VTMO 2026 - Grade 8 - Semifinal - SET 06 - Self-study

The line through $M=(-2,5)$ parallel to $y=2x+2024$ has equation $y=ax+b$, $a\ne0$. Find $a+b$.

The line $y=-ax+b$, with $a,b>0$, passes through $M=(1,2)$ and meets the coordinate axes at $A$ on $Ox$ and $B$ on $Oy$. When the area of triangle $OAB$ is as small as possible, find $a+b$.

For real numbers $x,y,z,t\in[1/3,1]$, find the greatest and least values of $P=\dfrac{y+z+t}{1+x}+\dfrac{x+z+t}{1+y}+\dfrac{x+y+t}{1+z}+\dfrac{x+y+z}{1+t}$.

Find all primes $p$ for which integers $a,b$ exist such that $\dfrac1p=\dfrac1{a^2}+\dfrac1{b^2}$.

For integer $x$, find the minimum of $P(x)=\dfrac{4x^4+81}{2x^2+6x+9}$.

How many ways can three distinct literature books and four distinct mathematics books be arranged on one shelf if the literature books must be consecutive?

Partition $1,2,\ldots,100$ into disjoint sets $A=\{a_1,\ldots,a_{50}\}$ and $B=\{b_1,\ldots,b_{50}\}$, where $a_1\lt \cdots\lt a_{50}$ and $b_1>\cdots>b_{50}$. Prove $\sum_{i=1}^{50}|a_i-b_i|=50^2$.

Integers $4,4,x,y,13$ are arranged so that $4\le4\le x\le y\le13$. How many different pairs $(x,y)$ make the mean of the five numbers an integer?

For which listed value of $a$ is $4y^2-24y-a$ the square of a polynomial?

In the five-pointed star shown, find the sum of its five tip angles $a+b+c+d+e$.

How many integers $x$ satisfy $(x-1)(x-3)(x-5)\cdots(x-2021)(x-2023)\lt 0$?

How many three-digit natural numbers have the property that any two adjacent digits differ in absolute value by at most $1$? Repeated digits are allowed, as in $221$ or $555$.

Rectangle $ABCD$ has $AB=7$ cm and $BC=24$ cm. Points $E,F$ on diagonal $AC$ satisfy $AE=CF=10$ cm. Lines $BE,BF$ meet $AD,CD$ at $G,H$, respectively. Find the area of quadrilateral $EFHG$ in cm².

Triangle $ABC$ is isosceles at $A$, and $M$ is the midpoint of $AB$. Take $F$ on $AC$ and $E$ on the ray opposite to $BA$ so that $BE=2CF$. Let $D$ be the reflection of $A$ across $F$. Lines $EF$ and $BC$ meet at $K$, and $DK$ meets $AB$ at $N$. Prove $MN=CF$.

Isosceles right triangle $ABC$ is right at $A$. Point $M$ is inside it and satisfies $\angle MAC:\angle MBA=1:2$. Given $AB=41$ cm and $AM=18$ cm, find the area of triangle $MAB$ in cm².

Integers $m,n,p,r$ are such that the equations $mx^3+nx^2+px+r=2021$, $mx^3+nx^2+px+r=2022$, and $mx^3+nx^2+px+r=2023$ have integer roots $a,b,c$, respectively. Prove $a+c=2b$.

When $1001$ is divided by an odd prime, the remainder is $1$. What is the remainder when $2012$ is divided by the same prime?

Fifty coins initially show the same side. In the first move $25$ coins are flipped, in the second $29$, and in the third $14$. At the end every coin shows the opposite side. How many coins were flipped in all three moves?

Let $f(x)=1+2x+3x^2+\cdots+2015x^{2014}$. Find $f(-1)$.

For natural numbers define $a*b=(a+1)(b+1)$. If $a>b$ and $a*b=13$, find $a$.