Prove

Problem 1

Given a circle with center $O$ and diameter $MN$ , prove:
(i) $\angle MAN = 90^{\circ} + \angle PQR$ ,
(ii) $\angle QPR + 2 \times \angle MAN = 360^{\circ}$ .

See Solution

Problem 2

Prove that two lines are parallel if and only if alternate exterior angles are congruent: $\angle 1 \cong \angle 8$ .

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Problem 3

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

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Problem 4

Prove that in a given diagram, (i) $A B$ is parallel to $P Q$ and (ii) $M P \times A M = B M \times M Q$ .

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Problem 5

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

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Problem 6

Show that the roots of $m x^{2}+2 x+1=m$ are always real for any real constant $m$ .

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Problem 7

Show that the roots of the equation $m x^{2}+2 x+1=0$ are real for a real constant $m$ .

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Problem 8

Prove that if $B P$ and $P Q$ are tangents to circles, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

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Problem 9

1. Prove $n^{7}-n$ is divisible by 42 for all positive integers $n$ . Show primes ≠ 2, 5 divide numbers like 1, 11, etc.
2. Prove if $p>3$ is prime, then $p^{2} \equiv 1(\bmod 24)$ .
3. Find the number of trailing zeros in $1000!$ .
4. If $p$ and $p^{2}+2$ are primes, prove $p^{3}+2$ is prime.
5. Prove $\operatorname{gcd}(2^{a}-1,2^{b}-1)=2^{\operatorname{gcd}(a, b)}-1$ for positive integers $a, b$ .

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Problem 10

Prove that $n^{7}-n$ is divisible by 42 for all positive integers $n$ and that primes other than 2 or 5 divide infinitely many of $1, 11, 111, 1111,$ etc.

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Problem 11

Prove that for any positive integer $n$ , $n^{7}-n$ is divisible by 42. Also, show $p^{2} \equiv 1(\bmod 24)$ for primes $p>3$ .

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Problem 12

Prove that $n^{7}-n$ is divisible by 42 for all positive integers $n$ and $p>3$ prime, $p^{2} \equiv 1(\bmod 24)$ .

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Problem 13

Any two parallel lines lie in the same plane. True or False?

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Problem 14

Is it true or false that 'If two rays are parallel, then the lines containing them must be coplanar'?

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Problem 15

Prove $x^{2}-x+2 \geq \frac{7}{4}$ for all values of $x$ .

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Problem 16

Prove that for a sector with radius $r$ cm and perimeter 50 cm, $\theta = \frac{360}{\pi} \left(\frac{25}{r} - 1\right)$ .

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Problem 17

Une piscine rectangulaire de 10 m sur 15 m est entourée d'une bande de gazon de $x$ m. Montre que la clôture fait $50 + 8x$ m et l'aire des allées de gazon est $50x + 4x^2$ . Calcule pour $x=2$ m.

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Problem 18

Soit $n$ un entier. Trouve l'entier précédent et suivant de $n$ . Prouve que la somme de 3 entiers consécutifs est un multiple de 3.

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Problem 19

Définis $R$ en fonction de $x$ pour le programme donné. Prouve que $R$ est le carré de $x$ . Factorise $64-25 x^{2}-(8-5 x)^{2}$ et $49 x^{2}+28 x+4-25$ .

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Problem 20

Le mât mesure 3,6 m. Vérifie si les triangles VEN et VNT sont rectangles avec les longueurs 4,2 m et 3,9 m. Justifie.

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Problem 21

A ring of weight 8.5 N is on a string with tension $T$ . Show that $T \sin \theta = 12$ and $T \cos \theta = 3.5$ , then find $\theta$ .

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Problem 22

Prove that $\sin 3\theta + \sin \theta = 4 \sin \theta - 4 \sin^3 \theta$ .

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Problem 23

Find the derivative of $f(x) = x^2$ using the limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

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Problem 24

Nyatakan sama ada semua segi empat tepat adalah segi empat sama: benar atau palsu?

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Problem 25

Buat kesimpulan umum secara induksi untuk urutan nombor $-\frac{1}{2}, \frac{3}{2}, \frac{25}{6}, \ldots$ mengikut pola yang diberikan.

See Solution

Problem 26

Demuestra la ley de Boyle: $p V=$ const, la ley de Charles: $V / T=$ const, y la ley de Gay-Lussac: $p / T=$ const.

See Solution

Problem 27

A particle with mass $m_{1}$ and momentum $p_{1}$ collides elastically with mass $m_{2}$ at rest. Show $E_{3}$ and $p_{3}$ formulas.

