W_2$. Under what conditions we have
$W=(W\cap W_1)\oplus(W\cap W_2)$?
Let $S$ be a set of $9$ distinct integers all of whose prime factors are at most $3.$ Prove that $S$ contains $3$ distinct integers such that their product is a perfect cube.
Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.
Let $ABC$ be a triangle. Let $A_1$ and $A_2$ be points on side $BC, B_1$ and $B_2$ be points on side $CA$ and $C_1$ and $C_2$ be points on side $AB$ such that $A_1A_2B_1B_2C_1C_2$ is a convex hexagon and that $B,A_1,A_2$ and $C$ are located in that order on side $BC$.
We say that triangles $AB_2C_1, BA_1C_2$ and $CA_2B_1$ are glueable if there exists a triangle $PQR$ and there exist $X,Y$ and $Z$ on sides $QR, RP$ and $PQ$ respectively, such that triangle $AB_2C_1$ is congruent in that order to triangle $PYZ$, triangle $BA_1C_2$ is congruent in that order to triangle $QXZ$ and triangle $CA_2B_1$ is congruent in that order to triangle $RXY$. Prove that triangles $AB_2C_1, BA_1C_2$ and $CA_2B_1$ are glueable if and only if the centroids of triangles $A_1B_1C_1$ and $A_2B_2C_2$ coincide.
Inside triangle $ABC$ consider random point $O$. Prove that: $$E_A \overrightarrow{OA}+E_B \overrightarrow{OB}+E_C\overrightarrow{OC}=\overrightarrow{O}$$
where $E_A,E_B,E_C$ the areas of triangle $BOC, COB, AOB$ respectively
Let $n$ be a positive integer. For any positive integer $k$, let $0_k=diag\{\underbrace{0, ...,0}_{k}\}$ be a $k \times k$ zero matrix. Let $Y=\begin{pmatrix}
0_n & A \\
A^t & 0_{n+1}
\end{pmatrix}$ be a $(2n+1) \times (2n+1)$ where $A=(x_{i, j})_{1\leq i \leq n, 1\leq j \leq n+1}$ is a $n \times (n+1)$ real matrix. Let $A^T$ be transpose matrix of $A$ i.e. $(n+1) \times n$ matrix, the element of $(j, i)$ is $x_{i, j}$.
(a) Let complex number $\lambda$ be an eigenvalue of $k \times k$ matrix $X$. If there exists nonzero column vectors $v=(x_1, ..., x_k)^t$ such that $Xv=\lambda v$. Prove that 0 is the eigenvalue of $Y$ and the other eigenvalues of $Y$ can be expressed as a form of $\pm \sqrt{\lambda}$ where nonnegative real number $\lambda$ is the eigenvalue of $AA^t$.
(b) Let $n=3$ and $a_1$, $a_2$, $a_3$, $a_4$ are $4$ distinct positive real numbers. Let $a=\sqrt[]{\sum_{1\leq i \leq 4}^{}a^{2}_{i}}$ and $x_{i,j}=a_i\delta_{i,j}+a_j\delta_{4,j}-\frac{1}{a^2}(a^2_{i}+a^2_{4})a_j$ where $1\leq i \leq 3, 1\leq j \leq 4$, $\delta_{i, j}=
\begin{cases}
1 \text{ if } i=j\\
0 \text{ if } i\neq j\\
\end{cases}\,$. Prove that $Y$ has 7 distinct eigenvalue.
Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.
Suppose $a_1, \dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties:
(a) For $i=1, \dots, n$, $a_i \in S$.
(b) For $i,j = 1, \dots, n$ (not necessarily distinct), $a_i - a_j \in S$.
(c) For any integers $x,y \in S$, if $x+y \in S$, then $x-y \in S$.
Prove that $S$ must be equal to the set of all integers.