Found problems: 85335
Let $V_1,V_2,V_3,V_4$ be four dimensional linear subspaces in $\mathbb{R}^8$ such that the intersection of any two contains only the zero vector. Prove that there exists a linear four dimensional subspace $W$ in $\mathbb{R}^8$ such that all four vector spaces $W\cap V_i$ are two dimensional.
Let $ ABCDE$ be a convex pentagon such that
\[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE.
\]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$.
[i]Proposed by Zuming Feng, USA[/i]
Let $p$ be a prime. For which $k$ can the set $\{1, 2, \dots , k\}$ be partitioned into $p$ subsets with equal sums of elements ?
Let $n \geq 3$ be an integer. A labelling of the $n$ vertices, the $n$ sides and the interior of a regular $n$-gon by $2n + 1$ distinct integers is called [i]memorable[/i] if the following conditions hold:
(a) Each side has a label that is the arithmetic mean of the labels of its endpoints.
(b) The interior of the $n$-gon has a label that is the arithmetic mean of the labels of all the vertices.
Determine all integers $n \geq 3$ for which there exists a memorable labelling of a regular $n$-gon consisting of $2n + 1$ consecutive integers.
Let $a,b,c$ be three positive real numbers such that $abc=1$. Prove that: \[ 1+\frac{3}{a+b+c}\ge{\frac{6}{ab+bc+ca}} . \]
Let $k$ be a positive integer. Prove that one can partition the set $\{ 0,1,2,3, \cdots ,2^{k+1}-1 \}$ into two disdinct subsets $\{ x_1,x_2, \cdots, x_{2k} \}$ and $\{ y_1, y_2, \cdots, y_{2k} \}$ such that $\sum_{i=1}^{2^k} x_i^m =\sum_{i=1}^{2^k} y_i^m$ for all $m \in \{ 1,2, \cdots, k \}$.
Let $ABC$ be a triangle inscribed in a circle of radius $O$. The angle bisectors $AD,BE,CF$ are concurrent at $I$. The points $M,N, P$ are respectively on $EF, FD$, and $DE$ such that $IM, IN, IP$ are perpendicular to $BC,CA,AB$ respectively. Prove that the three lines $AM,BN, CP$ are concurrent at a point on $OI$.
Nguyễn Minh Hà
A [i]massless[/i] spring with spring constant $k$ is vertically mounted so that bottom end is firmly attached to the ground, and the top end free. A ball with mass $m$ falls vertically down on the top end of the spring, becoming attached, so that the ball oscillates vertically on the spring. What equation describes the acceleration a of the ball when it is at a height $y$ above the [i]original[/i] position of the top end of the spring? Let down be negative, and neglect air resistance; $g$ is the magnitude of the acceleration of free fall.
(a) $a=mv^2/y+g$
(b) $a=mv^2/k-g$
(c) $a=(k/m)y-g$
(d) $a=-(k/m)y+g$
(e) $a=-(k/m)y-g$
Let $ABC$ be a triangle, satisfying $2AC=AB+BC$. If $O$ and $I$ are its circumcenter and incenter, show that $\angle OIB=90^{\circ}$.
Positive real numbers $x, y, z$ satisfy $\sqrt x + \sqrt y + \sqrt z = 1$. Prove that $$\frac{x^4+y^2z^2}{x^{\frac 52}(y+z)} + \frac{y^4+z^2x^2}{y^{\frac 52}(z+x)} + \frac{z^4+y^2x^2}{z^{\frac 52}(y+x)} \geq 1.$$
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 15)[/i]
Positive integers $a, b, c$ have greatest common divisor $1$. The triplet $(a, b, c)$ may be altered into another triplet such that in each step one of the numbers in the actual triplet is increased or decreased by an integer multiple of another element of the triplet. Prove that the triplet $(1,0,0)$ can be obtained in at most $5$ steps.
Two triangles are said to be [i]twins [/i] if one of them is an image of the other one under a parallel projection.
Prove that two triangles are twins if and only if either at least a side of one of them equals a side of another or both the triangles have equal segments that connect the corresponding vertices with some points on the opposite sides which divide these sides in the same ratio.
(E. Barabanov)
Prove that if the odd prime $p$ divides $a^{b}-1$, where $a$ and $b$ are positive integers, then $p$ appears to the same power in the prime factorization of $b(a^{d}-1)$, where $d=\gcd(b,p-1)$.
For any positive integer $a$, define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$. Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$.
Put $1,2,....,2018$ (2018 numbers) in a row randomly and call this number $A$. Find the remainder of $A$ divided by $3$.
[b]O[/b]n a blackboard a stranger writes the values of $s_7(n)^2$ for $n=0,1,...,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.
Prove that among any $51$ vertices of the $101$-regular polygon there are three that are the vertices of an isosceles triangle.
Let $a > b > c$ be sides of a triangle and $h_a,h_b,h_c$ be the corresponding altitudes.
Prove that $a+h_a > b+h_b > c+h_c$.
In $\triangle ABC$, $A,B,C$ are arithmetic sequence, and $c-a$ is equal to height on side $BC$, then $\sin\frac{C-A}{2}=$________.
Let $ P_1$ be a regular $ r$-gon and $ P_2$ be a regular $ s$-gon $ (r\geq s\geq 3)$ such that each interior angle of $ P_1$ is $ \frac {59}{58}$ as large as each interior angle of $ P_2$. What's the largest possible value of $ s$?
In the figure, point $P, Q,R,S$ lies on the side of the rectangle $ABCD$.
[img]https://1.bp.blogspot.com/-Ff9rMibTuHA/X9PRPbGVy-I/AAAAAAAAMzA/2ytG0aqe-k0fPL3hbSp_zHrMYAfU-1Y_ACLcBGAsYHQ/s426/2020%2BIndonedia%2BMO%2BProvince%2BP2%2Bq1.png[/img]
If it is known that the area of the small square is $1$ unit, determine the area of the rectangle $ABCD$.
One of two wizards, named Steven, was told the sum of three positive integers and the other, named Peter, their product. “If I knew”, said Steven, “that your number is greater than mine, I could find the integers”. “But my number is less than yours,” replied Peter, “and the integers are $X$, $Y$ and $Z$”. Find these numbers.
(L Borisov)
Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.
Let $n$ be a positive integer, and let $S_1,S_2,…,S_n$ be a collection of finite non-empty sets such that $$\sum_{1\leq i<j\leq n}{\frac{|S_i \cap S_j|}{|S_i||S_j|}} <1.$$
Prove that there exist pairwise distinct elements $x_1,x_2,…,x_n$ such that $x_i$ is a member of $S_i$ for each index $i$.
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T = aL + bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimate it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet?
$\textbf{(A) } 240 \qquad \textbf{(B) } 246 \qquad \textbf{(C) } 252 \qquad \textbf{(D) } 258 \qquad \textbf{(E) } 264$