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konkur-1403-math-v1.5

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$\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2 - ax - b = 0$, such that $3\beta^2 + 2\alpha^2 - \beta = 7$. What is the sum of the cubes of the roots?

$\boxed{4}$

$4$

hard

1403_farvardin-Sabz_Riazi-3

A child has 3 rectangular blocks of the same size but different colors. In how many ways can he arrange them to form one or several stacks?

$\boxed{15}$

$15$

medium

1403_ordibehesht-Riazi-22

A five-digit number is written using two consecutive digits less than $10$. If the sum of the digits of this number is $23n+1$, how many five-digit numbers with this property exist?

$\boxed{6}$

$6$

hard

1403_ordibehesht-Riazi-37

A parallelepiped is generated by the vectors $\vec{a} = (2, -3, 4)$, $\vec{b} = (-1, 2, 3)$, and $\vec{c} = (3, -2, 1)$ and the plane $P$ includes the vectors $\vec{b}$ and $\vec{c}$. What is the height of this parallelepiped perpendicular to the plane $P$?

$\boxed{\sqrt{5}}$

$\sqrt{5}$

medium

1403_ordibehesht-Riazi-36

Box $A$ contains 6 blue beads, 4 green beads, and 5 red beads. Box $B$ contains 5 blue beads, 3 green beads, and 6 red beads. A bead is randomly selected from box $A$ and placed in box $B$. Then a bead is randomly selected from box $B$. What is the probability that the selected bead from box $B$ is blue?

$\boxed{0.36}$

$0.36$

easy

1403_ordibehesht-Tajrobi-132

By adding $4$ units to the first and second terms of an arithmetic sequence, the first and second terms of a new arithmetic sequence is formed. What is the difference of the $n$-th terms of the two sequences?

$\boxed{4}$

$4$

easy

1403_ordibehesht-Tajrobi-138

Consider a statistical dataset with a standard deviation of $\frac{1}{5}$. If each data point in the dataset is first multiplied by three and then decreased by one, what is the variance of the resulting dataset?

$\boxed{\frac{20}{25}}$

$\frac{20}{25}$

easy

1403_farvardin-Sabz_Riazi-26

Consider an arithmetic sequence with positive terms, denoted by $a_n$. Let $b_n = na_n$ define a new sequence such that the fourth term of the sequence $b_n$ is equal to the fifteenth term of the sequence $a_n$, and $a_1b_1 = 4$. What is the geometric mean between $a_{47}$ and $b_7$?

$\boxed{140}$

$140$

medium

1403_farvardin-Sabz_Riazi-1

Consider the equation $$ \sin x + \cos x = \frac{1}{2 \sin x}. $$ Determine the ratio of the sum of the solutions to this equation to the smallest solution within interval $[0, 2\pi]$.

$\boxed{28}$

$28$

hard

1403_ordibehesht-Sabz_Riazi-5

Consider the function $f(x) = 2x - \frac{1}{x}$. First, shift the graph of this function three units to the right and two units up. Then, reflect the resulting graph across the $y$-axis. Determine the $x$-coordinate of the point where this final graph intersects the line $y = -2x - 3$.

$\boxed{-2}$

$-2$

hard

1403_ordibehesht-Sabz_Riazi-1

Consider the parabola given by $y = -mx^2 + mx + 1$ and the line given by $y = -m - x$. Determine the number of integer values of $m$ for which the parabola and the line do not intersect.

$\boxed{0}$

$0$

hard

1403_ordibehesht-Tajrobi-111

Consider the quadratic function $f(x) = x^2 + a(x - b)$, where $a$ and $b$ are constants. The graph of $f(x)$ is tangent to its derivative at $x = 2$. Find the value of $f(a)$.

$\boxed{10}$

$10$

1403_ordibehesht-Sabz_Riazi-14

Consider two propositions $A$ and $B$. Given that $\mathrm{P(A - B) + \mathrm{P(B)} = \mathrm{P(A' \cup B')} = 0.79}$, and $\mathrm{P(B \cap A')} = \mathrm{P(B)} (1 - \mathrm{P(A)})$, determine the probability of $\mathrm{P(A' | B)}$? If there are two possible answers, list them separated by a comma.

