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Point that is orthocenter, incenter, and centroid
EmersonSoriano   0
16 minutes ago
Source: 2017 Peru Southern Cone TST P9
Let $BXC$ be a triangle and $A_1, A_2, A_3$ points in the same plane such that $X$ is the orthocenter of triangle $A_1BC$, $X$ is the incenter of triangle $A_2BC$, and $X$ is the centroid of triangle $A_3BC$. If line $A_1A_3$ is parallel to $BC$, prove that $A_2$ is the midpoint of segment $A_1A_3$.
0 replies
EmersonSoriano
16 minutes ago
0 replies
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Least common multiple from $n+1$ to $n+2016$.
EmersonSoriano   0
36 minutes ago
Source: 2017 Peru Southern Cone TST P7
For each positive integer $n$, define
$$P_n = (n+1)(n+2)(n+3)\dots(n+2016)$$and
$$Q_n = \text{lcm}(n+1, n+2, n+3, \dots, n+2016),$$meaning $Q_n$ is the least common multiple of the numbers $n+1, n+2, n+3, \dots, n+2016$. Determine whether or not there exists a constant $C$ such that
$$ \frac{P_n}{Q_n} < C, $$for every positive integer $n$.
0 replies
EmersonSoriano
36 minutes ago
0 replies
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Moving tokens from the leftmost end to the rightmost end.
EmersonSoriano   0
40 minutes ago
Source: 2017 Peru Southern Cone TST P6
Let $n$ and $\ell$ be positive integers with $\ell > 7$. There are $n$ tokens placed initially in the leftmost cell of a horizontal row consisting of $\ell$ cells. A move consists of shifting any token $1$, $2$, $3$, $4$, $5$, or $6$ positions to the right. Andrés and Beto alternate turns, starting with Andrés. The winner is the player who moves a token into the rightmost cell. Determine who has a winning strategy in terms of $n$ and $\ell$.
0 replies
EmersonSoriano
40 minutes ago
0 replies
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A point on the midline of BC.
EmersonSoriano   0
an hour ago
Source: 2017 Peru Southern Cone TST P5
Let $ABC$ be an acute triangle with circumcenter $O$. Draw altitude $BQ$, with $Q$ on side $AC$. The parallel line to $OC$ passing through $Q$ intersects line $BO$ at point $X$. Prove that point $X$ and the midpoints of sides $AB$ and $AC$ are collinear.
0 replies
EmersonSoriano
an hour ago
0 replies
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Mundane Primes
bryanguo   10
N an hour ago by awesomeming327.
Source: 2023 HMIC P2
A prime number $p$ is mundane if there exist positive integers $a$ and $b$ less than $\tfrac{p}{2}$ such that $\tfrac{ab-1}{p}$ is a positive integer. Find, with proof, all prime numbers that are not mundane.
10 replies
bryanguo
Apr 25, 2023
awesomeming327.
an hour ago
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Set of 2n real numbers divided into two groups
EmersonSoriano   0
an hour ago
Source: 2017 Peru Southern Cone TST P4
Let $n$ be a fixed positive integer. Find the greatest real constant $C_n$ that has the following property: Any $2n$ real numbers, not necessarily distinct, that lie in the interval $[100,101]$, can be partitioned into two groups with sums $S_1$ and $S_2$ such that
$$1 \ge \frac{S_2}{S_1} \ge C_n.$$
0 replies
EmersonSoriano
an hour ago
0 replies
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intersting system with integer positive numbers
teomihai   2
N an hour ago by teomihai
Let next positiv integer numbers: $a ,b ,c, d $ .
If $a^4=b^3 $ , $c^5=d^6 $ and $ c-a=37 $ .
Find $ b-d$.
2 replies
teomihai
2 hours ago
teomihai
an hour ago
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Non-overlapping L-tromino tiles
EmersonSoriano   0
an hour ago
Source: 2017 Peru Southern Cone TST P3
An $L$-tromino is a figure made up of three squares, obtained by removing one square from a $2\times 2$ board.

