A function $f$ is continuous on the closed interval $[ 2,5 ]$ with $f ( 2 ) = 17$ and $f ( 5 ) = 17$. Which of the following additional conditions guarantees that there is a number $c$ in the open interval $( 2,5 )$ such that $f ^ { \prime } ( c ) = 0$ ?
(A) No additional conditions are necessary.
(B) $f$ has a relative extremum on the open interval $( 2,5 )$.
(C) $f$ is differentiable on the open interval $( 2,5 )$.
(D) $\int _ { 2 } ^ { 5 } f ( x ) d x$ exists.

Let $f$ be a function that is continuous on the closed interval $[ 2,4 ]$ with $f ( 2 ) = 10$ and $f ( 4 ) = 20$. Which of the following is guaranteed by the Intermediate Value Theorem?
(A) $f ( x ) = 13$ has at least one solution in the open interval $( 2,4 )$.
(B) $f ( 3 ) = 15$
(C) $f$ attains a maximum on the open interval $( 2,4 )$.
(D) $f ^ { \prime } ( x ) = 5$ has at least one solution in the open interval $( 2,4 )$.
(E) $f ^ { \prime } ( x ) > 0$ for all $x$ in the open interval $( 2,4 )$.

The function $h$ is differentiable, and for all values of $x$, $h ( x ) = h ( 2 - x )$. Which of the following statements must be true?
I. $\int _ { 0 } ^ { 2 } h ( x ) d x > 0$
II. $h ^ { \prime } ( 1 ) = 0$
III. $h ^ { \prime } ( 0 ) = h ^ { \prime } ( 2 ) = 1$
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III

Let $x$, $y$ and $z$ be three real numbers. We consider the following implications $\left( P _ { 1 } \right)$ and $\left( P _ { 2 } \right)$:
$$\begin{array} { l l } \left( P _ { 1 } \right) & ( x + y + z = 1 ) \Rightarrow \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \geqslant \frac { 1 } { 3 } \right) \\ \left( P _ { 2 } \right) & \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \geqslant \frac { 1 } { 3 } \right) \Rightarrow ( x + y + z = 1 ) \end{array}$$

Part A

Is the implication $\left( P _ { 2 } \right)$ true?

Part B

In space, we consider the cube $A B C D E F G H$ and we define the orthonormal coordinate system $( A ; \overrightarrow { A B } , \overrightarrow { A D } , \overrightarrow { A E } )$.

1. a. Verify that the plane with equation $x + y + z = 1$ is the plane $( B D E )$. b. Show that the line $( A G )$ is orthogonal to the plane $( B D E )$. c. Show that the intersection of the line $( A G )$ with the plane $( B D E )$ is the point $K$ with coordinates $\left( \frac { 1 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.
2. Is the triangle $B D E$ equilateral?
3. Let $M$ be a point in space. a. Prove that if $M$ belongs to the plane $( B D E )$, then $A M ^ { 2 } = A K ^ { 2 } + M K ^ { 2 }$. b. Deduce that if $M$ belongs to the plane $( B D E )$, then $A M ^ { 2 } \geqslant A K ^ { 2 }$. c. Let $x$, $y$ and $z$ be arbitrary real numbers. By applying the result of the previous question to the point $M$ with coordinates $( x ; y ; z )$, show that the implication $\left( P _ { 1 } \right)$ is true.

The manufacturer of padlocks of the brand ``K'' wishes to print a logo for his company. This logo has the shape of a stylized capital letter K, inscribed in a square ABCD, with side length one unit of length, and respecting the following conditions C1 and C2:

• Condition C1: the letter K must consist of three lines:
• one of the lines is the segment $[AD]$;
• a second line has endpoints at point A and a point E on segment $[DC]$;
• the third line has endpoint at point B and a point G located on the second line.
• Condition C2: the area of each of the three surfaces delimited by the three lines drawn in the square must be between 0.3 and 0.4, with the unit of area being that of the square. These areas are denoted $r$, $s$, $t$.

We place ourselves in the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD})$.
Part A: study of Proposal A
In this proposal, the three lines are segments and the three areas are equal: $r = s = t = \frac{1}{3}$. Determine the coordinates of points E and G.