$\boxed{0.3 , 0.7}$

$0.3$ , $0.7$

hard

1403_farvardin-Sabz_Riazi-24

Determine the number of natural numbers within the domain of the function $y = -\frac{1}{3-x}$ for which the graph of the function lies above $y = 0$ and below $y = -4$.

$\boxed{2}$

$2$

easy

1403_ordibehesht-Tajrobi-114

Determine the number of three-digit positive integers $a$ for which the linear Diophantine equation $ax + 84y = 97$ has integer solutions.

$\boxed{258}$

$258$

hard

1403_ordibehesht-Sabz_Riazi-28

Find the value of following $$ \frac{\sin^4 \alpha + 4 \cos^2 \alpha}{1 + \cos^2 \alpha} - \frac{\cos^4 \alpha + 4 \sin^2 \alpha}{1 + \sin^2 \alpha} $$?

$\boxed{\cos 2\alpha}$

$\cos 2\alpha$

easy

1403_ordibehesht-Riazi-13

Find the value of following expression $$ \frac{3 \cos(248^\circ) - 2 \sin(158^\circ)}{\sin(202^\circ) - \cos(292^\circ)}. $$

$\boxed{2.5}$

$2.5$

easy

1403_ordibehesht-Tajrobi-119

For any real value of $a \neq 0$, the function $ f(x) = \begin{cases} bx + c & x < a \\ \frac{1}{x} & x \geq a \end{cases} $ is differentiable on $\mathbb{R}$. What is the value of $ac$?

$\boxed{2}$

$2$

easy

1403_ordibehesht-Riazi-20

For how many integer values of $m$ is the function $ y = \frac{mx + 2}{x - 1 + m} $ decreasing on the interval $(1, +\infty)$ (with $ m \neq 2 $)?

$\boxed{2}$

$2$

medium

1403_ordibehesht-Riazi-19

For how many single-digit integer $a$'s, does the equation $\sqrt{x} + \sqrt{x - a} = a$ have an integer solution for $x$?

$\boxed{6}$

$6$

medium

1403_ordibehesht-Riazi-7

For the following data sets, the mean of the first quartile is 9 and the mean of the third quartile is 39. If the mean of the data between the first and third quartiles is 26, which value of $a$ makes the mean of the entire data set greater than the third quartile? Data: $9, 23, 29, 1, 3, 42, a, a, 2a + 1, 9, 23, 29, 1, 3, 42, 18$

$\boxed{54.5}$

$54.5$

medium

1403_ordibehesht-Tajrobi-129

For three real numbers $x, y$, and $z$, how many of the following statements cannot be proved by the method of contradiction? \n (a) $\frac{x}{y} + \frac{y}{x} \geq 2$ \n (b) $x^2 + y^2 + z^2 \geq xy + yz + zx$ \n (c) $x^2 + y^2 + 2 \geq (x+1)(y+1)$ \n (d) $x^2 + xz + z^2 \geq 0$

$\boxed{1}$

$1$

hard

1403_ordibehesht-Sabz_Riazi-19

For which value of $a$ does the inverse function of $f(x) = x^3 + 6x^2 + ax + 1$ intersect the line $y = -10x + 10$ at the point where $y = 1$?

$\boxed{12}$

$12$

easy

1403_ordibehesht-Riazi-8

From a population, a sample of $11$, $9$, $8$, $8$, $7$, $6$, $5$, $5$, and $4$ has been taken. If the population standard deviation is half of the sample standard deviation, what is the 95\% confidence interval for the mean of this population? ($\sqrt{20} \approx 4/5$)

$\boxed{(6.25, 7.75)}$

$(6.25, 7.75)$

1403_farvardin-Sabz_Riazi-28

Given $f(x) = \sqrt{x}(3x^2 + 5)$, find the value of $$ \lim_{h \to 0} \frac{f^2(1-h) - f^2(1+2h)}{h}. $$

$\boxed{-480}$

$-480$

hard

1403_ordibehesht-Sabz_Riazi-12

Given that the polynomial $f(x) = x^3 + ax^2 + bx + 1$ is divisible by both $x - 1$ and $x + 2$. Determine the remainder when $f(x + 2a) + f(x + 2b)$ is divided by $x - 2$?