We have a $7\times 7$ board consisting of $112$ unit segments. A configuration of several $L$-trominoes is optimal if the $L$-trominoes do not overlap, each one covers exactly three squares of the board, and moreover, no unit segment of the board belongs to two $L$-trominoes. Below is an optimal configuration of 5 $L$-trominoes:


[center]IMAGE[/center]

Determine the largest possible value of $n$ for which there exists an optimal configuration of L-trominoes on the $7\times 7$ board.
0 replies
EmersonSoriano
an hour ago
0 replies
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Hand shaken
mathservant   0
an hour ago
A total of 3300 handshakes were made at a party attended by 600 people. It was observed
that the total number of handshakes among any 300 people at the party is at least N. Find
the largest possible value for N.
0 replies
mathservant
an hour ago
0 replies
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¿10^n-1 is a divisor of 11^n-1?
EmersonSoriano   0
2 hours ago
Source: 2017 Peru Southern Cone TST P2
Determine if there exists a positive integer $n$ such that $10^n - 1$ is a divisor of $11^n - 1$.
0 replies
EmersonSoriano
2 hours ago
0 replies
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Diagonal of a pentagon that divides it into a triangle and a cyclic quadrilatera
EmersonSoriano   0
2 hours ago
Source: 2017 Peru Southern Cone TST P1
We say that a diagonal of a convex pentagon is good if it divides the pentagon into a triangle and a circumscribable quadrilateral. What is the maximum number of good diagonals that a convex pentagon can have?

Clarification: A polygon is circumscribable if there is a circle tangent to each of its sides.
0 replies
EmersonSoriano
2 hours ago
0 replies
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Gheorghe Țițeica 2025 Grade 9 P3
AndreiVila   1
N Mar 29, 2025 by AlgebraKing
Source: Gheorghe Țițeica 2025
Consider the plane vectors $\overrightarrow{OA_1},\overrightarrow{OA_2},\dots ,\overrightarrow{OA_n}$ with $n\geq 3$. Suppose that the inequality $$\big|\overrightarrow{OA_1}+\overrightarrow{OA_2}+\dots +\overrightarrow{OA_n}\big|\geq \big|\pm\overrightarrow{OA_1}\pm\overrightarrow{OA_2}\pm\dots \pm\overrightarrow{OA_n}\big|$$takes place for all choiches of the $\pm$ signs. Show that there exists a line $\ell$ through $O$ such that all points $A_1,A_2,\dots ,A_n$ are all on one side of $\ell$.

Cristi Săvescu
1 reply
AndreiVila
Mar 28, 2025
AlgebraKing
Mar 29, 2025
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Gheorghe Țițeica 2025 Grade 9 P3
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vector
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Source: Gheorghe Țițeica 2025
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AndreiVila
208 posts
AndreiVila
#1 h  Mar 28, 2025, 9:16 PM • 1 Y
Y by MS_asdfgzxcvb
Consider the plane vectors $\overrightarrow{OA_1},\overrightarrow{OA_2},\dots ,\overrightarrow{OA_n}$ with $n\geq 3$. Suppose that the inequality $$\big|\overrightarrow{OA_1}+\overrightarrow{OA_2}+\dots +\overrightarrow{OA_n}\big|\geq \big|\pm\overrightarrow{OA_1}\pm\overrightarrow{OA_2}\pm\dots \pm\overrightarrow{OA_n}\big|$$takes place for all choiches of the $\pm$ signs. Show that there exists a line $\ell$ through $O$ such that all points $A_1,A_2,\dots ,A_n$ are all on one side of $\ell$.

Cristi Săvescu
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AlgebraKing
4 posts
AlgebraKing
#2 h  Mar 29, 2025, 2:49 PM
Y by
Denote $\sum_{i=1}^n \overrightarrow{OA_i}=\overrightarrow{OX}$. If we let $d$ be a line perpendicular to $OX$ at $O$, the claim is that all the vectors lie on the same side of $d$ as $X$. Simply notice that by the given condition the following holds for all $i$. \[\big|\overrightarrow{OX}\big|\ge\big|\overrightarrow{OX}-2\overrightarrow{OA_i}\big|\]Therefore, if we suppose that $A_i$ is not on the same side of $d$ as $X$ and let $B_i$ be the point such that $\overrightarrow{OB_i}=-2\overrightarrow{OA_i}$ we get \[\angle XOB_i>90^\circ\implies XB_i>OX\]Which is a contradiction to the above.
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