SAT 1000 problems series - Upcoming frequently

SAT1000 - P159

If $f(x)$ and $g(x)$ are functions defined at $\mathbb R$, and equation $x-f(g(x))=0$ has real roots.

Then which cannot be the function $g(f(x))$?

$A.\ x^2+x-\dfrac{1}{5}$

$B.\ x^2+x+\dfrac{1}{5}$

$C.\ x^2-\dfrac{1}{5}$

$D.\ x^2+\dfrac{1}{5}$

SAT1000 - P592

As shown above, a $5 \times 5 \times 5$ cube has been punched through so that there will always be two $1 \times 1$ square holes when looking from up, front and left.

Find the surface area of this solid. (Including the interior part)

SAT1000 - P158

There exist function $f(x)$ such that $\forall x \in \mathbb R$, the property follows:

$A.\ f(\sin 2x)=\sin x$

$B.\ f(\sin 2x)=x^2+x$

$C.\ f(x^2+1)=|x+1|$

$D.\ f(x^2+2x)=|x+1|$

SAT1000 - P333

Given that $f(x)=\cos 2x, g(x)=\sin x$.

Find all possible real number $a$ and positive integer $n$, so that $f(x)+a g(x)=0$ has exactly $2013$ roots on the interval $(0,n \pi)$.

How to submit:

• First, find the number of all possible solutions $(a,n)$. Let $N$ denote the number of solutions.

• Then sort the solutions by $a$ from smallest to largest, if $a$ is the same, then sort by $n$ from smallest to largest.

• Let the sorted solutions be: $(a_1,n_1), (a_2,n_2), (a_3,n_3), \cdots ,(a_N,n_N)$, then $M=\displaystyle \sum_{k=1}^N k(a_k+n_k)$.

For instance, if the solution is: $(-1,2), (-1,1), (1,3), (0,4)$

Then the sorted solution will be: $(-1,1), (-1,2), (0,4), (1,3)$

Then $N=4$, $M=\displaystyle \sum_{k=1}^4 k(a_k+n_k)= 1 \times (-1+1) + 2 \times (-1+2) + 3 \times (0+4) + 4 \times (1+3) =30$.

For this problem, submit $\lfloor M+N \rfloor$.

SAT1000 - P966

If function $f(x)=x-\dfrac{1}{3} \sin 2x + a \sin x$ is monotonic increasing for all $x \in \mathbb R$, what's the range of $a$?

$A.\ [-1,1]$

$B.\ [-1, \dfrac{1}{3}]$

$C.\ [-\dfrac{1}{3}, \dfrac{1}{3}]$

$D.\ [-1, -\dfrac{1}{3}]$

SAT1000 - P594

As shown above, in triangular pyramid $A-BCD$, the cross section $AEF$ passes through the center of the inscribed sphere $O$ (i.e. The sphere which is tangent to all of the faces of the solid) of $A-BCD$, and it intersects with $BC, DC$ at point $E,F$ respectively.

If the cross section divides the pyramid into two parts whose volumes are equal, $S_1$ is the surface area of solid $A-BEFD$, $S_2$ is the surface area of solid $A-EFC$, what is always true for $S_1$ and $S_2$?

SAT1000 - P759

As shown above, the line: $x-3y+m=0\ (m \neq 0)$ intersects with the two asymptotes of the hyperbola:

$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\ (a>0,b>0)$ at point $A,B$.

If point $P$ is at $(m,0)$ and $|PA|=|PB|$, find the eccentricity of the hyperbola.

Let $E$ denote the eccentricity, submit $\lfloor 1000E \rfloor$.

SAT1000 - P760

As shown above, $F_1, F_2$ are left and right focus of the hyperbola: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\ (a,b>0)$ respectively, and $B(0,b)$.

Line $F_1B$ intersects with the two asymptotes of the hyperbola at $P,Q$, and the perpendicular bisector of $PQ$ intersects with x-axis at point $M$.

If $|MF_2|=|F_1F_2|$, then find the eccentricity of the hyperbola.

Let $E$ denote the eccentricity, submit $\lfloor 1000E \rfloor$.

SAT1000 - P765

Let $A,B$ be the end points of the major axis of the ellipse $C:\dfrac{x^2}{3}+\dfrac{y^2}{m}=1$.

If there exists point $M$ on the ellipse so that $\angle AMB = \dfrac{2\pi}{3}$, find the range of $m$.

These pictures show the two cases:

SAT1000 - P767

As shown above, the hyperbola: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\ (a>0,b>0)$ has right focus point $F$, right vertex $A$.

