Found problems: 85335
Let $m$ and $n$ be integers greater than $1$ and $a_1 ,a_2 ,\ldots, a_{m+1}$ be real numbers. Prove that there exist real $n\times n$ matrices $A_1 ,A_2,\ldots, A_m$ such that
(i) $\det(A_j) =a_j$ for $j=1,2,\ldots,m$ and
(ii) $\det(A_1 +A_2 +\ldots+A_m)=a_{m+1}.$
Are there six different positive odd numbers $a,b,c,d,e,f$ such that
\[ 1/a + 1/b + 1/c + 1/d + 1/e + 1/f = 1?\]
Let $\, k_1 < k_2 < k_3 < \cdots \,$ be positive integers, no two consecutive, and let $\, s_m = k_1 + k_2 + \cdots + k_m \,$ for $\, m = 1,2,3, \ldots \; \;$. Prove that, for each positive integer $\, n, \,$ the interval $\, [s_n, s_{n+1}) \,$ contains at least one perfect square.
A small hard solid sphere of mass m and negligible radius is connected to a thin rod of length L and
mass 2m. A second small hard solid sphere, of mass M and negligible radius, is fired perpendicularly
at the rod at a distance h above the sphere attached to the rod, and sticks to it.
A: h = 0
B: h = L/3
C: h = L/2
D: h = L
E: Any L will work
Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.
Find all primes $p,q$ and even $n>2$ such that $p^n+p^{n-1}+...+1=q^2+q+1$.
Let $p$ be a prime number and let $M$ be a convex polygon. Suppose that there are precisely $p$ ways to tile $m$ with equilateral triangles with side $1$ and squares with side $1$. Show there is some side of $M$ of length $p-1$.
Three cevians are drawn in a triangle that do not intersect at one point. In this case, $4$ triangles and $3$ quadrangles were formed. Find the sum of the areas of the quadrilaterals if the area of each of the four triangles is $8$.
In rectangle $ABCD$, $AB=6$, $AD=30$, and $G$ is the midpoint of $\overline{AD}$. Segment $AB$ is extended $2$ units beyond $B$ to point $E$, and $F$ is the intersection of $\overline{ED}$ and $\overline{BC}$. What is the area of $BFDG$?
$ \textbf{(A)}\ \frac{133}{2}\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ \frac{135}{2}\qquad\textbf{(D)}\ 68\qquad\textbf{(E)}\ \frac{137}{2}$
Call an $n$-tuple $(a_1, . . . , a_n)$ [i]occasionally periodic [/i] if there exist a nonnegative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+p+j}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, . . . , a_n)$ with elements from set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in \{1, 2, . . . , k\}$.
The altitudes $AA_1, BB_1$ and $CC_1$ were drawn in the triangle $ABC$. Point $K$ is a projection of point $B$ on $A_1C_1$. Prove that the symmmedian $\vartriangle ABC$ from the vertex $B$ divides the segment $B_1K$ in half.
(Anton Trygub)
Given two positive integers $a$ and $b$, an [i]legal move[/i] consists in choosing a proper divisor of one of them and adding it to $a$ or adding it to $b$. Two players, Agustin and Ian, take turns making an legal move; Agustin plays first. Whoever gets a number greater than or equal to $2015$ wins the game.
[list=a]
[*]Determine which of the players has a winning strategy if $a=3, b=5$.
[*]Determine which of the players has a winning strategy if $a=6, b=7$.
[/list]
For positive numbers $a, b, c$ holds $(a + c) (b^2 + a c) = 4a$.
Determine the maximum value of $b + c$ and find all triplets of numbers $(a, b, c)$ for which expression takes this value
In a convex quadrilateral $ABCD$, both $\angle ADB=2\angle ACB$ and $\angle BDC=2\angle BAC$. Prove that $AD=CD$.
Suppose that $P$ is the polynomial of least degree with integer coefficients such that $$P(\sqrt{7} + \sqrt{5}) = 2(\sqrt{7} - \sqrt{5})$$Find $P(2)$.
Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set
$A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$
$B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$
Prove that $2|A|\geq |B|$.
The area of a circle inscribed in a regular hexagon is $ 100\pi$. The area of hexagon is:
$ \textbf{(A)}\ 600\qquad
\textbf{(B)}\ 300\qquad
\textbf{(C)}\ 200\sqrt{2}\qquad
\textbf{(D)}\ 200\sqrt{3}\qquad
\textbf{(E)}\ 120\sqrt{5}$
Let $E$ be the midpoint of the side $AB$ of the parallelogram $ABCD$. Let $F$ be the projection of $B$ on $AC$. Prove that the triangle $ABF$ is isosceles
Is it possible for the projection of the set of points $(x, y, z)$ with $0 \leq x, y, z \leq 1$ onto some two-dimensional plane to be a simple convex pentagon?
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?
$\textbf{(A) } 9
\qquad\textbf{(B) } 11
\qquad\textbf{(C) } 12
\qquad\textbf{(D) } 13
\qquad\textbf{(E) } 15
$
Given a prime number $p$ and let $\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}$ be $n$ distinct vectors of length $p$ with integer coordinates in an $\mathbb{R}^3$ Cartesian coordinate system. Suppose that for any $1\leqslant j<k\leqslant n$, there exists an integer $0<\ell <p$ such that all three coordinates of $\overline{v_j} -\ell \cdot \overline{v_k} $ is divisible by $p$. Prove that $n\leqslant 6$.
Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.
Solve logarithmical equation $x^{\log _{3} {x-1}} + 2(x-1)^{\log _{3} {x}}=3x^2$
A set of $1985$ points is distributed around the circumference of a circle and each of the points is marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with $-1$ is less than $662$, there must be at least one good point.
An ant is moving on the cooridnate plane, starting form point $(0,-1)$ along a straight line until it reaches the $x$- axis at point $(x,0)$ where $x$ is a real number. After it turns $90^o$ to the left and moves again along a straight line until it reaches the $y$-axis . Then it again turns left and moves along a straight line until it reaches the $x$-axis, where it once more turns left by $90^o$ and moves along a straight line until it finally reached the $y$-axis.
Can both the length of the ant's journey and distance between it's initial and final point be:
(a) rational numbers ?
(b) integers?
Justify your answers
PS. Collected [url=https://artofproblemsolving.com/community/c949609_2019_nigerian_senior_mo_round_4]here[/url]