Chủ đề này có 7 trả lời

[DarkBlood]

Question 1. Write $2013$ as a sum of $m$ prime number. The smallest value of $m$ is:

$A.\ 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B.\ 3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C.\ 4 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D.\ 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E.$ None of the above.

 

 

Question 2. How many natural numbers $n$ are there so that $n^2+2014$ is a perfect square?

$A.\ 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B.\ 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C.\ 3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D.\ 4 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E.$ None of the above.

 

 

Question 3. The largest integer not exceeding $[(n+1)\alpha]-[n\alpha],$ where $n$ is a natural number, $\alpha =\frac{\sqrt{2013}}{\sqrt{2014}},$ is:

$A.\ 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B.\ 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C.\ 3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D.\ 4 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E.$ None of the above.

 

 

Question 4. Let $A$ be an even number but not divisor by 10. The last two digits of $A^{20}$ are:

$A.\ 46\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B.\ 56 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C.\ 66 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D.\ 76 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E.$ None of the above.

 

 

Question 5. The number of integer solutions $x$ of the equation below.

$(12x-1)(6x-1)(4x-1)(3x-1)=330$

$A.\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B.\ 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C.\ 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D.\ 3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E.$ None of the above.

 

 

Question 6. Let $ABC$ be a triangle with area $1\ cm^2.$ Points $D,\ E,\ F$ lie on the sides $AB,\ BC,\ CA,$ respectively. Prove the

$min\left \{ S_{ADF};\ S_{BED};\ S_{CEF} \right \}\leq \frac{1}{4}\ cm^2$

 

 

Question 7. Let $ABC$ be a triangle with $\widehat{A}=90^{\circ},$ $\widehat{B}=60^{\circ},\ BC=1\ cm.$ Draw outside of $\bigtriangleup ABC,$ three equilateral triangle $ABD,\ ACE,\ BCF.$ Determine the area of $\bigtriangleup DEF.$

 

 

Question 8. Let $ABCDE$ be a convex pentagon. Given that $S_{ABC}= S_{BCD}=S_{CDE}= S_{DEA}= S_{EAB}=2\ cm^2.$ Find $S_{ABCDE}.$

 

 

Question 9. Solve the following  system in positive numbers

$\left\{\begin{matrix} x+y\leq 1\\ \frac{2}{xy}+\frac{1}{x^2+y^2}=10 \end{matrix}\right.$

 

 

Question 10. Consider the set of all rectangles with a given perimeter $p.$ Find the largest value of

$M=\frac{S}{2S+p+2}$

Where $S$ is denoted the area of the rectangle.

 

 

Question 11.  The positive numbers $a,\ b,\ c,\ d,\ e$ are such that the following identidy hold for all real number $x.$

$(x+a)(x+b)(x+c)=x^3+3dx^2+3x+e^3$

Find the smallest value of $d.$

 

 

Question 12. If $f(x)=ax^2+bx+c$ safisfies the condition

$|f(x)|<1,\ \forall\ x\in [-1,\ 1]$

Prove the the equation $f(x)=2x^2-1$ has two real roots.

 

 

Question 13. Solve the system of equations

$\left\{\begin{matrix} \frac{1}{x}+\frac{1}{y}=\frac{1}{6}\\ \frac{3}{x}+\frac{2}{y}=\frac{5}{6} \end{matrix}\right.$

 

 

Question 14. Solve the system of equations

$\left\{\begin{matrix} x^3+y=x^2+1\\ 2y^3+z=2y^2+1\\ 3z^3+x=3z^2+1 \end{matrix}\right.$

 

 

Question 15. Denote by $\mathbb{Q}$ and $\mathbb{N}^*$ the set of all rational and positive integer numbers, respectively. Suppose the $\frac{ax+b}{x}\in \mathbb{Q}$ for every $x\in \mathbb{N}^*.$ Prove that there exist integers $A,\ B,\ C$ such that

$\frac{ax+b}{x}=\frac{Ax+B}{Cx}$ for all $x\in \mathbb{N}^*$

 

___________

P/s: Do đi thi không được mang đề về, mình phải chép lại, có sai sót gì mong các bạn thông cảm.

Mấy bài trắc nghiệm giải ra luôn nha

Bài viết đã được chỉnh sửa nội dung bởi Hoang Huy Thong: 24-03-2013 - 18:44

[BlackSelena]

Nhân đây mình cũng nhắc luôn, vì đây là cuộc thi bằng tiếng Anh nên các bạn hãy trình bày lời giải bằng tiếng Anh nhé, coi như là luyện tập !