$\boxed{11}$

$11$

hard

1403_ordibehesht-Sabz_Riazi-2

Given the function $f(x) = a + \sqrt{2a - 3 - x}$, determine the value of $f^{-1}(6)$ if the equation $f^{-1} \circ f(x) = f \circ f^{-1}(x)$ has exactly one solution.

$\boxed{-6}$

$-6$

hard

1403_farvardin-Sabz_Riazi-13

Given the function $y = f(x)$, the asymptotes intersect only at the point $A=(2, a)$. For the function $y = 2 - 3f\left(\frac{x}{2}\right)$, the asymptotes intersect at the point $A'=(b, -3)$. Determine the value of $a + b$.

$\boxed{\frac{17}{3}}$

$\frac{17}{3}$

hard

1403_ordibehesht-Sabz_Riazi-11

Given the functions $f(x) = x^2 - 3|x|$ and $g(x) = x^3 + 2|x|$, determine the ratio of the derivative of $f \circ g^\prime$ to the derivative of $g \circ f^\prime$ evaluated at $x = -1$.

$\boxed{3}$

$3$

1403_ordibehesht-Sabz_Riazi-13

Given the linear function $f(x) = ax + b$, where $a$ and $b$ are constants, determine the value of $f^{-1}(-a)$ if the graph of the function $\frac{f}{f \circ f}$ coincides with the graph of its inverse function.

$\boxed{b}$

$b$

medium

1403_farvardin-Sabz_Riazi-12

Given the quadratic function $y = 25\alpha x^2 + 4x + \beta$, where $\alpha$ and $\beta$ are the roots and $\beta > \alpha$, determine the $i$-th quadrant of the plane in which the vertex of this parabola

$\boxed{1}$

$1$

medium

1403_ordibehesht-Tajrobi-113

How many solutions does the trigonometric equation $ \sin 2x - 4 \sin^2 x \cos x = 0 $ have in the interval $(-\pi, \pi)$?

$\boxed{5}$

$5$

easy

1403_ordibehesht-Tajrobi-120

How many surjective functions can be written from the set $A = \{1, 2, 3, 4\}$ to the set $B'$ such that $B' \subseteq A$?

$\boxed{256}$

$256$

hard

1403_ordibehesht-Sabz_Riazi-27

If $$ \frac{1}{\sin^2 x} - \frac{1}{\cos^2 x} = \frac{3}{2}, $$ what is the value of $\cos 4x$?

$\boxed{-\frac{7}{9}}$

$-\frac{7}{9}$

medium

1403_farvardin-Sabz_Riazi-14

If $\lim_{{x \to \frac{1}{2}}^+} \frac{a + 3 \left\lfloor -x \right\rfloor}{1 - 2x} = -\infty$ what is the value of $\lim_{{x \to \frac{1}{2}}} \left\lfloor \frac{x}{a} - x \right\rfloor$?

$\boxed{-1}$

$-1$

hard

1403_ordibehesht-Tajrobi-123

If $ f(x) = \begin{cases} (1-a) \lfloor x \rfloor + (3a^2 - 1) \lfloor -x \rfloor & x \notin \mathbb{Z} \\ b \sin \left( \frac{\pi}{a} \right) & x \in \mathbb{Z} \end{cases} $ is continuous on the set of real numbers, what is the value of $\frac{a}{b}$?

