125- In the figure, the side length of the square is 2 units. Two equilateral triangles are constructed on two adjacent sides of the square. What is the area of triangle $ABC$?
[Figure: A square with two equilateral triangles constructed on adjacent sides, forming triangle ABC]
$\sqrt{6}\ (1$ $1+\sqrt{3}\ (2$ $2+\sqrt{3}\ (3$ $4\ (4$

126- An equilateral triangle has a side length of 4 units, which is the diameter of a square. What is the shortest distance from the other vertices of the rectangle to the side of this triangle?
$2-\sqrt{3}\ (1$ $\sqrt{3}-1\ (2$ $\dfrac{1}{2}\sqrt{3}\ (3$ $1\ (4$

127- Inside a regular quadrilateral solid with lateral edge $\sqrt{6}$ units and base edge $2\sqrt{6}$ units, the largest possible sphere is placed. What is the radius of this sphere in units?
$1\ (1$ $\dfrac{4}{3}\ (2$ $\dfrac{3}{2}\ (3$ $2\ (4$

128- In trapezoid $ABCD$, the perpendicular bisectors of sides $AB$ and $CD$ intersect at point $M$. If $BC > AD$, which of the following inequalities is correct?
$A\hat{M}B > B\hat{M}C\ (1$ $C\hat{A}B > C\hat{A}D\ (2$ $B\hat{M}C > A\hat{M}D\ (3$ $C\hat{M}D > A\hat{M}B\ (4$

129- In the figure, $O$ is the orthocenter of triangle $ABC$. The angle $A\hat{O}D$ equals which of the following?
[Figure: Triangle ABC inscribed in a circle, with O as orthocenter and D a point on the circle]
$O\hat{B}C\ (1$ $C\hat{A}D\ (2$ $O\hat{A}C\ (3$ $A\hat{D}O\ (4$
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132- Point O and line d are outside plane P. In which case does there exist exactly one line passing through point O that is parallel to plane P and intersects line d?
(1) $d \subset P$ (2) $d \parallel P$ (3) $d \cap P \neq \phi$ (4) The plane passing through O and d is parallel to P

137- Point $S(2,1)$ is the vertex of a parabola whose axis of symmetry is parallel to the $x$-axis, and it passes through point $(5,0)$. What is the equation of its directrix?
(1) $y = \dfrac{1}{4}$ (2) $y = \dfrac{1}{2}$ (3) $y = \dfrac{3}{4}$ (4) $y = \dfrac{3}{2}$
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138- By rotating the coordinate axes by an appropriate angle, the equation of the hyperbola $\sqrt{3}\,xy + y^2 = 1$ is written in which form?
(1) $3x^2 - y^2 = 2$ (2) $2x^2 - 3y^2 = 2$ (3) $3x^2 + y^2 = 2$ (4) $2x^2 + 3y^2 = 2$

143- Which natural number satisfies the statement ``every natural number can be written as the sum of at most how many consecutive numbers''?
(1) $56$ (2) $64$ (3) $72$ (4) $74$

144- How many ordered pairs $(a, b)$ can we choose with integer and positive coordinates, so that the sum of the first coordinates and the sum of the second coordinates of the two chosen pairs are both even?
(1) $3$ (2) $4$ (3) $5$ (4) $6$

149. If $A$ is the adjacency matrix of graph $G$, and the entries in row $\hat{n}$ and column $\hat{n}$ of matrix $A^2$ are $«4,4,2,2,2»$, then graph $G$ has how many cycles?
(1) $3$ (2) $4$ (3) $5$ (4) $6$

151. In dividing a natural number $a$ by a natural number $b$, the quotient is $21$ and the remainder is $37$. How many members of the set of answers that are multiples of $5$ does $a$ have?
(1) $1$ (2) $2$ (3) $3$ (4) $4$

161- In the figure, if the string, pulley, and surfaces are frictionless, what is the ratio $\dfrac{T}{T_1}$ of the tension forces? $\left(g = 10\,\dfrac{\text{m}}{\text{s}^2}\right)$
[Figure: A pulley fixed to the ceiling with string tension $T$. A mass of $10\,\text{kg}$ hangs from the pulley. Below it, string tension $T_1$ connects to two masses: $20\,\text{kg}$ on the left and $20\,\text{kg}$ on the right.]
(1) $1.5$
(2) $2$
(3) $2.5$
(4) $3$
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165. A 1 kg object is released in vacuum conditions and reaches the ground after 4 seconds. The work done by gravity in the third second of fall is how many Joules? ($g = 10\,\frac{m}{s^2}$)
(1) $150$ (2) $250$ (3) $400$ (4) $450$