For every pair of non-zero integers $(a, b)$, we denote by $\operatorname{gcd}(a, b)$ the greatest common divisor of $a$ and $b$. The plane is equipped with a coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

1. Example. Let $\Delta_{1}$ be the line with equation $y = \frac{5}{4}x - \frac{2}{3}$. a. Show that if $(x, y)$ is a pair of integers then the integer $15x - 12y$ is divisible by 3. b. Does there exist at least one point on the line $\Delta_{1}$ whose coordinates are two integers? Justify.

Generalization: We now consider a line $\Delta$ with equation $(E): y = \frac{m}{n}x - \frac{p}{q}$ where $m, n, p$ and $q$ are non-zero integers such that $\operatorname{gcd}(m, n) = \operatorname{gcd}(p, q) = 1$. Thus, the coefficients of equation $(E)$ are irreducible fractions and we say that $\Delta$ is a rational line. The purpose of the exercise is to determine a necessary and sufficient condition on $m, n, p$ and $q$ for a rational line $\Delta$ to contain at least one point whose coordinates are two integers.

1. We suppose here that the line $\Delta$ contains a point with coordinates $(x_{0}, y_{0})$ where $x_{0}$ and $y_{0}$ are integers. a. By noting that the number $ny_{0} - mx_{0}$ is an integer, prove that $q$ divides the product $np$. b. Deduce that $q$ divides $n$.
2. Conversely, we suppose that $q$ divides $n$, and we wish to find a pair $(x_{0}, y_{0})$ of integers such that $y_{0} = \frac{m}{n}x_{0} - \frac{p}{q}$. a. We set $n = qr$, where $r$ is a non-zero integer. Prove that we can find two integers $u$ and $v$ such that $qru - mv = 1$. b. Deduce that there exists a pair $(x_{0}, y_{0})$ of integers such that $y_{0} = \frac{m}{n}x_{0} - \frac{p}{q}$.
3. Let $\Delta$ be the line with equation $y = \frac{3}{8}x - \frac{7}{4}$. Does this line have a point whose coordinates are integers? Justify.

Part A: a volume calculation without a coordinate system We consider an equilateral pyramid SABCD (pyramid with a square base whose lateral faces are all equilateral triangles). The diagonals of the square ABCD measure 24 cm. We denote O the center of the square ABCD. We will admit that $\mathrm { OS } = \mathrm { OA }$.

1. Without using a coordinate system, prove that the line (SO) is orthogonal to the plane (ABC).
2. Deduce the volume, in $\mathrm { cm } ^ { 3 }$, of the pyramid SABCD.

Part B: in a coordinate system We consider the orthonormal coordinate system ( $\mathrm { O } ; \overrightarrow { \mathrm { OA } } , \overrightarrow { \mathrm { OB } } , \overrightarrow { \mathrm { OS } }$ ).

1. We denote P and Q the midpoints of the segments [AS] and [BS] respectively. a. Justify that $\vec { n } ( 1 ; 1 ; - 3 )$ is a normal vector to the plane (PQC). b. Deduce a Cartesian equation of the plane (PQC).
2. Let H be the point of the plane (PQC) such that the line (SH) is orthogonal to the plane (PQC). a. Give a parametric representation of the line (SH). b. Calculate the coordinates of the point H. c. Show then that the length SH, in unit of length, is $\frac { 2 \sqrt { 11 } } { 11 }$.
3. We will admit that the area of the quadrilateral PQCD, in unit of area, is equal to $\frac { 3 \sqrt { 11 } } { 8 }$. Calculate the volume of the pyramid SPQCD, in unit of volume.

Part C: fair sharing For the birthday of her twin daughters Anne and Fanny, Mrs. Nova has made a beautiful cake in the shape of an equilateral pyramid whose diagonals of the square base measure 24 cm. She is about to share it equally by placing her knife on the apex. That is when Anne stops her and proposes a more original cut: ``Place the blade on the midpoint of an edge, parallel to a side of the base, then cut towards the opposite side''. Is this the case? Justify the answer.

An association assigns each registered child a 6-digit number $c_1 c_2 c_3 c_4 c_5 k$ where:

• $c_1 c_2$ represents the last two digits of the child's birth year;
• $c_3 c_4 c_5$ are three digits chosen by the association;
• $k$ is a check digit computed as follows:

• we compute the sum $S = c_1 + c_3 + c_5 + a \times (c_2 + c_4)$ where $a$ is an integer between 1 and 9;
• we perform the Euclidean division of $S$ by 10, the remainder obtained is the check digit $k$.

