SMO 2024 Junior Round 1

If $x^2+4x+16=0$, what is the value of $x^3$?

Let $a$ be a real number that satisfies $-1\lt a\lt 0$. Which of the following is true?

How many non-congruent triangles are there whose sides have integer lengths and the longest side has length $10$ units?

In the diagram below, the points $B$ and $E$ lie on $AF$ and $DF$ respectively, and $AE$ and $BD$ intersect at $C$. If $AB=AC$, $BD=BF$ and $EA=EF$, find $\angle BAC$.

Let $x,y,z$ be real numbers such that $x\ne0$, $y-z\ne0$ and $z+x\ne0$. If $\frac2x=\frac4{y-z}=\frac5{z+x}$, find the value of $\frac{7x-y}{y+2z}$.

Let $N$ be a 2-digit whole number. When $2692$ is divided by $N$, the remainder is $13$, and when $2978$ is divided by $N$, the remainder is $14$. Find the sum of all the possible values of $N$.

If $x$ is a positive integer such that $2^x+2^{89}=512^{10}$, find the value of $x$.

The diagram shows a right-angled triangle $ABC$. The sides $AC$ and $AB$ are in the ratio $3:5$. The point $D$ lies on $BC$ such that $AD$ is perpendicular to $BC$. Furthermore, $DB$ is $8$ cm longer than $CD$. What is the length of $BC$ in cm?

Let $n$ be a positive integer. Suppose that $a_1,a_2,a_3,\ldots$ is a sequence of numbers defined by
$$a_1=\sqrt{(n+3)(n-1)+4},\qquad a_k=\sqrt{(n+2k+1)a_{k-1}+4}\quad\text{for }k\ge2.$$
If $a_{100}=2024$, find the value of $n$.

Let $N$ be the smallest positive integer such that the sum of its digits is $2024$. What is the sum of the digits of the number $N+2$?

If $a$ and $b$ are non-zero real numbers such that $\frac1b-\frac1a=4$, find the value of $\frac{3a+7ab-3b}{a-3ab-b}$.

If the 5-digit whole number $\overline{11ab6}$ is a perfect square, find the value of $a+b$.

Let $N=1\times2+2\times3+3\times4+\cdots+30\times31$. Find the value of $N$.

Let $x$ and $y$ be non-zero real numbers where $x\ne y$. If $x^2+\sqrt3\,y=4$ and $y^2+\sqrt3\,x=4$, find the value of $(y/x)^2+(x/y)^2$.

Find the sum of all 2-digit even numbers $N$ with the following property: $N$ is a multiple of the product of its two digits.

In the diagram below, $AEB$ is an isosceles right-angled triangle and $ABG$ is a $30^\circ$-$60^\circ$-$90^\circ$ right-angled triangle with $\angle GAB=30^\circ$. The sides $AG$ and $BE$ intersect at $H$. If the area of triangle $AHE$ is $50$ cm$^2$, find the area of triangle $BGH$ in cm$^2$.

Find the smallest positive integer $n$ such that $\sqrt n-\sqrt{n-1}\lt \frac1{99}$.

Let $a,b,c$ be real numbers such that $a+b+c=8$ and $ab+bc+ca=0$. Find the maximum value of $3(a+b)$.

In the table below, every row and column is an infinite arithmetic progression. How many times does the number $2025$ appear in the table?

In the diagram below, $ABC$ is a right-angled triangle. Points $D,E$ lie on $AB$ while points $F,G$ lie on $BC$ such that $\triangle EFG$ and $\triangle DGC$ are right-angled isosceles triangles. It is given that $DC=3EG$ and the area of $\triangle DGC$ is $1$ cm$^2$. What is the area of $\triangle ADC$ in cm$^2$?