Line $l_1$ passes through $F$ and it intersects with the hyperbola at point $B,C$, $l_2$ passes through $B$ and $l_2 \perp AC$, $l_3$ passes through $C$ and $l_2 \perp AB$, $l_2, l_3$ intersects at point $D$.

If the distance from $D$ to line $l_1$ is less than $a+\sqrt{a^2+b^2}$, what is the range of the slope of the hyperbola's asymptotes?

SAT1000 - P766

As shown above, the left and right focus of the ellipse: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a>b>0)$ are $F_1(-c,0), F_2(c,0)$.

If there exists point $P$ such that $\dfrac{a}{\sin \angle PF_1F_2}=\dfrac{c}{\sin \angle PF_2F_1}$, find the range of the eccentricity of the ellipse.

The range can be expressed as $(l,r)$. Submit $\lfloor 1000(2r-l) \rfloor$.

SAT1000 - P606

As shown above, in $\triangle ABC$, $AB=BC=2$, $\angle ABC = \dfrac{2\pi}{3}$.

If $P$ is outside plane $ABC$ and point $D$ is on segment $AC$, so that $PD=DA, PB=BA$, then find the maximum volume of pyramid $PBCD$.

Let $V$ denote the volume of $PBCD$, submit $\lfloor 10000V \rfloor$.

SAT1000 - P648

As shown above, in pyramid $ABCD$, $AD \perp BC$, if $AD=BC=2$, $AB+BD=AC+CD=4$, then find the maximum volume for pyramid $ABCD$.

Let $V$ denote the maximum volume. Submit $\lfloor 1000V \rfloor$.

SAT1000 - P787

As shown above, the focus of the parabola $C: y^2=2x$ is $F$. Two lines $l_1, l_2$ which are parallel to the x-axis intersect with $C$ at $A,B$ and intersect with the directix at $P,Q$. $M$ is the midpoint of $AB$.

If $S_{\triangle PQF}=2S_{\triangle ABF}$, find the locus of point $M$.

If the locus can be expressed as $f(x,y)=0$, then when $y=50$, submit the sum of all possible value(s) for $x$.

Note: $S_{\triangle ABC}$ denotes the area of $\triangle ABC$.

SAT1000 - P801

As shown above, the ellipse $E$ has equation: $\dfrac{x^2}{16}+\dfrac{y^2}{12}=1$, and point $C$ is the center of the circle: $x^2+y^2-4x+2=0$ If $P$ is a point on ellipse $E$, $l_1, l_2$ both pass through $P$ and they are both tangent to circle $C$, and the product of the slope of $l_1, l_2$ is equal to $\dfrac{1}{2}$, then find the all possible coordinate(s) of point $P$.

How to submit:

• First, find the number of all possible solutions. Let $N$ denote the number of solutions.

• Then Sort the solutions by x-coordinate from smallest to largest, if the x-coordinate is the same, then sort by y-coordinate from smallest to largest.

• Let the sorted solutions be: $(x_1,y_1), (x_2,y_2), (x_3,y_3), \cdots ,(x_N,y_N)$, then $M=\displaystyle \sum_{k=1}^N k(x_k+y_k)$.

For instance, if the solution is: $(-1,2), (-1,1), (1,3), (0,4)$

Then the sorted solution will be: $(-1,1), (-1,2), (0,4), (1,3)$

Then $N=4$, $M=\displaystyle \sum_{k=1}^4 k(x_k+y_k)= 1 \times (-1+1) + 2 \times (-1+2) + 3 \times (0+4) + 4 \times (1+3) =30$.

For this problem, submit $\lfloor 1000(M+N) \rfloor$.

SAT1000 - P802

As shown above, the parabola $C_1: x^2=y$, circle $C_2: x^2+{(y-4)}^2=1$, and $M$ is the center of circle $C_2$.

Point $P$ is a point on $C_1$ (not at $(0,0)$), and $l_1, l_2$ are two lines tangent to $C_2$ and they intersects with $C_1$ at point $A,B$ respectively. Line $l$ passes through $M$ and $P$.

If $l \perp AB$, find the equation of line $l$.

The equation can be expressed as: $y=\pm k x+b\ (k>0)$. Submit $\lfloor 1000(k+b) \rfloor$.

Can you do some SAT algebra questions? @Alice Smith

SAT1000 - P156

Let $f_1(x)=x^2, f_2(x)=2(x-x^2), f_3(x)=\dfrac{1}{3}|\sin 2\pi x|$.

If $I_k=\displaystyle \sum_{i=1}^{99} |f_k(\dfrac{i}{99})-f_k(\dfrac{i-1}{99})|$, compare $I_1, I_2, I_3$.

