ITMC 2026 Grade 7 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.

Year 2026 assigned because the source does not state an exam year.

If $A=2019$ and $B=2562$, find $2562A-2019B$.

If $\dfrac1{x-2019}=5$, find $\dfrac1{x-2020}$.

Two sides of a square have lengths $2-5x$ and $6-3x$, as shown. Find the area of the square.

An isosceles triangle $ABC$ has $\angle ABC=86^\circ$. Which listed value is possible for $\angle ACB$?

In the figure, $AB=18$, $BC=19$, $CA=17$. $D$ lies on $AB$, $E$ on $CA$, and $F$ on $DE$ such that $\angle DFB=\angle DBF$ and $\angle EFC=\angle ECF$. Find the perimeter of triangle $ADE$.

A right triangle has side lengths 14, 48, 50. What is the length of its longest altitude?

Find the product of 30% of 30 and 70% of 70.

There are three prime numbers between 2000 and 2019. What is their sum?

What are the last two digits of $7^{2019}+2019$?

Which listed number can be the sum of seven consecutive integers?

A palindrome is a positive integer whose digits read the same forwards and backwards; for example, 13431, 2552 and 77. How many palindromes lie between 2019 and 2562?

What is the maximum possible number of intersection points of a circle and the boundary of a triangle?

If $f(x)=4x$, find $f(x+1)-f(x-1)$.

For a real number $x$, $[x]$ is the greatest integer not exceeding $x$ (the integer part). The fractional or decimal 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) $0<\{x\}<1$ for all real $x$; (2) $\{x\}=0$ exactly when $x$ is an integer; (3) $x-1<[x]\le x<[x]+1$; (4) $[n+x]=n+[x]$ for positive integer $n$ and real $x$; (5) $[-x]=-[x]-1$ if $x$ is not an integer, and $[-x]=-[x]$ if $x$ is an integer. Source correction: property (1) is printed as $0<\{x\}<1$ for all real $x$, contradicting both the definition and property (2) at integers. The correct bound is $0\le\{x\}<1$. The solution uses the definition and this corrected bound.

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

For a real number $x$, $[x]$ is the greatest integer not exceeding $x$ (the integer part). The fractional or decimal 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) $0<\{x\}<1$ for all real $x$; (2) $\{x\}=0$ exactly when $x$ is an integer; (3) $x-1<[x]\le x<[x]+1$; (4) $[n+x]=n+[x]$ for positive integer $n$ and real $x$; (5) $[-x]=-[x]-1$ if $x$ is not an integer, and $[-x]=-[x]$ if $x$ is an integer. Source correction: property (1) is printed as $0<\{x\}<1$ for all real $x$, contradicting both the definition and property (2) at integers. The correct bound is $0\le\{x\}<1$. The solution uses the definition and this corrected bound.

Real numbers $x,y$ satisfy $[\sqrt{x}]=7$ and $[\sqrt{y}]=11$. Find the smallest possible value of $[x-y]$.

In the figure, $ABH$ is equilateral, $BCDH$ is a square, and $DEFGH$ is a regular pentagon. Find $\angle GAH$.

Trapezoid $ABCD$ has $AB\parallel CD$, $AB=5$, $CD=3$. $E$ is the midpoint of $BC$. What fraction of the trapezoid is shaded, as shown?

A circle is inscribed in a right triangle. Its point of tangency on the hypotenuse divides that side into lengths 14 and 15. Find the area of the triangle.

How many ordered pairs of positive integers $(a,b)$ satisfy $2a+5b=62$?

Let $N$ be the smallest positive integer for which $N/2$ is a perfect square and $N/3$ a perfect cube. Which listed digit appears in $N$?