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Problem 28

True or false: For a population proportion confidence interval, we use $\mathrm{z}$ -distribution, not $\mathrm{t}$ -distribution.

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Problem 29

Verify that $(A B)^{T} = B^{T} A^{T}$ for matrices $A=\begin{bmatrix}1 & 2 & -1 \\ 3 & 0 & 2 \\ 4 & 5 & 0\end{bmatrix}$ and $B=\begin{bmatrix}1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 1 & 3\end{bmatrix}$ .

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Problem 30

Find the value of $x$ if $\mathrm{m} \angle \mathrm{PQR}=6x-7$ and $\mathrm{m} \angle \mathrm{RQT}=20x+5$ sum to 180°.

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Problem 31

函数 $f(x)$ 定义在 $\mathbf{R}$ 上，满足 $f(x+y)=f(x) \cdot f(y)$ ，且存在 $x_{1} \neq x_{2}$ 使得 $f\left(x_{1}\right) \neq f\left(x_{2}\right)$ 。求 $f(0)$ 并证明 $f(x)>0$ 对任意 $x \in \mathbf{R}$ 成立。

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Problem 32

Prove that in cyclic quadrilateral $ABCD$ , the lines $XY$ and $ZW$ are parallel, where $X$ , $Y$ , $Z$ , and $W$ are feet of perpendiculars.

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Problem 33

Demonstrați că în triunghiul dreptunghic, lungimea medianei pe ipotenuză este $A O=\frac{B C}{2}$ .

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Problem 34

Si $a \in \mathbb{X}, b \in \mathbb{R}$ y $c \in \mathbb{R}$ , evalúa las proposiciones sobre $a$ , $b$ y $c$ .

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Problem 35

Sean A y B subconjuntos de un conjunto Re. ¿Cuáles son las afirmaciones verdaderas sobre ellos?

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Problem 36

Dadas las matrices cuadradas $A$ y $B$ de orden $n$ y una constante real $k$ , determina si las siguientes afirmaciones son verdaderas.

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Problem 37

Considerați punctele coliniare $M, O, N$ și punctele $P$ și $Q$ de o parte și de alta a dreptei $M N$ . Demonstrați: a) $M Q \equiv N P$ ; b) $M P \equiv N Q$ ; c) $\triangle M P Q \equiv \triangle N Q P$ ; d) $\triangle M P N \equiv \triangle N Q M$ .

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Problem 38

9) Sif estinationale. - Verifica si son verdaderas las siguientes afirmaciones para $a, b, c \in \mathbb{R}$ : a) $(a \leq b) \Rightarrow(a c \leq b c)$ b) $(a \leq b \wedge c>0) \Rightarrow(a c \geq b c)$ c) $(a b=0) \Rightarrow(a=0 \wedge b=0)$ d) $(a b=c) \Rightarrow(a=c \vee b=c)$ e) $(a \geq b \wedge c<0) \Rightarrow(a c \leq b c)$

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Problem 39

Compare the $15^{\text {th }}$ term of the arithmetic sequence $150,650,1150,1650,...$ and geometric sequence $4,12,36,108,...$ . Show your work.

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Problem 40

Une piscine rectangulaire de 10m x 15m a une bande de gazon de $x$ m. Montre que la clôture mesure $50 + 8x$ .

See Solution

Problem 41

Soit $n$ un entier. a) Trouve l'entier précédent $n$ . b) Trouve l'entier suivant $n$ . Prouve que la somme de 3 entiers consécutifs est un multiple de 3.

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Problem 42

For $n \geq 100$ , show that in two piles of cards numbered $n$ to $2n$ , at least one pile has two cards summing to a perfect square.

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Problem 43

Show that the limit $\operatorname{limit}_{x \rightarrow 2} \frac{|x-2|}{x-2}$ does not exist.

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Problem 44

Show that $p \Rightarrow(q \wedge r) \equiv(p \Rightarrow q) \wedge(p \Rightarrow r)$ is always true.

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Problem 45

Un voilier a un mât de $3,6 \mathrm{~m}$ avec un étai de $4,2 \mathrm{~m}$ et des haubans de $3,9 \mathrm{~m}$ . Est-ce que les triangles VEN et VNT sont rectangles ? Justifie.

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Problem 46

Is the transformation from $y=3^{x}$ to $y=5 \cdot 3^{-(x-2)}+9$ unique? List transformations to prove or disprove the claim.