166. Inside a container, 200 grams of ice is at $-10$ degrees Celsius. At least how many grams of water at $20^\circ$C must be added to melt all the ice? (Heat exchange occurs only between water and ice, $L_f = 336\,\frac{J}{g}$, $C_{\text{ice}} = 2\,\frac{J}{g \cdot k}$, $C_{\text{water}} = \frac{1}{2}$)
(1) $50$ (2) $200$ (3) $850$ (4) $1200$

167. In the figure, two rods each 50 cm long with equal cross-sections are connected together. When the thermal conductivity of aluminum is three times the thermal conductivity of iron, the temperature at the junction of the two rods is how many degrees Celsius?
[Figure: Iron rod (50 cm) and Aluminum rod (50 cm) connected, with water at $100^\circ$C on the iron side and water at $20^\circ$C on the aluminum side]
(1) $80$ (2) $40$ (3) $45$ (4) $30$
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168- The ($P-V$) diagram of a perfect gas that goes from state i to state f via three paths a, b, and c, according to the figure below. If the internal energy change of the gas is $\Delta u$ and the heat absorbed by the gas is $Q$, which relation is correct?
\begin{minipage}{0.35\textwidth} [Figure: P-V diagram showing three paths a, b, c from state i to state f, with P on vertical axis and V on horizontal axis] \end{minipage} \begin{minipage}{0.6\textwidth} (1) $Q_c > Q_b > Q_a > \circ$
(2) $Q_a > Q_b > Q_c > \circ$
(3) $\Delta u_a = \Delta u_b = \Delta u_c < \circ$
(4) $\Delta u_a = \Delta u_b = \Delta u_c = \circ$ \end{minipage}

169- If we reduce the cold source temperature of a heat engine that performs $100$ joules of work with a Carnot cycle by $20\%$, and increase $\eta$ by $20\%$, the temperature of the cold source of this engine is how many degrees Celsius?
(1) $500$ (2) $327$ (3) $300$ (4) $227$

170- The ($P-V$) diagram of one mole of a perfect monatomic gas is shown below. The heat exchanged by the gas with the environment in process abc is how many joules? $\left(R = 8\ \dfrac{\text{J}}{\text{mol.k}}\right)$
\begin{minipage}{0.4\textwidth} [Figure: P-V diagram with points a, b, c; P axis shows $\frac{5}{3}\times10^5$ and $10^5$ (pa); V axis shows 1, 3, 5 (lit); path goes from a to b horizontally at higher pressure, then from b to c vertically down] \end{minipage} \begin{minipage}{0.55\textwidth} (1) $1100$
(2) $3300$
(3) $\dfrac{1700}{3}$
(4) $\dfrac{2300}{3}$ \end{minipage}

171- According to the figure below, ray SI after reflecting from flat mirrors is reflected along path $I'R$. The angle $\beta$ is how many times the angle $\alpha$?
\begin{minipage}{0.4\textwidth} [Figure: Two flat mirrors meeting at an angle, with incident ray I making angle $\alpha$ with the lower mirror, ray reflecting off both mirrors, angle $\beta$ shown between rays at upper mirror, and final reflected ray $I'R$] \end{minipage} \begin{minipage}{0.55\textwidth} (1) $1$
(2) $2$
(3) $\dfrac{3}{2}$
(4) It depends on the angle of incidence of mirror (1). \end{minipage}

172- In a convex mirror, the object distance from the mirror is $75$ centimeters. If the focal length of the mirror is $20$ centimeters, the image length is how many times the object length?
(1) $3$ (2) $\dfrac{2}{3}$ (3) $\dfrac{1}{3}$ (4) $\dfrac{1}{4}$
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173- In the figure, a light ray refracts at point A in medium with refractive index $n_1$ to point B in medium with refractive index $n_2$, where $n_2$ is the refractive index of the second medium. If $AI = IB = L$ and the speed of light in the first medium is $V_1$, what is the travel time of light from A to B?
[Figure: Light ray traveling from point A through medium $n_1$, refracting at point I on surface N, then continuing through medium $n_2$ to point B]
\begin{flushright} (1) $\dfrac{L}{V_1}\!\left(1+\dfrac{n_2}{n_1}\right)$
(2) $\dfrac{L}{V_1}\!\left(1+\dfrac{n_1}{n_2}\right)$
(3) $\dfrac{2L}{V_1}\!\left(1-\dfrac{n_2}{n_1}\right)$
(4) $\dfrac{2L}{V_1}\!\left(1-\dfrac{n_1}{n_2}\right)$ \end{flushright}

174- A lens forms a real image of an object placed 15 cm in front of it, on a screen 3 cm from the lens. What is the focal length of the lens in centimeters?
\begin{flushright} (1) $40$ (2) $30$ (3) $15$ (4) $10$ \end{flushright}