$\boxed{2}$

$2$

hard

1403_ordibehesht-Riazi-17

If $ f(x) = \sqrt{x + 8} - \sqrt{x} $ and $ g(x) = \frac{1}{\sqrt{x + 8} + \sqrt{x}} $, what is the value of the expression $ f'(1) g(1) - g'(1) f(1) $?

$\boxed{0}$

$0$

medium

1403_ordibehesht-Riazi-18

If $\alpha$ is the larger root and $\beta$ is the smaller root of the equation $\sqrt{2x+6} = 4|x|-3$ (where $\beta < 0$, $\alpha > 0$), what is the result of $8\beta^2 + 11\beta - \alpha$?

$\boxed{-3}$

$-3$

hard

1403_farvardin-Sabz_Riazi-5

If $\log_{2}(x^2 + 2x + 4) + \log_{2}(x - 2) = 3$, what is the value of $\log_{\sqrt[3]{2}}x$?

$\boxed{4}$

$4$

easy

1403_ordibehesht-Riazi-9

If $\vec{a} = (1, -3, 4)$, $\vec{b} = (3, -4, 2)$, and $\vec{c} = (-1, 1, 4)$, and the orthogonal projection of $\vec{a}$ onto the span of $\vec{b} + \vec{c}$ is $\vec{p}i + \vec{q}j + \vec{r}k$, what is the value of $\vec{p} + \vec{q} + \vec{r}$?

$\boxed{3\frac{4}{7}}$

$3\frac{4}{7}$

hard

1403_ordibehesht-Sabz_Riazi-39

If $\vec{a} = \vec{i} \times (\vec{i} \times \vec{j}) - 3\vec{j} \times (\vec{j} \times \vec{k}) + 2\vec{k} \times (\vec{k} \times \vec{i})$, what is the value of the cosine of the angle that $\vec{a}$ makes with the positive direction of the $z$-axis?

$\boxed{\frac{3}{\sqrt{14}}}$

$\frac{3}{\sqrt{14}}$

hard

1403_ordibehesht-Sabz_Riazi-40

If $A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 0 & 1 \\ -1 & 1 & 2 \end{bmatrix}$, what is the value of $|A||A|$?

$\boxed{256}$

$256$

easy

1403_ordibehesht-Sabz_Riazi-32

If $A = \begin{bmatrix} 1 & 1 & -1 \\ 0 & 2 & 0 \\ -2 & 1 & 1 \end{bmatrix}$, which one is the third row of the matrix $A^3$?

$\boxed{\begin{bmatrix} -10 & 1 & 7 \end{bmatrix}}$

$\begin{bmatrix} -10 & 1 & 7 \end{bmatrix}$

medium

1403_ordibehesht-Riazi-34

If $A = \begin{bmatrix} x & 1 & 3 \\ 0 & 1 & -1 \\ -1 & 1 & 1 \end{bmatrix} \begin{bmatrix} -1 & 2 & 3 \\ 2 & -1 & 1 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix} $, what is the value of $x$ so the sum of the elements on the main diagonal of matrix $A$ equal to the sum of the elements in its third column?

$\boxed{2}$

$2$

hard

1403_ordibehesht-Sabz_Riazi-30

If $A^3 = \begin{bmatrix} 3 & 2 \\ 2 & 1 \end{bmatrix}$ and $A^2 = \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}$, what is the sum of the elements of matrix $A$?

$\boxed{3}$

$3$

medium

1403_ordibehesht-Sabz_Riazi-31

If $A$ and $B$ are two rank-2 square matrices and $AB = \begin{bmatrix} 3 & 2 \\ \frac{1}{2} & \frac{1}{3} \end{bmatrix}$, what is the product of the non-diagonal entries of the matrix $A \begin{bmatrix} 0 & 3 \\ -6 & -1 \end{bmatrix} B - \frac{3}{2}A \begin{bmatrix} 2 & 2 \\ -4 & \frac{4}{3} \end{bmatrix} B$?

$\boxed{9}$

9

medium

1403_ordibehesht-Riazi-33

If $B = \frac{\frac{2}{\sqrt{2}} + \sqrt{14}}{\frac{8}{\sqrt{2}} + \sqrt{14}}$, what is the value of $3B + 1$?