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Problem 47

Show that if for each $x \in A$ , there exists an open set $U$ with $U \subset A$ , then $A$ is open in $X$ .

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Problem 48

In a coordinate plane, $\triangle ABC$ is a right triangle with $A(-4,1)$ and $B(2,1)$ . Area is 9 sq. units. Find point $C$ .

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Problem 49

In a coordinate plane, $\triangle ABC$ has $A(-4,1)$ and $B(2,1)$ with area 9 sq. units. Find other $C$ coordinates besides $(2,4)$ .

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Problem 50

Verify the $\mathrm{K}_{\mathrm{a}}$ value of benzoic acid given $\mathrm{H}_{3} \mathrm{O}^{+}$ concentration is $3.08 \times 10^{-3} \mathrm{M}$ in $0.150 \mathrm{M}$ solution.

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Problem 51

Derive the equation $2as = v^2 - u^2$ for final velocity $v$ , initial velocity $u$ , acceleration $a$ , and displacement $s$ .

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Problem 52

Derive the formula for final velocity: $V_f = V_i + a t$ .

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Problem 53

Prove the system $y' = A y$ is solvable when $A = \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}$ is upper triangular.

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Problem 54

Show that the system $\mathbf{y}^{\prime}=A \mathbf{y}$ can be solved directly for a $2 \times 2$ upper triangular matrix $A$ with constant entries. Use the form $\left[\begin{array}{l} x \\ y \end{array}\right]^{\prime}=\left[\begin{array}{ll} a & b \\ 0 & c \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]$ and solve the second equation first.

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Problem 55

Prove that $(3a+2b+3c)^{3}=(a+b)^{3}+(b+2c)^{3}+(c+2a)^{3}+450$ for constants $a$ , $b$ , $c$ .

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Problem 56

Prove that $\cos 2x \cos \frac{x}{2} - \cos 3x \cos \frac{9x}{2} = \sin 5x \sin \frac{5x}{2}$ .

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Problem 57

Prove the following for subsets $A, B \subset S$ : (a) $A \subset B$ iff $A \cup B = B$ ; (b) $A \subset B$ iff $A \cap B = A$ ; (c) $A \subset C(B)$ iff $A \cap B = \varnothing$ ; (d) $C(A) \subset B$ iff $A \cup B = S$ ; (e) $A \subset B$ iff $C(B) \subset C(A)$ ; (f) $A \subset C(B)$ iff $B \subset C(A)$ .

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Problem 58

Prove that $\operatorname{Cos}^{2} \theta+\operatorname{Sin}^{2} \theta=1$ . Do all rough work on the answer scripts.

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Problem 59

在 $\square A B C D$ 中, $A C, B D$ 交于点 $O$ , 点 $E, F$ 在 $A C$ 上, $A E=C F$ 。证明四边形 $E B F D$ 是平行四边形和菱形。

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Problem 60

Prove that $c^{2}=a^{2}+b^{2}-2ab\cos C$ .

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Problem 61

Show that $\cos^{2} \theta + \sin^{2} \theta = 1$ .

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Problem 62

Show that $\cos ^{2} \theta+\sin ^{2} \theta=1$ is true for any angle $\theta$ .

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Problem 63

Numărul $\overline{x y z}$ satisface $\overline{x y z}=\overline{x y}+\overline{y z}+\overline{z x}$ . a) Poate $x=2$ ? Justifică. b) Găsește $\overline{x y z}$ .

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Problem 64

Find the center $(h, k)$ and radius $r$ of the circle from $(x-h)^{2}+(y-k)^{2}=r^{2}$ . Derive the general equation of a circle.

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Problem 65

Identify the pattern in the sequence $0, 3, 18, 57$ and extend it. Prove your conclusion using induction.

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Problem 66

Solve the equation $(n+1)^{2} \cdot n !+(n+1) !=(n+2) !$ for n.

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Problem 67

1. (a) Show that $y^{2}=\frac{1}{2} x^{2}$ if $\log \left(x^{2}+y^{2}\right)-\log 3=\log \left(x^{2}-y^{2}\right)$ . (b) Find $m n$ such that $\log _{3} m+3 \log _{27} n=5$ . (c) If $\ln p=7$ , what is $\log _{p} \sqrt{e}$ ?

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Problem 68

Arjun needs a \$350,000 loan for 20 years at 4.2\% interest, compounded monthly. Find his monthly repayments, approx. \$2160.

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Problem 69

Arjun needs a \ $350,000 loan for 20 years at 4.2\% interest, compounded monthly. Find his monthly repayment of about$ 2160$.