$A.\ I_1<I_2<I_3$

$B.\ I_2<I_1<I_3$

$C.\ I_1<I_3<I_2$

$D.\ I_3<I_2<I_1$

SAT1000 - P370

Given that in $\triangle ABC$, $\sin 2A+\sin(A-B+C)=\sin(C-A-B)+\dfrac{1}{2}$, $S$ is the area of $\triangle ABC$, $1 \leq S \leq 2$, let $a,b,c$ be the opposite side of angle $A,B,C$ respectively.

Which inequality always holds?

$A.\ bc(b+c)>8$

$B.\ ab(a+b)>16 \sqrt{2}$

$C.\ 6 \leq abc \leq 12$

$D.\ 12 \leq abc \leq 24$

SAT1000 - P527

Given that $a,b \in \mathbb R, a+b=2, b>0$, then find the minimum value of $\dfrac{1}{2|a|}+\dfrac{|a|}{b}$.

Let $M$ be the minimum value. Submit $\lfloor 1000M \rfloor$.

SAT1000 - P548

Given that $x,y \in \mathbb R$ and $x^2+y^2 \leq 1$, find the minimum value of $|2x+y-2|+|6-x-3y|$.

Let $M$ be the minimum value. Submit $\lfloor 1000M \rfloor$.

SAT1000 - P807

As shown above, parabola $C$ has equation: $y^2=4x$ and its focus is $F$.

Line $l_1, l_2$ both passes through $F$, $l_1$ intersects with $C$ at $A,B$, $l_2$ intersects with $C$ at $D,E$, $l_1 \perp l_2$.

Then find the minimum value of $\overrightarrow{AD} \cdot \overrightarrow{EB}$.

Let $M$ be the minimum value. Submit $\lfloor 1000M \rfloor$.

SAT1000 - P808

As shown above, the ellipse has equation: $\dfrac{x^2}{4}+y^2=1$.

If line $l$ passes through point $C(m,0)$ and is tangent to circle: $x^2+y^2=1$.

$l$ intersects with the ellipse at point $A,B$, then find the maximum value of $|AB|$.

Let $M$ be the maximum value. Submit $\lfloor 1000M \rfloor$.

SAT1000 - P810

As shown above, the ellipse $C: \dfrac{x^2}{4}+\dfrac{y^2}{3}=1$, $O(0,0), P(2,1)$, line $l$ intersects with $C$ at point $A,B$, and line $OP$ bisects segment $AB$.

If the area of $\triangle APB$ reaches the maximum value, then find the equation of line $l$.

The equation of the line can be expressed as $y=kx+b$, submit $\lfloor 1000(k+b) \rfloor$.

SAT1000 - P811

As shown above, given that ellipse $E: \dfrac{x^2}{t}+\dfrac{y^2}{3}=1\ (t>3)$, point $A$ is the left vertex of $E$, and line $l$ whose slope is $k\ (k>0)$ intersects with $E$ at point $A,M$, point $N$ is a point on $E$ such that $MA \perp NA$.

If $2|AM|=|AN|$, find the range of $k$ as $t$ changes.

If the range can be expressed as $(l,r)$, submit $\lfloor 1000(2r-l) \rfloor$.

SAT1000 - P819

As shown above, given that $F(1,0)$ is the right focus of the ellipse: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\ (a>b>0)$, $O$ is the origin.

If for all lines passing through $F$ which intersect with the ellipse at point $A,B$, the following inequality always holds:

$|OA|^2+|OB|^2<|AB|^2$

Then find the range of $a$.

If the range can be expressed as: $(l,+\infty)$, submit $\lfloor 1000l \rfloor$.

SAT1000 - P296

Find the range of the function $f(x)=\dfrac{\sin x-1}{\sqrt{3-2\cos x-2\sin x}}\ (x \in [0,2\pi])$.

If the range can be expressed as $[l,r]$, submit $\lfloor 1000(2r-l) \rfloor$.

SAT1000 - P320

Given the function $f(x)= \sin(\omega x+\phi)\ (\omega>0, |\phi| \leq \dfrac{\pi}{2})$, $f(-\dfrac{\pi}{4})=0, f'(\dfrac{\pi}{4})=0$, and $f(x)$ is strictly monotone on the interval $(\dfrac{\pi}{18}, \dfrac{5\pi}{36})$, then find the maximum value of $\omega$.

SAT1000 - P321

Given the function $f(x)=\sin(\omega x + \dfrac{\pi}{3})\ (\omega>0)$, $f(\dfrac{\pi}{6})=f(\dfrac{\pi}{3})$.

If $f(x)$ has minimum value but no maximum value on the interval $(\dfrac{\pi}{6},\dfrac{\pi}{3})$, then find the value of $\omega$.