$\boxed{\sqrt{7}}$

$\sqrt{7}$

easy

1403_ordibehesht-Tajrobi-136

If $b$ is a real number and $\lim_{x \to \pi^+} \frac{\sqrt{a + \cos x}}{\sin 2x} = b$, what is the value of $ab\sqrt{2}$?

$\boxed{\frac{1}{2}}$

$\frac{1}{2}$

medium

1403_farvardin-Sabz_Riazi-17

If $f(x) = \begin{cases} \sqrt{x^2 + 3} + 2a & \text{if } |x| \leq 1 \\ ax^2 + 5 & \text{if } |x| \geq 1 \end{cases}$ is a function, what is the value of $f(a)$?

$\boxed{32}$

$32$

easy

1403_ordibehesht-Tajrobi-139

If $f(x) = \frac{4}{x^2 + 2x - 3}$ and $g(x) = \frac{1}{x-1}$, what is the point of intersection of the asymptotes of the graphs of $f - g$?

$\boxed{(-3, 0)}$

$(-3, 0)$

easy

1403_ordibehesht-Riazi-16

If $f(x) = \left\{ \left( \frac{1}{9}, -1 \right), \left( \frac{1}{3}, 1 \right), \left( -\frac{1}{4}, 3 \right), \left( \frac{1}{4}, -3 \right) \right\}$ and $g(x) = -|x|\sqrt{x}$ and $\left( f \circ g \right)^{-1}(a) = -3$, find the value of $a$?

$\boxed{-\frac{1}{8}}$

$-\frac{1}{8}$

easy

1403_ordibehesht-Tajrobi-112

If $f(x) = x^2 - \lfloor x \rfloor$ and $f(af(\sqrt{5})) = 2$, find the values of $a$?

$\boxed{-\frac{1}{3}}$

$-\frac{1}{3}$

easy

1403_ordibehesht-Riazi-6

If $m$ is the smallest positive member of the set $\{407r + 592s \mid r, s \in \mathbb{Z}\}$, what is the sum of the digits of $m$?

$\boxed{10}$

$10$

medium

1403_ordibehesht-Riazi-38

If $n(A \cup B) = 57$ and $n(A \cap B) = 3n(A - B) = 4n(B - A)$, what is the number of elements in set $A$?

$\boxed{48}$

$48$

easy

1403_ordibehesht-Tajrobi-137

If $x > 1$, determine the minimum value of $A = \log_9 x + 2\log_{x^2} 3$.

$\boxed{\sqrt{2}}$

$\sqrt{2}$

medium

1403_farvardin-Sabz_Riazi-9

If $y = \frac{x+2}{4} - \frac{\sqrt{x+1}}{2}$ is the inverse of the function $y = ax + a \sqrt{x}$, find the value of $a$.

$\boxed{4}$

$4$

easy

1403_ordibehesht-Tajrobi-117

If the function $f(x) = b[x^2 - ax] - 2a$ is non-zero and continuous in $\mathbb{R}$, what is the value of $\frac{a}{f(b)}$?

$\boxed{-\frac{1}{2}}$

-\frac{1}{2}$

easy

1403_ordibehesht-Tajrobi-124

If the statement $p$ is true, the statement $q$ is false, and $r$ is an arbitrary statement, which statement is logically equivalent to $(p \Rightarrow r) \Rightarrow (r \Rightarrow q)$?

$\boxed{\sim r}$

$\sim r$

easy

1403_ordibehesht-Riazi-2

In a $k$-regular graph of order 5, what is the minimum possible number of odd-length cycles in the complement of this graph?

$\boxed{1}$

$1$

hard

1403_ordibehesht-Sabz_Riazi-23

In an isosceles triangle with a base length of 16 and the length of the median to the base equal to half the base, what is the length of median corresponding to each leg of the triangle?