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Problem 70

4. Într-un pătrat $A B C D$ , cu punctele $E, F, G$ pe laturi, arată că triunghiul $D E G$ este isoscel și află unghiul dintre $D F$ și $E G$ .

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Problem 71

Prove the identity: $\frac{1}{\cos x}-\frac{\cos x}{1+\sin x}=\tan x$ using trigonometric identities.

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Problem 72

Prove the identity: $\frac{1}{\cos x}-\frac{\cos x}{1+\sin x}=\tan x$ .

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Problem 73

1. Într-o livadă sunt 160 de pomi. În prima zi s-au plantat $60\%$ meri și $40\%$ nuci, iar în a doua zi 76 pomi. Află procentul din prima zi și câți meri și nuci s-au plantat.
2. Consideră $E(x)=x^{2}-(x+1)^{2}-(x+2)^{2}+(x+3)^{2}$ . Arată că $E(x)=4$ pentru orice $x$ și calculează suma $S=1^{2}-2^{2}-3^{2}+4^{2}+\ldots+40^{2}$ .
3. În sistemul de axe $x O y$ , $A(3,2)$ , $B$ simetricul lui $A$ față de $O x$ , și $C$ simetricul lui $A$ față de $O$ . Arată că $C=(-3,-2)$ și determină funcția corespunzătoare lui $BC$ .
4. Semicercul de centru $D$ este tangent la laturile $A B$ și $A C$ ale triunghiului isoscel $A B C$ . Demonstrează că $D$ este mijlocul lui $B C$ și află raza semicercului, știind că $B C=30 \mathrm{~cm}$ și $A D=20 \mathrm{~cm}$ .

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Problem 74

1. Acum doi ani, mama avea de 5 ori vârsta fiicei. În 6 ani, fiica va avea 1/3 din vârsta mamei. Poate mama avea 40 de ani? Determină vârsta fiicei.

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Problem 75

Check if triangles $\triangle ABC$ and $\triangle XYZ$ are congruent using transformations: reflect and translate.

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Problem 76

Find angle $O Q P$ given points $P, Q, R$ on a circle with $m\angle Q P R=35^{\circ}$ and $m\angle O R P=30^{\circ}$ .

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Problem 77

Prove that $2^{4x+1} + 32^{\frac{4}{5}(x+1)}$ is divisible by 9.

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Problem 78

Prove that $\frac{5+2 \sqrt{3}}{2+\sqrt{3}} = 4-\sqrt{3}$ .

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Problem 79

Given $(2 h-3 k):(h+2 k)=3: 5$ , show $h=3 k$ and find $h: k$ .

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Problem 80

Prove that $2^{n+1}+3^{2n-1}$ is divisible by 7 for all positive integers $n$ .

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Problem 81

Soit $I$ et $J$ deux intervalles ouverts, avec $(I \cap \mathbb{Q}) \cap (J \cap \mathbb{Q}) = \varnothing$ . Prouvez que $I \cap J = \varnothing$ .

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Problem 82

Démontrez que les nombres suivants sont irrationnels : 1. $\sqrt{x}+\sqrt{y}$ avec $x,y$ rationnels positifs irrationnels. 2. $\sqrt{2}+\sqrt{3}+\sqrt{5}$ .

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Problem 83

Prove that $\sin 60^{\circ} \cdot \tan 45^{\circ} \cdot \tan 30^{\circ} - \sin 240^{\circ} \cdot \tan 315^{\circ} \cdot \tan 210^{\circ} = 0$ .

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Problem 84

Prove the identity: $\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}=\frac{2}{\sin x}$ .

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Problem 85

Soient $A$ et $B$ deux parties non-vides de $\mathbb{R}$ avec $a \leq b$ pour tout $a \in A$ et $b \in B$ . Montrez que $A$ est majoré, $B$ est minoré et $\sup (A) \leq \inf (B)$ .

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Problem 86

A. Verify $(3,125+3,168)+4,375=3,125+(3,168+4,375)$ using the associative property. B. Show $3,125+3,168+4,365=11,078-(550+692)$ . C. Prove $3,168+4,375+3,125=(3,168+3,125)$ . D. Confirm $(4,168+3,125)+4,375=3,125+(1,168+3,168)$ .

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Problem 87

Beweisen Sie durch Induktion, dass $2n + 1 \leq n^2 \leq 2^n$ für alle $n \geq 4$ gilt.

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Problem 88

Jika $r=25$ , maka $r^{2}=625$ . Jika $r^{2}=625$ , maka $r=25$ .