If $\omega=\dfrac{a}{b}$, where $a,b$ are positive coprime integers. submit $a+b$.

SAT1000 - P322

Given the function $f(x)=\sin^2 \dfrac{\omega x}{2}+\dfrac{1}{2} \sin \omega x - \dfrac{1}{2}\ (\omega>0)$, $x \in \mathbb R$.

If $f(x)=0$ has no roots for $x \in (\pi, 2\pi)$, then find the range of $\omega$.

$A.\ (0,\dfrac{1}{8}]$

$B.\ (0,\dfrac{1}{4}] \cup [\dfrac{5}{8},1]$

$C.\ (0,\dfrac{5}{8}]$

$D.\ (0,\dfrac{1}{8}] \cup [\dfrac{1}{4},\dfrac{5}{8}]$

SAT1000 - P838

As shown above, the ellipse has equation: $x^2+\dfrac{y^2}{2}=1$, $F(0,1)$, line $l: y=-\sqrt{2} x + 1$ intersects with the ellipse at point $A,B$, and $P$ is a point on the ellipse so that $\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OP}=\textbf 0$.

Point $P,Q$ are symmetry about point $O$, and it turns out $A,P,B,Q$ are on the same circle.

The radius of the circle is $r$. Submit $\lfloor 1000r \rfloor$.

SAT1000 - P839

As shown above, point $F$ is the focus of the parabola $C: y^2=4x$, and line $l$ passing through $K(-1,0)$ intersects with $C$ at point $A,B$, and $A,D$ are symmetry about the x-axis.

If $\overrightarrow{FA} \cdot \overrightarrow{FB}=\dfrac{8}{9}$, then find the equation of the incircle of $\triangle BDK$.

The equation can be expressed as: $(x-a)^2+(y-b)^2=r^2$. Submit $\lfloor 1000(a+b+r) \rfloor$.

SAT1000 - P842

As shown above, the ellipse has equation: $\dfrac{x^2}{4}+\dfrac{y^2}{2}=1$, and line $l$ passing through $P(0,1)$ intersects with the ellipse at point $A,B$.

Then there exists a fixed point $Q$ so that the following equation always holds as $l$ rotates:

$\dfrac{|QA|}{|QB|}=\dfrac{|PA|}{|PB|}$

Then find the coordinate of $Q$.

The coordinate of $Q$ is $(x_0,y_0)$. Submit $\lfloor 1000(2y_0-x_0) \rfloor$.

SAT1000 - P843

As shown above, the ellipse has equation: $\dfrac{x^2}{2}+y^2=1$, line $l: y=k_1 x-\dfrac{\sqrt{3}}{2}$ intersects with the ellipse at point $A,B$.

Point $C$ is on the ellipse and line $OC$ has slope $k_2$, $k_1 k_2=\dfrac{\sqrt{2}}{4}$.

$M$ is a point on ray $OC$, $|MC| : |AB| = 2 : 3$, and the radius of circle $M$ is $|MC|$, $OS,OT$ are two tangent lines of circle $M$ and $S,T$ are tangent points.

Then find the maximum value of $\angle SOT$ (in radians), and find the slope of $l$ when $\angle SOT$ reaches the maximum.

Let $\theta$ be the maximum value of $\angle SOT$ (in radians), $k$ is the slope of $l$. Submit $\lfloor 1000(\theta+|k|)\rfloor$.

SAT1000 - P844

As shown above, $F(1,0)$ is the focus of the parabola $C: y^2=4x$, line $l: y=k(x-1)$ intersects with $C$ at point $A,B$, and the perpendicular bisector of $AB$ intersects with $C$ at point $M,N$.

If $A,M,B,N$ are on the same circle, then find the value of $\lfloor 1000|k| \rfloor$.

SAT1000 - P845

As shown above, the parabola $E: y^2=x$ and circle $M: (x-4)^2+y^2=r^2$ for $r>0$ intersect at points $A$, $B$, $C$, and $D$; and their relative positions are shown in the figure. Lines $AC$ and $BD$ intersect at point $P$.

Find the coordinates $(x_0,0)$ of $P$ as the area of quadrilateral $ABCD$ reaches the maximum when $r$ varies. Submit $\lfloor 1000x_0 \rfloor$.

The coordinate of point $P$ is $(x_0,0)$. Submit $\lfloor 1000x_0 \rfloor$.

SAT1000 - P846

As shown above, $ON, NM$ are rigid rods and $DN=ON=1,MN=3$, $O(0,0)$ is fixed on the coordinate plane, and $D$ is restricted along the x-axis. Then as $D$ moves horizontally, point $M$ will rotate around point $O$. Curve $C$ is the locus of point $M$.