$\boxed{4\sqrt{10}}$

$4\sqrt{10}$

medium

1403_ordibehesht-Riazi-32

In graph $G$, $|V(G)|=8$ and $|E(G)|=24$. What is the minimum possible value of $\delta(G)$?

$\boxed{3}$

$3$

easy

1403_ordibehesht-Riazi-40

In how many subsets of the set $A = \{ x \in \mathbb{N} \mid \frac{1}{x} > 0.1 \}$, the difference between the largest and smallest element less than 5?

$\boxed{416}$

$416$

hard

1403_farvardin-Sabz_Riazi-20

In how many ways can the set $A = \{a, b, c, d, e\}$ be partitioned into three subsets?

$\boxed{25}$

$25$

easy

1403_farvardin-Sabz_Riazi-25

In the graph $G$ with the vertex set $V(G) = \{a, b, c, d, e, f\}$, we have: $N_G(a) = \{b, c, d\}$, $N_G(b) = \{a, c\}$, $N_G(c) = \{a, b\}$, $N_G(d) = \{a, f\}$, $N_G(e) = \{\}$, $N_G(f) = \{d\}$. How many natural-length paths are there in this graph?

$\boxed{17}$

$17$

hard

1403_ordibehesht-Sabz_Riazi-22

Line $3y +2x = 9$ is tangent to the circle $x^2 + y^2 + 3x + ay = c$ at the point $(0, 3)$. What is the value of $a$?

$\boxed{-1.5}$

$-1.5$

medium

1403_ordibehesht-Tajrobi-140

Point $A'$ is the image of point $A$ reflected across line $d$. If $AA' = 6$ and $O$ is a point on line $d$ such that $OA = 5$, then what is the distance of point $A$ from line $OA'$?

$\boxed{4.8}$

$4.8$

hard

1403_farvardin-Sabz_Riazi-37

Point $A$ is located on the circle $C(O,4)$. In a homothety with center $A$ and ratio $-\frac{1}{2}$, circle $C$ is transformed into circle $C'(O',R')$. In a counterclockwise rotation with center $O$ and angle $60^\circ$, circle $C'$ is transformed into circle $C''(O'',R')$. What is the length of the internal common tangent of circles $C'$ and $C''$?

$\boxed{2\sqrt{5}}$

$2\sqrt{5}$

hard

1403_farvardin-Sabz_Riazi-38

Reflect the graph of the function $$ f(x) = \begin{cases} (x - 1)^3 & \text{if } x \geq 0 \\ -1 + \sqrt[3]{-x} & \text{if } x < 0 \end{cases} $$ across the $y$-axis. Then, shift the resulting graph two units to the right and one unit down. Determine the distance between the points where the transformed graph intersects the $x$-axis.

$\boxed{10}$

$10$

hard

1403_ordibehesht-Sabz_Riazi-3

Seven people with names $A_1, A_2, \ldots, A_7$ want to compete in a shooting contest. What is the probability that exactly two people stand between $A_1$ and $A_2$?

$\boxed{\frac{4}{21}}$

$\frac{4}{21}$

hard

1403_farvardin-Sabz_Riazi-22

Students from two schools, $A$ and $B$, took part in an exam. $60\%$ of the students from school $A$ and $70\%$ of the students from school $B$ passed the exam. If the number of students from school $A$ is $\frac{3}{2}$ times the number of students from school $B$, what is the probability that a randomly selected student who passed the exam is from school $A$?

$\boxed{\frac{9}{16}}$

$\frac{9}{16}$

easy

1403_ordibehesht-Riazi-25

The area of a cross-section of a right circular cylinder that passes through its axis is twice the area of a cross-section perpendicular to its height. What is the ratio of the lateral surface area to the total surface area of the cylinder?

$\boxed{\frac{\pi}{\pi + 1}}$

$\frac{\pi}{\pi + 1}$

hard

1403_farvardin-Sabz_Riazi-34

The average rate of change of the function $f(x) = (x^2 + 1)^3(ax + 1)$ over the interval $[-1, 0]$ is equal to $-11$. What is the rate of change of this function at the point $x = -2a$?

