Question 1
Let $a,b,c$ be real numbers with $a\ge b\ge c>0$. Prove that $(a-b+c)\left(\frac1a-\frac1b+\frac1c\right)\ge1$.
VTMO 2026 - Grade 8 - Final - SET 3 - Self-study
Question 2
In trapezoid $ABCD$, $AB\parallel CD$ and $AB\lt CD$. The lines $DA$ and $CB$ extended meet at $I$. Let $E$ be a point on $CD$, and let $IE$ meet $AB$ at $F$. Take any point $P$ on the ray opposite to ray $BD$. The line $PF$ meets $AD$ at $M$, and $PE$ meets $BC$ at $N$. Prove that $MN\parallel AB$.
Question 3
Triangle $ABC$ is isosceles at $A$. Points $M,N$ move on the rays opposite to $BA,CA$ respectively, with $CN=2BM$. Point $D$ lies on the segment $MN$ with $DN=2DM$.
a) Prove that $D$ always moves on a fixed line.
b) Let $I,J$ be the midpoints of $BC,MN$ respectively. The line $IJ$ meets the line through $A$ parallel to $BC$ at $K$. Prove that $\angle IKD$ is constant.
a) Prove that $D$ always moves on a fixed line.
b) Let $I,J$ be the midpoints of $BC,MN$ respectively. The line $IJ$ meets the line through $A$ parallel to $BC$ at $K$. Prove that $\angle IKD$ is constant.
Question 4
a) Solve
$$\frac5{x^2+2}+\frac6{x^2+3}+\frac9{x^2+6}-\frac{3x^2+5}{x^2+4}=1.$$
b) Nonzero real numbers $x,y$ satisfy $xy+y+1\ne0$ and
$$\frac{x}{x^2-x+1}-\frac{2x^2}{x^4-x^2+1}+\frac{4x^4}{x^8-x^4+1}-\frac{8x^8}{x^{16}+x^8+1}=\frac{y}{xy+y+1}.$$
Prove that $x=y$.
$$\frac5{x^2+2}+\frac6{x^2+3}+\frac9{x^2+6}-\frac{3x^2+5}{x^2+4}=1.$$
b) Nonzero real numbers $x,y$ satisfy $xy+y+1\ne0$ and
$$\frac{x}{x^2-x+1}-\frac{2x^2}{x^4-x^2+1}+\frac{4x^4}{x^8-x^4+1}-\frac{8x^8}{x^{16}+x^8+1}=\frac{y}{xy+y+1}.$$
Prove that $x=y$.
Question 5
A tanker $B$ is $100$ km due north of a cargo ship $A$. The cargo ship travels northwest at $20$ km/h, and the tanker travels southeast at $30$ km/h. When the two ships are closest, how many kilometres has the cargo ship travelled?