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Problem 89

Est-ce que $p^{2}$ est un nombre premier si $p$ est un nombre premier?

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Problem 90

Is the statement "A number can only be divisible by exactly one number" true or false? Explain your choice.

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Problem 91

Is it true that in an isosceles right triangle, the hypotenuse is $\sqrt{2}$ times one leg's length? A. Yes B. No C. Maybe D. Sometimes E. Not applicable

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Problem 92

Explain why, when multiplying powers like $a^m \cdot a^n$ , we add the exponents to get $a^{m+n}$ .

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Problem 93

Is it true that if two chords in the same circle are congruent, their minor areas are also congruent? A. Yes B. No C. Maybe D. Sometimes E. Not applicable

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Problem 94

Identify the two true statements: 1) A line is one-dimensional. 2) A plane has an endpoint. 3) The intersection of two planes can be a line. 4) Parallel lines intersect at 45 degrees.

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Problem 95

If $B$ is the midpoint of $\overline{AC}$ , $D$ is the midpoint of $\overline{CE}$ , and $\overline{AB} \cong \overline{DE}$ , prove that $AE=4AB$ .

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Problem 96

Calculate the products of matrices $S$ and $T$ : $S T$ and $T S$ to show multiplication is not commutative. Fill in the boxes.

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Problem 97

Billy needs help matching customers to cars based on arrival times and licenses. Use clues to find out who rented the Mustang.

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Problem 98

Show that matrix multiplication is not commutative by calculating $S T$ and $T S$ for the matrices $S=\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}$ and $T=\begin{bmatrix} 0 & 4 \\ -4 & 3 \end{bmatrix}$ .

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Problem 99

Prove that $2 x^{4}-6 x^{3}+3 x^{2}+3 x-2$ is divisible by $x^{2}-3 x+2$ without division.

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Problem 100

Is the equation $4x - 1 + 8x = 7$ true? Yes or No.

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Prove

Problem 1

Given a circle with center OOO and diameter MNMNMN, prove:(i) ∠MAN=90∘+∠PQR\angle MAN = 90^{\circ} + \angle PQR∠MAN=90∘+∠PQR,(ii) ∠QPR+2×∠MAN=360∘\angle QPR + 2 \times \angle MAN = 360^{\circ}∠QPR+2×∠MAN=360∘.

Problem 2

Prove that two lines are parallel if and only if alternate exterior angles are congruent: ∠1≅∠8 \angle 1 \cong \angle 8 ∠1≅∠8.

Problem 3

Prove that if BP B P BP and PQ P Q PQ are tangents, then (i) AB∥PQ A B \parallel P Q AB∥PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 4

Prove that in a given diagram, (i) AB A B AB is parallel to PQ P Q PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 5

Prove that if BP B P BP and PQ P Q PQ are tangents, then (i) AB∥PQ A B \parallel P Q AB∥PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 6

Show that the roots of mx2+2x+1=m m x^{2}+2 x+1=m mx2+2x+1=m are always real for any real constant m m m.

Problem 7

Show that the roots of the equation mx2+2x+1=0 m x^{2}+2 x+1=0 mx2+2x+1=0 are real for a real constant m m m.

Problem 8

Prove that if BP B P BP and PQ P Q PQ are tangents to circles, then (i) AB∥PQ A B \parallel P Q AB∥PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 9

Problem 10

Prove that n7−n n^{7}-n n7−n is divisible by 42 for all positive integers n n n and that primes other than 2 or 5 divide infinitely many of 1,11,111,1111, 1, 11, 111, 1111, 1,11,111,1111, etc.

Problem 11

Prove that for any positive integer n n n, n7−n n^{7}-n n7−n is divisible by 42. Also, show p2≡1( mod 24) p^{2} \equiv 1(\bmod 24) p2≡1(mod24) for primes p>3 p>3 p>3.

Problem 12

Prove that n7−n n^{7}-n n7−n is divisible by 42 for all positive integers n n n and p>3 p>3 p>3 prime, p2≡1( mod 24) p^{2} \equiv 1(\bmod 24) p2≡1(mod24).

Problem 13

Any two parallel lines lie in the same plane. True or False?

Problem 14

Is it true or false that 'If two rays are parallel, then the lines containing them must be coplanar'?

Problem 15

Prove x2−x+2≥74 x^{2}-x+2 \geq \frac{7}{4} x2−x+2≥47​ for all values of x x x.

Problem 16

Prove that for a sector with radius rrr cm and perimeter 50 cm, θ=360π(25r−1)\theta = \frac{360}{\pi} \left(\frac{25}{r} - 1\right)θ=π360​(r25​−1).

