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Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

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Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

3 M G

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JBMO Combinatorics vibes

Sadigly 1

N 44 minutes ago by Royal_mhyasd

Source: Azerbaijan Senior NMO 2018

Numbers $1,2,3...,100$ are written on a board. $A$ and $B$ plays the following game: They take turns choosing a number from the board and deleting them. $A$ starts first. They sum all the deleted numbers. If after a player's turn (after he deletes a number on the board) the sum of the deleted numbers can't be expressed as difference of two perfect squares,then he loses, if not, then the game continues as usual. Which player got a winning strategy?

1 reply

Sadigly

Yesterday at 9:53 PM

Royal_mhyasd

44 minutes ago

line JK of intersection points of 2 lines passes through the midpoint of BC

parmenides51 3

N an hour ago by cursed_tangent1434

Source: Rioplatense Olympiad 2018 level 3 p4

Let $ABC$ be an acute triangle with $AC> AB$ . be $\Gamma$ the circumcircle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smallest arc $BC$ of this circle. Let $E$ and $F$ points of the segments $AB$ and $AC$ respectively such that $AE = AF$ . Let $P \neq A$ be the second intersection point of the circumcircle circumscribed to $AEF$ with $\Gamma$ . Let $G$ and $H$ be the intersections of lines $PE$ and $PF$ with $\Gamma$ other than $P$ , respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines $AB$ and $AC$ respectively. Show that the $JK$ line passes through the midpoint of $BC$

3 replies

1 viewing

parmenides51

Dec 11, 2018

cursed_tangent1434

an hour ago

Six variables

Nguyenhuyen_AG 2

N an hour ago by arqady

Let $a,\,b,\,c,\,x,\,y,\,z$ be six positive real numbers. Prove that
$\frac{a}{b+c} \cdot \frac{y+z}{x} + \frac{b}{c+a} \cdot \frac{z+x}{y} + \frac{c}{a+b} \cdot \frac{x+y}{z} \geqslant 2+\sqrt{\frac{8abc}{(a+b)(b+c)(c+a)}}.$

2 replies

Nguyenhuyen_AG

Yesterday at 5:09 AM

arqady

an hour ago

Brilliant guessing game on triples

Assassino9931 2

N an hour ago by Mirjalol

Source: Al-Khwarizmi Junior International Olympiad 2025 P8

There are $100$ cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from $1$ to $100$ . In a move, Madina is allowed to choose any $3$ cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?

Shubin Yakov, Russia

2 replies

Assassino9931

Saturday at 9:46 AM

Mirjalol

an hour ago

ISI UGB 2025 P5

SomeonecoolLovesMaths 4

N an hour ago by Shiny_zubat

Source: ISI UGB 2025 P5

Let $a,b,c$ be nonzero real numbers such that $a+b+c \neq 0$ . Assume that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c}$ Show that for any odd integer $k$ , $\frac{1}{a^k} + \frac{1}{b^k} + \frac{1}{c^k} = \frac{1}{a^k+b^k+c^k}.$

4 replies

SomeonecoolLovesMaths

Yesterday at 11:15 AM

Shiny_zubat

an hour ago

ISI UGB 2025 P2

SomeonecoolLovesMaths 6

N an hour ago by quasar_lord

Source: ISI UGB 2025 P2

If the interior angles of a triangle $ABC$ satisfy the equality, $\sin ^2 A + \sin ^2 B + \sin^2 C = 2 \left( \cos ^2 A + \cos ^2 B + \cos ^2 C \right),$ prove that the triangle must have a right angle.

6 replies

SomeonecoolLovesMaths

Yesterday at 11:16 AM

quasar_lord

an hour ago

ISI UGB 2025 P6

SomeonecoolLovesMaths 3

N an hour ago by Shiny_zubat

Source: ISI UGB 2025 P6

Let $\mathbb{N}$ denote the set of natural numbers, and let $\left( a_i, b_i \right)$ , $1 \leq i \leq 9$ , be nine distinct tuples in $\mathbb{N} \times \mathbb{N}$ . Show that there are three distinct elements in the set $\{ 2^{a_i} 3^{b_i} \colon 1 \leq i \leq 9 \}$ whose product is a perfect cube.

3 replies

SomeonecoolLovesMaths

Yesterday at 11:18 AM

Shiny_zubat

an hour ago

Shortest number theory you might've seen in your life

AlperenINAN 5

N an hour ago by Royal_mhyasd

Source: Turkey JBMO TST 2025 P4

Let $p$ and $q$ be prime numbers. Prove that if $pq(p+1)(q+1)$ is a perfect square, then $pq + 1$ is also a perfect square.

5 replies

AlperenINAN

Yesterday at 7:51 PM

Royal_mhyasd

an hour ago

d+2 pts in R^d can partition

EthanWYX2009 0

3 hours ago

Source: Radon's Theorem

Show that: any set of $d + 2$ points in $\mathbb R^d$ can be partitioned into two sets whose convex hulls intersect.

0 replies

EthanWYX2009

3 hours ago

0 replies

hard inequality omg

tokitaohma 4

N 3 hours ago by arqady

1. Given $a, b, c > 0$ and $abc=1$
Prove that: $\sqrt{a^2+1} + \sqrt{b^2+1} + \sqrt{c^2+1} \leq \sqrt{2}(a+b+c)$

2. Given $a, b, c > 0$ and $a+b+c=1$
Prove that: $\dfrac{\sqrt{a^2+2ab}}{\sqrt{b^2+2c^2}} + \dfrac{\sqrt{b^2+2bc}}{\sqrt{c^2+2a^2}} + \dfrac{\sqrt{c^2+2ca}}{\sqrt{a^2+2b^2}} \geq \dfrac{1}{a^2+b^2+c^2}$

4 replies

tokitaohma

Yesterday at 5:24 PM

arqady

3 hours ago

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