$\boxed{8}$

$8$

medium

1403_ordibehesht-Tajrobi-126

The average rate of change of the function $f(x) = x\sqrt{x-4}$ over the interval $[4, 8]$ is how many times the instantaneous rate of change of the function at $x = 8$?

$\boxed{1}$

$1$

hard

1403_ordibehesht-Sabz_Riazi-15

The difference between the roots of the equation $x^2 + 2kx + 5 = 0$ is equal to $\frac{4k}{3}$. What is the value of $\lfloor\frac{k^2}{2}\rfloor$?

$\boxed{4}$

$4$

medium

1403_ordibehesht-Riazi-4

The distance of point $A(a, a+3, 4)$ from the plane $P: x=2$ is $5$ units. What is the minimum distance from $A$ to the origin?

$\boxed{5}$

$5$

hard

1403_ordibehesht-Sabz_Riazi-38

The foci of an ellipse are on the $x$-axis at $x = 3$ and $x = -3$. If the eccentricity of the ellipse is $\frac{1}{3}$, what is the length of the minor axis of this ellipse?

$\boxed{12\sqrt{2}}$

$12\sqrt{2}$

easy

1403_ordibehesht-Riazi-35

The foci of an ellipse are the points $(1,3)$ and $(1,-5)$. If the length of the minor axis of this ellipse is $6$, what is its eccentricity?

$\boxed{0.8}$

$0.8$

hard

1403_ordibehesht-Sabz_Riazi-29

The focus of the parabola $y^2 = 2x - 4y$ and its intersection points with the y axis form the vertices of a triangle. What is the area of this triangle?

$\boxed{3}$

$3$

hard

1403_ordibehesht-Sabz_Riazi-35

The function $f(x) = x^2 - 2|x - 1|$ is strictly monotonic in the interval $(a, +\infty)$. What is the minimum value of $a$?

$\boxed{-1}$

$-1$

hard

1403_ordibehesht-Sabz_Riazi-16

The function $y = (x-1) \left| x \right|$ is strictly decreasing in the interval $(a,b)$. What is the value of $a + b$?

$\boxed{\frac{1}{2}}$

$\frac{1}{2}$

easy

1403_ordibehesht-Tajrobi-115

The length of the longest side of a right-angled triangle is $a$, and its smallest angle measures $30^\circ$. The volume of the solid formed by rotating this triangle around its shortest side is how many times $\pi a^3$?

$\boxed{\frac{1}{8}}$

$\frac{1}{8}$

medium

1403_farvardin-Sabz_Riazi-33

The line $7y - x = 5$ is tangent to the curve $y = \frac{ax - 1}{3x + 1}$ in the first quadrant of the coordinate plane. What is the value of $a$?

$\boxed{4}$

$4$

medium

1403_ordibehesht-Tajrobi-125

The mean of the first set with 4 data points is equal to the mean of the second set of data, which has 5 data points. If we exchange one data point from the first set with one data point from the second set, the new means of the sets will be equal. If the variance of the first set before the exchange is $1.25$, what is the variance of the first set after the exchange?

$\boxed{1.25}$

$1.25$

easy

1403_ordibehesht-Riazi-24

The members of the set $A = \{1, 5, 9, \ldots, 85\}$ form an arithmetic sequence. What is the minimum number of elements that can be selected from this set such the sum of at least two element of this subset is 90?

$\boxed{13}$

$13$

hard

1403_ordibehesht-Sabz_Riazi-26

The number $a$ is the smallest three-digit number that, when divided by the numbers 7 and 8, the remainders are 5 and 7, respectively. What is the sum of the digits of $a$?

$\boxed{4}$

$4$

hard

1403_ordibehesht-Sabz_Riazi-20

The points $(-4, 3)$ and $(-1.5, -4)$ lie on a quadratic function. What is the sum of the roots of this function?