Problem 17

Une piscine rectangulaire de 10 m sur 15 m est entourée d'une bande de gazon de xxx m. Montre que la clôture fait 50+8x50 + 8x50+8x m et l'aire des allées de gazon est 50x+4x250x + 4x^250x+4x2. Calcule pour x=2x=2x=2 m.

Problem 18

Soit nnn un entier. Trouve l'entier précédent et suivant de nnn. Prouve que la somme de 3 entiers consécutifs est un multiple de 3.

Problem 19

Définis RRR en fonction de xxx pour le programme donné. Prouve que RRR est le carré de xxx. Factorise 64−25x2−(8−5x)264-25 x^{2}-(8-5 x)^{2}64−25x2−(8−5x)2 et 49x2+28x+4−2549 x^{2}+28 x+4-2549x2+28x+4−25.

Problem 20

Le mât mesure 3,6 m. Vérifie si les triangles VEN et VNT sont rectangles avec les longueurs 4,2 m et 3,9 m. Justifie.

Problem 21

A ring of weight 8.5 N is on a string with tension TTT. Show that Tsin⁡θ=12T \sin \theta = 12Tsinθ=12 and Tcos⁡θ=3.5T \cos \theta = 3.5Tcosθ=3.5, then find θ\thetaθ.

Problem 22

Prove that sin⁡3θ+sin⁡θ=4sin⁡θ−4sin⁡3θ\sin 3\theta + \sin \theta = 4 \sin \theta - 4 \sin^3 \thetasin3θ+sinθ=4sinθ−4sin3θ.

Problem 23

Find the derivative of f(x)=x2f(x) = x^2f(x)=x2 using the limit definition: f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}f′(x)=limh→0​hf(x+h)−f(x)​.

Problem 24

Nyatakan sama ada semua segi empat tepat adalah segi empat sama: benar atau palsu?

Problem 25

Buat kesimpulan umum secara induksi untuk urutan nombor −12,32,256,…-\frac{1}{2}, \frac{3}{2}, \frac{25}{6}, \ldots−21​,23​,625​,… mengikut pola yang diberikan.

Problem 26

Demuestra la ley de Boyle: pV=p V=pV= const, la ley de Charles: V/T=V / T=V/T= const, y la ley de Gay-Lussac: p/T=p / T=p/T= const.

Problem 27

A particle with mass m1m_{1}m1​ and momentum p1p_{1}p1​ collides elastically with mass m2m_{2}m2​ at rest. Show E3E_{3}E3​ and p3p_{3}p3​ formulas.

Problem 28

True or false: For a population proportion confidence interval, we use z\mathrm{z}z-distribution, not t\mathrm{t}t-distribution.

Problem 29

Problem 30

Find the value of xxx if m∠PQR=6x−7\mathrm{m} \angle \mathrm{PQR}=6x-7m∠PQR=6x−7 and m∠RQT=20x+5\mathrm{m} \angle \mathrm{RQT}=20x+5m∠RQT=20x+5 sum to 180°.

Problem 31

Problem 32

Prove that in cyclic quadrilateral ABCDABCDABCD, the lines XYXYXY and ZWZWZW are parallel, where XXX, YYY, ZZZ, and WWW are feet of perpendiculars.

Problem 33

Demonstrați că în triunghiul dreptunghic, lungimea medianei pe ipotenuză este AO=BC2A O=\frac{B C}{2}AO=2BC​.

Problem 34

Si a∈X,b∈Ra \in \mathbb{X}, b \in \mathbb{R}a∈X,b∈R y c∈Rc \in \mathbb{R}c∈R, evalúa las proposiciones sobre aaa, bbb y ccc.

Problem 35

Sean A y B subconjuntos de un conjunto Re. ¿Cuáles son las afirmaciones verdaderas sobre ellos?

Problem 36

Dadas las matrices cuadradas AAA y BBB de orden nnn y una constante real kkk, determina si las siguientes afirmaciones son verdaderas.

Problem 37

Problem 38

Problem 39

Compare the 15th 15^{\text {th }}15th term of the arithmetic sequence 150,650,1150,1650,...150,650,1150,1650,...150,650,1150,1650,... and geometric sequence 4,12,36,108,...4,12,36,108,...4,12,36,108,.... Show your work.

Problem 40

Une piscine rectangulaire de 10m x 15m a une bande de gazon de xxx m. Montre que la clôture mesure 50+8x50 + 8x50+8x.

