ITMC 2026 Grade 8 National Preliminary sample-r1

Printed positive marks are preserved per question. The source uses different blank-answer and incorrect-answer rules by section, including an optional bonus penalty; the import does not reproduce that complete official scoring scheme.

Which identity is incorrect?

Two sides of an equilateral triangle are $2x+5$ and $6x+3$. Find its perimeter.

Evaluate $\dfrac{2562^2-543^2}{2019}$.

In the figure, $ABCD$ is a trapezium, $AB\parallel CD$, $AB=5$ and $CD=3$. What fraction of the trapezium is shaded?

An isosceles triangle has two sides of lengths 25 and 62. Find its perimeter.

In the figure, $ABCD$ is a square and $CDEFG$ a regular pentagon. Find $\angle AED$.

Add all prime numbers between 50 and 60 to form P. What is the largest prime divisor of P?

If $\dfrac{A}{x^2-25}=\dfrac{x}{x-5}$ with $x\ne\pm5$, find $A$.

Find the last two digits of $7^{2019}+2019$.

A palindrome is a positive integer unchanged when its digits are reversed, such as 13431,2552,77. How many four-digit palindromes are less than 2019?

Which number can be the sum of seven consecutive integers?

Use the distinct digits 0–9 once each to form five two-digit numbers. What is the largest possible sum of those five numbers?

For real $x$, $[x]$ denotes the greatest integer not exceeding $x$ (the integer part). The decimal or fractional part is $\{x\}=x-[x]$, so $x=[x]+\{x\}$. Examples: $[3]=3$, $[6.25]=6$, $[-6.25]=-7$, $[22/7]=3$. The source lists: (1) $0\lt \{x\}\lt 1$ for every real $x$; (2) $\{x\}=0$ exactly when $x$ is an integer; (3) $x-1\lt [x]\le x\lt [x]+1$; (4) $[n+x]=n+[x]$ for positive integers $n$; (5) $[-x]=-[x]-1$ if $x$ is not an integer, and $[-x]=-[x]$ if $x$ is an integer. Editorial correction: property (1) must read $0\le\{x\}\lt 1$; its printed strict lower bound contradicts (2). The definition of $[x]$ is used below.

Find $[x]$ if $x-\dfrac1{2020}\lt -2019\lt x$.

For real $x$, $[x]$ denotes the greatest integer not exceeding $x$ (the integer part). The decimal or fractional part is $\{x\}=x-[x]$, so $x=[x]+\{x\}$. Examples: $[3]=3$, $[6.25]=6$, $[-6.25]=-7$, $[22/7]=3$. The source lists: (1) $0\lt \{x\}\lt 1$ for every real $x$; (2) $\{x\}=0$ exactly when $x$ is an integer; (3) $x-1\lt [x]\le x\lt [x]+1$; (4) $[n+x]=n+[x]$ for positive integers $n$; (5) $[-x]=-[x]-1$ if $x$ is not an integer, and $[-x]=-[x]$ if $x$ is an integer. Editorial correction: property (1) must read $0\le\{x\}\lt 1$; its printed strict lower bound contradicts (2). The definition of $[x]$ is used below.

For positive integers $n$ satisfying $[\sqrt n]=5$, find the sum of all possible $n$.

Rectangle $ABCD$ has area 2560 and $AB=40$. Points $E,G$ lie on $CD,BC$ respectively, and interior point $F$ makes $CEFG$ a rectangle as shown. Find the perimeter of hexagon $ABGFED$.

Find the coefficient of $x^4$ in $(1-x)(1+x)(1-x+x^2)(1+x+x^2)$.

In trapezoid $ABCD$, $AB\parallel CD$ and diagonals meet at $E$. If $[ABE]=25$ and $[ADE]=20$, find $[ABCD]$.

Here $[ABE]$ and $[ADE]$ denote the areas of the corresponding triangles, and $[ABCD]$ denotes the area of the trapezoid.

In the figure $AC=BC$, $\angle ABC=62^\circ$, $\angle ADB=90^\circ$ and $\angle CAD=x^\circ$. Find $x$.

A dealer’s profit, based on cost, is $x\%$. If cost decreased by 8% but selling price stayed fixed, profit based on the new cost would be $(x+10)\%$. Find $x$.

How many positive integers $n$ make both $n$ and $n+99$ perfect squares?