If line $l$ intersects with $l_1:x-2y=0$ at point $P$, $l_2:x+2y=0$ at point $Q$, and $l$ is tangent to curve $C$.

Then find the minimum area of $\triangle OPQ$ when line $l$ moves and rotates.

Let $S$ be the minimum area. Submit $\lfloor 1000S \rfloor$.

SAT1000 - P491

Given that $f(x)=\dfrac{1}{1+x}$. $\{a_n\}$ is a sequence whose terms are all positive so that $a_1=1, a_{n+2}=f(a_n)$.

If $a_{2010}=a_{2012}$, find the value of $\lfloor 1000(a_{20}+a_{11}) \rfloor$.

SAT1000 - P492

$\{a_n\}$ is a sequence such that $a_{n+1}+(-1)^n a_n=2n-1$, then find $\displaystyle \sum_{k=1}^{60} a_k$.

SAT1000 - P496

Given that $a_n=n^2(\cos^2 \dfrac{n\pi}{3}-\sin^2 \dfrac{n\pi}{3})\ (n \in \mathbb N^+)$, let $S_n=\displaystyle \sum_{k=1}^{n} a_k$.

$b_n=\dfrac{S_{3n}}{n \cdot 4^n}\ (n \in \mathbb N^+)$, $T_n=\displaystyle \sum_{k=1}^{n} b_k$.

$T_{10} = \dfrac{p}{q}$, where $p,q$ are positive coprime integers.

Submit $\lfloor p-q+2(S_{100}+S_{201}+S_{302}) \rfloor$.

SAT1000 - P500

$\{a_n\}$ is a sequence such that $a_1=m\ (m \in \mathbb N^+)$, $a_{n+1}=\begin{cases} \dfrac{a_n}{2} ,\ a_n \equiv 0 \pmod{2} \\ 3 a_n+1 ,\ a_n \equiv 1 \pmod{2} \end{cases}$ If $a_6=1$, find the sum of all possible value(s) for $m$.

SAT1000 - P461

Given that $\{a_n\}$ is a Geometric Sequence, its common ratio $q=\sqrt{2}$, $S_n=\displaystyle \sum_{k=1}^n a_k$.

Let $T_n=\dfrac{17 S_n - S_{2n}}{a_{n+1}}\ (n \in \mathbb N^+)$. If $T_{m}$ is the maximum term of sequence $\{T_n\}$, find $m$.

SAT1000 - P153

$f(x)$ is a function defined at $[0,1]$ such that:

• $f(0)=f(1)=0$.

• $\forall x,y \in [0,1]\ (x \neq y), |f(x)-f(y)| < \dfrac{1}{2} |x-y|$.

If $\forall x,y \in [0,1], |f(x)-f(y)|<k$, find the minimum value of $k$.

Let $K$ be the minimum value. Submit $\lfloor 1000K \rfloor$.

SAT1000 - P329

As shown above, the circle has radius $r=1\ m$, $O$ is its center. At $t=0$, $O$ is at $(0,-1)$, and it is moving at $v=1\ m/s$ upwards along the y-axis. Let $x(t)$ be the length of the arc above the x-axis, $f(t)=\cos (x(t))$.

For $0 \leq t \leq 1$, what is the best graph for $f(t)$?

SAT1000 - P275

Without using calculator, Find $\dfrac{\cos 10 \degree}{\tan 20 \degree}+\sqrt{3} \sin 10 \degree \tan 70 \degree - 2 \cos 40 \degree$.

Let $A$ denote the answer. Submit $\lfloor 1000A \rfloor$.

SAT1000 - P276

If $\tan \alpha=2 \tan \dfrac{\pi}{5}$, find $\dfrac{\cos(\alpha-\dfrac{3\pi}{10})}{\sin(\alpha-\dfrac{\pi}{5})}$.

Let $A$ denote the answer. Submit $\lfloor 1000A \rfloor$.

SAT1000 - P277

Without using calculator, Find $4 \cos 50 \degree - \tan 40 \degree$.

Let $A$ denote the answer. Submit $\lfloor 1000A \rfloor$.

SAT1000 - P866

For all real number $x,y$, which is always true?

$A.\ \lfloor -x \rfloor=-\lfloor x \rfloor$

$B.\ \lfloor 2x \rfloor=2\lfloor x \rfloor$

$C.\ \lfloor x+y \rfloor \leq \lfloor x \rfloor + \lfloor y \rfloor$

$D.\ \lfloor x-y \rfloor \leq \lfloor x \rfloor - \lfloor y \rfloor$

SAT1000 - P873