Problem 41

Soit nnn un entier. a) Trouve l'entier précédent nnn. b) Trouve l'entier suivant nnn. Prouve que la somme de 3 entiers consécutifs est un multiple de 3.

Problem 42

For n≥100n \geq 100n≥100, show that in two piles of cards numbered nnn to 2n2n2n, at least one pile has two cards summing to a perfect square.

Given a circle with center $O$ and diameter $MN$ , prove:
(i) $\angle MAN = 90^{\circ} + \angle PQR$ ,
(ii) $\angle QPR + 2 \times \angle MAN = 360^{\circ}$ .

Prove that two lines are parallel if and only if alternate exterior angles are congruent: $\angle 1 \cong \angle 8$ .

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

Prove that in a given diagram, (i) $A B$ is parallel to $P Q$ and (ii) $M P \times A M = B M \times M Q$ .

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

Show that the roots of $m x^{2}+2 x+1=m$ are always real for any real constant $m$ .

Show that the roots of the equation $m x^{2}+2 x+1=0$ are real for a real constant $m$ .

Prove that if $B P$ and $P Q$ are tangents to circles, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

Prove that $n^{7}-n$ is divisible by 42 for all positive integers $n$ and that primes other than 2 or 5 divide infinitely many of $1, 11, 111, 1111,$ etc.

Prove that for any positive integer $n$ , $n^{7}-n$ is divisible by 42. Also, show $p^{2} \equiv 1(\bmod 24)$ for primes $p>3$ .

Prove that $n^{7}-n$ is divisible by 42 for all positive integers $n$ and $p>3$ prime, $p^{2} \equiv 1(\bmod 24)$ .

Prove $x^{2}-x+2 \geq \frac{7}{4}$ for all values of $x$ .

Prove that for a sector with radius $r$ cm and perimeter 50 cm, $\theta = \frac{360}{\pi} \left(\frac{25}{r} - 1\right)$ .

Une piscine rectangulaire de 10 m sur 15 m est entourée d'une bande de gazon de $x$ m. Montre que la clôture fait $50 + 8x$ m et l'aire des allées de gazon est $50x + 4x^2$ . Calcule pour $x=2$ m.

Soit $n$ un entier. Trouve l'entier précédent et suivant de $n$ . Prouve que la somme de 3 entiers consécutifs est un multiple de 3.

Définis $R$ en fonction de $x$ pour le programme donné. Prouve que $R$ est le carré de $x$ . Factorise $64-25 x^{2}-(8-5 x)^{2}$ et $49 x^{2}+28 x+4-25$ .

A ring of weight 8.5 N is on a string with tension $T$ . Show that $T \sin \theta = 12$ and $T \cos \theta = 3.5$ , then find $\theta$ .

Prove that $\sin 3\theta + \sin \theta = 4 \sin \theta - 4 \sin^3 \theta$ .

Find the derivative of $f(x) = x^2$ using the limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

Buat kesimpulan umum secara induksi untuk urutan nombor $-\frac{1}{2}, \frac{3}{2}, \frac{25}{6}, \ldots$ mengikut pola yang diberikan.

Demuestra la ley de Boyle: $p V=$ const, la ley de Charles: $V / T=$ const, y la ley de Gay-Lussac: $p / T=$ const.

A particle with mass $m_{1}$ and momentum $p_{1}$ collides elastically with mass $m_{2}$ at rest. Show $E_{3}$ and $p_{3}$ formulas.

True or false: For a population proportion confidence interval, we use $\mathrm{z}$ -distribution, not $\mathrm{t}$ -distribution.

Find the value of $x$ if $\mathrm{m} \angle \mathrm{PQR}=6x-7$ and $\mathrm{m} \angle \mathrm{RQT}=20x+5$ sum to 180°.

Prove that in cyclic quadrilateral $ABCD$ , the lines $XY$ and $ZW$ are parallel, where $X$ , $Y$ , $Z$ , and $W$ are feet of perpendiculars.

Demonstrați că în triunghiul dreptunghic, lungimea medianei pe ipotenuză este $A O=\frac{B C}{2}$ .

Si $a \in \mathbb{X}, b \in \mathbb{R}$ y $c \in \mathbb{R}$ , evalúa las proposiciones sobre $a$ , $b$ y $c$ .

Dadas las matrices cuadradas $A$ y $B$ de orden $n$ y una constante real $k$ , determina si las siguientes afirmaciones son verdaderas.