$\boxed{\frac{3}{2}}$

$\frac{3}{2}$

easy

1403_ordibehesht-Riazi-3

The roots of the function $f(x) = 2mx^2 + (4+m)x - m + 4$ and the intersection point of the function with the $y$-axis are the vertices of a triangle with an area of $\frac{1}{2}$. What is the product of different values of $m$ for $m > 0.$

$\boxed{\frac{256}{9}}$

$\frac{256}{9}$

hard

1403_farvardin-Sabz_Riazi-2

The tangent line to the curve $ y = x^3 + ax^2 + bx - 1 $ at the point $ (-1, -4) $ intersects the curve. What is the result of $\frac{a}{b}$?

$\boxed{0.6}$

$0.6$

medium

1403_ordibehesht-Riazi-21

The values $a$, $1 + 2a$, and $5 - a$ are in arithmetic progression. If $a$ is the first term of this sequence, what is the ninth term of the sequence?

$\boxed{14.75}$

$14.75$

easy

1403_ordibehesht-Riazi-1

We consider the line segment $AB = 2$ in the plane. How many points are at a distance of $\sqrt{2}$ from $A$ such that their distance from $B$ is equal to $2\sqrt{2}$?

$\boxed{2}$

$2$

medium

1403_farvardin-Sabz_Riazi-29

What is the minimum number of elements we need to choose from the set $\{3, 4, \cdots, 9, 12, 13, \cdots, 20\}$ to be sure that at least two of them have a common divisor which is not $1$?

$\boxed{8}$

$8$

medium

1403_ordibehesht-Riazi-39

What is the number of non-negative and real solutions of the equation $x_1 + \sqrt{x_2} + x_3 + x_4 = 3$?

$\boxed{20}$

$20$

hard

1403_ordibehesht-Sabz_Riazi-25

What is the probability that the statement $(p \lor q) \Rightarrow (r \lor s)$ is true?

$\boxed{\frac{13}{16}}$

$\frac{13}{16}$

hard

1403_farvardin-Sabz_Riazi-21

What is the solution of $\lim_{x \to 1} \frac{\sqrt[3]{\sqrt[3]{x} -2} + 1}{\sqrt{\sqrt{x} + 3} - 2}$?

$\boxed{\frac{8}{9}}$

$\frac{8}{9}$

hard

1403_farvardin-Sabz_Riazi-16

What is the sum of the left-hand and right-hand limits of the function $ f(x) = \frac{x-2}{x^2-\lfloor x^2 \rfloor} $ at the point $ x = 2 $?

$\boxed{\frac{1}{4}}$

$\frac{1}{4}$

easy

1403_ordibehesht-Riazi-15

What is the sum of the roots of the equation $(\sqrt[3]{x} - 1 + \sqrt[3]{x^{-1}})(\sqrt[3]{x} + 1) = 6x\sqrt[3]{x^2}$?

$\boxed{\frac{1}{6}}$

$\frac{1}{6}$

hard

1403_farvardin-Sabz_Riazi-4

What is the sum of the solutions of the equation $ \cos 2x + \sin^2 x = 0 $ in the interval $ [-3\pi, \pi] $?

$\boxed{-4\pi}$

$-4\pi$

easy

1403_ordibehesht-Riazi-14

What is the sum of the values of $m$ for which the line $y = mx$ is tangent to the circle $x^2 + y^2 - 10x - 10y + 49 = 0$?

$\boxed{\frac{25}{12}}$

$\frac{25}{12}$

hard

1403_ordibehesht-Sabz_Riazi-36

What is the value of the following expression when $x = \frac{\pi}{12}$? $$ 3 \cos 4x + \sqrt{2} \sin x - \sqrt{2} \cos x $$

$\boxed{\frac{1}{2}}$

$\frac{1}{2}$

medium

1403_ordibehesht-Riazi-12

What is the value of the relative minimum of the function $ y = x^3 - 12x + 2 $?

$\boxed{-14}$

$-14$

easy

1403_ordibehesht-Tajrobi-127