[duaconcuachua98]

Question $13$:

We have $\left\{\begin{matrix} \frac{1}{x}+\frac{1}{y}=\frac{1}{6} & \\ \frac{3}{x}+\frac{2}{y}=\frac{5}{6} & \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} \frac{1}{x}+\frac{1}{y}=\frac{1}{6} & \\ \frac{2}{x}+\frac{2}{y}+\frac{1}{x}=\frac{5}{6} & \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} \frac{1}{x}=\frac{3}{6} & \\ \frac{1}{y}=\frac{-2}{6} & \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} x=2 & \\ y=-3 & \end{matrix}\right.$

[ilovelife]

Q14:
$\left\{\begin{matrix} x^3+y=x^2+1\\ 2y^3+z=2y^2+1\\ 3z^3+x=3z^2+1 \end{matrix}\right.
\iff
\left\{\begin{matrix} x^3-x^2=1-y\\
2y^3-2y^2=1-z\\
3z^3-3z^2=1-x \end{matrix}\right.$

Case 1: $x=1 \implies y=z=1$

Case 2: $x>1 \implies 1-y=x^3-x^2>0 \iff 1>y  \\ \implies 2(y^3-y^2)=1-z<0 \iff z>1 \implies 1-x>0 \iff x<1 \implies \text{contradiction}$

Case 3: $x<1$, similar to Case 1, we get the contradiction

So, $x=y=z=1$

 

Q12:From the condition, we have:

$\left\{\begin{matrix}
|f(1)|=|a+b+c|<1\\
|f(-1)|=|a-b+c|<1\\
|f(0)|=|c|<1
\end{matrix}\right.$

$\implies 2>|f(1)|+|f(-1)|\ge 2|a+c| \iff |a+c|<1\ (1)$

Case 1: $0>c>-1 \implies a<2 \implies (a-2)(c+1)<0\ (1')$

Case 2: $1>c\ge 0 \implies a<1 \implies (a-2)(c+1)<0\ (1'')$

We also have: $f(x)=2x^2-1 \iff (a-2)x^2+bx + (c+1)=0\ (2)$

Combine (1),(1'),(1") and (2) $\implies f(x)=2x^2-1$ has 2 real root as required 

Tiếng anh mình không giỏi đâu

Bài viết đã được chỉnh sửa nội dung bởi ilovelife: 24-03-2013 - 20:32

[namcpnh]

Có đề THPT không em?

[buiminhhieu]

Q9: we have $\frac{2}{xy}+\frac{1}{x^{2}+y^{2}}=\frac{1}{2xy}+\frac{1}{x^{2}+y^{2}}+\frac{3}{2xy}\geq \frac{4}{(x+y)^{2}}+\frac{3}{2.\frac{(x+y)^{2}}{4}}\geq 10$

Dấu = xảy ra $x=y=$$\frac{1}{2}$

[Nguyen Duc Thuan]

Senior Section

 

Question 1. How many three-digit perfect squares are there

such that if each digit is increased by one, the resulting number

is also a perfect square?

(A): 1; (B): 2; (C): 4; (D): 8; (E) None of the above.

 

Question 2. The smallest value of the function

$f(x) = \left | x \right |+\left | \frac{1-2013x}{2013-x} \right |$

where $x\in \left [ -1;1 \right ]$ is:

(A):$\frac{1}{2012}$       B: $\frac{1}{2013}$             C:$\frac{1}{2014}$           D: $\frac{1}{2015}$            (E): None of the above.                           

 

 Question 3. What is the largest integer not exceeding $8x^2+6x-1$

where $x=\frac{1}{2}\left ( \sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}} \right )$

(A): 1; (B): 2; (C): 3; (D): 4; (E) None of the above.

 

 

Question 4. Let $x_0=[\alpha ];x_1=[2\alpha ]-[\alpha ];x_2=[3\alpha ]-[2\alpha ];x_3=[4\alpha ]-[\alpha ]...$ 

where $\alpha =\frac{\sqrt{2013}}{\sqrt{2014}}$ .The value of $x_9$ is:

(A): 2; (B): 3; (C): 4; (D): 5; (E): None of the above.

 

Question 5. The number n is called a composite number if it

can be written in the form n = ab; where a; b are positive

integers greater than 1.

Write number 2013 in a sum of m composite numbers. What

is the largest value of m?

(A): 500; (B): 501; (C): 502; (D): 503; (E): None of the

above.

 

Em mới có bấy nhiêu, mn thông cảm!

[Zaraki]

Đáp án phần thi Junior và Senior

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