Search found 30 matches
- Mon Jun 14, 2021 7:13 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34113
Problem 11
In a $\triangle ABC$, $D$ and $E$ are respectively on $AB$ and $AC$ such that $DE\parallel BC$. $P$ is the intersection of $BE$ and $CD$. $M$ is the second intersection of $(APD)$ and $(BCD)$, $N$ is the second intersection of $(APE)$ and $(BCE)$. $w$ is the circle passing through $M$ and $N$ and ta...
- Wed Jun 02, 2021 1:23 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34113
Re: Special Problem Marathon
$f(0)=0$ and $f(x) \in \{1, -1\}$ for any $x \in \mathbb{Z} - \{0\}$ should work in this case.
- Wed Jun 02, 2021 1:15 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34113
Re: Special Problem Marathon
Solution (P5): Claim-1: $A, B, D, E$ and $A, C, D, F$ are concyclic. Proof: $\measuredangle BDE = \measuredangle CDE= 90^o - \measuredangle OCD=\measuredangle BAC =\measuredangle BAE$. Thus, $A, B, D, E$ are concyclic. Similarly, $A, C, D, F$ are concyclic. We use directed length. It suffices to sh...
- Tue Jun 01, 2021 11:16 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34113
Re: Special Problem Marathon
Problem 4:
Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$$f(x + y) + f(x)^2f(y) = f(y)^3 + f(x + y)f(x)^2$$
for all $x, y \in \mathbb{Z}$.
Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$$f(x + y) + f(x)^2f(y) = f(y)^3 + f(x + y)f(x)^2$$
for all $x, y \in \mathbb{Z}$.
- Tue Jun 01, 2021 11:12 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34113
Re: Special Problem Marathon
Solution (P3): By AM-GM , we have $$4(a^3+b^3+c^3+d^3) \geq (a^3+b^3+c^3+d^3)+(ab^2+bc^2+cd^2+da^2)+(ac^2+a^2c)+(bd^2+b^2d)+(ad^2+ba^2+cb^2+dc^2) \geq (a^2+b^2+c^2+d^2)(a+b+c+d)$$ and $$16(a^3+b^3+c^3+d^3) \geq (a^3+b^3+c^3+d^3)+3(ab^2+bc^2+cd^2+da^2)+3(ac^2+a^2c)+3(bd^2+b^2d)+3(ad^2+ba^2+cb^2+dc^2...
- Tue Jun 01, 2021 8:49 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34113
Re: Special Problem Marathon
Problem 2: There are $N$ boxes labelled $B_1, B_2, \ldots, B_N$ which are filled with balls of $N$ different colours $C_1, C_2, \ldots, C_N$. Further, it is known that for each colour we can partition the boxes into two sets, such that the total number of balls of that colour in both sets is the sa...
- Tue Jun 01, 2021 8:42 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34113
Re: Special Problem Marathon
Let \(G\) be the centroid of a right-angled triangle \(ABC\) with \(\angle BCA = 90^\circ\). Let \(P\) be the point on ray \(AG\) such that \(\angle CPA=\angle CAB\), and let \(Q\) be the point on ray \(BG\) such that \(\angle CQB =\angle ABC\). Prove that the circumcircles of triangles \(AQG\) and...
- Mon Mar 01, 2021 1:03 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2020 #1
- Replies: 1
- Views: 6163
Re: IMO 2020 #1
Let the circumcircle of $\triangle PAB$ intersects line $DA$ and $BC$ at points $P_1$ and $P_2$, respectively. Now, $\measuredangle P_1PA = \measuredangle P_1BA = \measuredangle PBA - \measuredangle PBP_1 = \measuredangle PBA - \measuredangle PAP_1 = \measuredangle PBA - \measuredangle PAD = \measur...
- Mon Mar 01, 2021 12:52 pm
- Forum: National Math Camp
- Topic: National Math Camp 2020 Exam 2 Problem 4
- Replies: 1
- Views: 2707
Re: National Math Camp 2020 Exam 2 Problem 4
It is easy to check that at least one of the two adjacent numbers is even. So, there are at least $1008$ even numbers on the circle. It is also easy to check that there exists at least one number which is divisible by $4$, because there exists at least one pair of consecutive numbers both of which a...
- Mon Mar 01, 2021 12:35 pm
- Forum: National Math Camp
- Topic: National Math Camp 2020 Exam 2 Problem 3
- Replies: 2
- Views: 2759
Re: National Math Camp 2020 Exam 2 Problem 3
We have the following claim. Claim : For some subset of the quadratic residue set $a^2$ (mod $2n-1$), we can pick $b_i$ such that the number of all possible values of $(a^2 + b_i)^2$ (mod $2n-1$) has decreased by at least $1$. Proof : Let $m_1, m_2$ be such that $m_1^2$ and $m_2^2$ are not same modu...