Compare the $15^{\text {th }}$ term of the arithmetic sequence $150,650,1150,1650,...$ and geometric sequence $4,12,36,108,...$ . Show your work.

Une piscine rectangulaire de 10m x 15m a une bande de gazon de $x$ m. Montre que la clôture mesure $50 + 8x$ .

Soit $n$ un entier. a) Trouve l'entier précédent $n$ . b) Trouve l'entier suivant $n$ . Prouve que la somme de 3 entiers consécutifs est un multiple de 3.

For $n \geq 100$ , show that in two piles of cards numbered $n$ to $2n$ , at least one pile has two cards summing to a perfect square.

Show that the limit $\operatorname{limit}_{x \rightarrow 2} \frac{|x-2|}{x-2}$ does not exist.

Show that $p \Rightarrow(q \wedge r) \equiv(p \Rightarrow q) \wedge(p \Rightarrow r)$ is always true.

Un voilier a un mât de $3,6 \mathrm{~m}$ avec un étai de $4,2 \mathrm{~m}$ et des haubans de $3,9 \mathrm{~m}$ . Est-ce que les triangles VEN et VNT sont rectangles ? Justifie.

Is the transformation from $y=3^{x}$ to $y=5 \cdot 3^{-(x-2)}+9$ unique? List transformations to prove or disprove the claim.

Show that if for each $x \in A$ , there exists an open set $U$ with $U \subset A$ , then $A$ is open in $X$ .

In a coordinate plane, $\triangle ABC$ is a right triangle with $A(-4,1)$ and $B(2,1)$ . Area is 9 sq. units. Find point $C$ .

In a coordinate plane, $\triangle ABC$ has $A(-4,1)$ and $B(2,1)$ with area 9 sq. units. Find other $C$ coordinates besides $(2,4)$ .

Verify the $\mathrm{K}_{\mathrm{a}}$ value of benzoic acid given $\mathrm{H}_{3} \mathrm{O}^{+}$ concentration is $3.08 \times 10^{-3} \mathrm{M}$ in $0.150 \mathrm{M}$ solution.

Derive the equation $2as = v^2 - u^2$ for final velocity $v$ , initial velocity $u$ , acceleration $a$ , and displacement $s$ .

Derive the formula for final velocity: $V_f = V_i + a t$ .

Prove the system $y' = A y$ is solvable when $A = \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}$ is upper triangular.

Prove that $(3a+2b+3c)^{3}=(a+b)^{3}+(b+2c)^{3}+(c+2a)^{3}+450$ for constants $a$ , $b$ , $c$ .

Prove that $\cos 2x \cos \frac{x}{2} - \cos 3x \cos \frac{9x}{2} = \sin 5x \sin \frac{5x}{2}$ .

Prove that $\operatorname{Cos}^{2} \theta+\operatorname{Sin}^{2} \theta=1$ . Do all rough work on the answer scripts.

在 $\square A B C D$ 中, $A C, B D$ 交于点 $O$ , 点 $E, F$ 在 $A C$ 上, $A E=C F$ 。证明四边形 $E B F D$ 是平行四边形和菱形。

Prove that $c^{2}=a^{2}+b^{2}-2ab\cos C$ .

Show that $\cos^{2} \theta + \sin^{2} \theta = 1$ .

Show that $\cos ^{2} \theta+\sin ^{2} \theta=1$ is true for any angle $\theta$ .

Numărul $\overline{x y z}$ satisface $\overline{x y z}=\overline{x y}+\overline{y z}+\overline{z x}$ . a) Poate $x=2$ ? Justifică. b) Găsește $\overline{x y z}$ .

Find the center $(h, k)$ and radius $r$ of the circle from $(x-h)^{2}+(y-k)^{2}=r^{2}$ . Derive the general equation of a circle.

Identify the pattern in the sequence $0, 3, 18, 57$ and extend it. Prove your conclusion using induction.

Solve the equation $(n+1)^{2} \cdot n !+(n+1) !=(n+2) !$ for n.

Arjun needs a \ $350,000 loan for 20 years at 4.2\% interest, compounded monthly. Find his monthly repayment of about$ 2160$.

4. Într-un pătrat $A B C D$ , cu punctele $E, F, G$ pe laturi, arată că triunghiul $D E G$ este isoscel și află unghiul dintre $D F$ și $E G$ .

Prove the identity: $\frac{1}{\cos x}-\frac{\cos x}{1+\sin x}=\tan x$ using trigonometric identities.

Prove the identity: $\frac{1}{\cos x}-\frac{\cos x}{1+\sin x}=\tan x$ .