When an employee enters the 6-digit number of a child, an input error can be detected when the sixth digit is not equal to the check digit calculated from the first five digits.

1. In this question only, we choose $a = 3$. a. Can the number 111383 be that of a child registered with the association? b. The employee, confusing a brother and sister, exchanges their birth years: 2008 and 2011. Thus, the number $08c_3c_4c_5k$ is transformed into $11c_3c_4c_5k$. Is this error detected thanks to the check digit?
2. We denote $c_1c_2c_3c_4c_5k$ the number of a child. We seek the values of the integer $a$ for which the check digit systematically detects the typing error when the digits $c_3$ and $c_4$ are swapped. We therefore assume that the digits $c_3$ and $c_4$ are distinct. a. Show that the check digit does not detect the error of swapping the digits $c_3$ and $c_4$ if and only if $(a-1)(c_4 - c_3)$ is congruent to 0 modulo 10. b. Determine the integers $n$ between 0 and 9 for which there exists an integer $p$ between 1 and 9 such that $np \equiv 0 \pmod{10}$. c. Deduce the values of the integer $a$ which allow, thanks to the check digit, to systematically detect the swap of the digits $c_3$ and $c_4$.

Exercise 4 (Candidates who have not followed the specialization course)
We connect the centres of each face of a cube ABCDEFGH to form a solid IJKLMN. More precisely, the points $\mathrm { I } , \mathrm { J } , \mathrm { K } , \mathrm { L } , \mathrm { M }$ and N are the centres respectively of the square faces ABCD, BCGF, CDHG, ADHE, ABFE and EFGH (thus the midpoints of the diagonals of these squares).

1. Without using a coordinate system (and thus coordinates) in the reasoning, justify that the lines (IN) and (ML) are orthogonal.

In what follows, we consider the orthonormal coordinate system $( \mathrm { A } ; \overrightarrow { \mathrm { AB } } ; \overrightarrow { \mathrm { AD } } ; \overrightarrow { \mathrm { AE } } )$ in which, for example, the point N has coordinates $\left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; 1 \right)$.

1. a. Give the coordinates of the vectors $\overrightarrow { \mathrm { NC } }$ and $\overrightarrow { \mathrm { ML } }$. b. Deduce that the lines (NC) and (ML) are orthogonal. c. From the previous questions, deduce a Cartesian equation of the plane (NCI).
2. a. Show that a Cartesian equation of the plane (NJM) is: $x - y + z = 1$. b. Is the line (DF) perpendicular to the plane (NJM)? Justify. c. Show that the intersection of the planes (NJM) and (NCI) is a line for which you will give a point and a direction vector. Name the line thus obtained using two points from the figure.

Exercise 3 — Part A
In a plane P, consider a triangle ABC right-angled at A. Let $d$ be the line orthogonal to plane P and passing through point B. Consider a point D on this line distinct from point B.
1. Show that the line (AC) is orthogonal to the plane (BAD).
A bicoin is called a tetrahedron whose four faces are right triangles.
2. Show that the tetrahedron ABCD is a bicoin.
3. a. Justify that the edge $[CD]$ is the longest edge of the bicoin ABCD.
b. Let I be the midpoint of edge $[CD]$. Show that point I is equidistant from the 4 vertices of the bicoin ABCD.

``In a non-equilateral triangle, the Euler line is the line that passes through the following three points:

• the center of the circumscribed circle of this triangle (circle passing through the three vertices of this triangle).
• the centroid of this triangle located at the intersection of the medians of this triangle.
• the orthocenter of this triangle located at the intersection of the altitudes of this triangle''.

The purpose of the exercise is to study an example of an Euler line. We consider a cube ABCDEFGH with side length one unit. The space is equipped with the orthonormal coordinate system $( \mathrm { A } ; \overrightarrow { \mathrm { AB } } ; \overrightarrow { \mathrm { AD } } ; \overrightarrow { \mathrm { AE } } )$. We denote I the midpoint of segment [AB] and J the midpoint of segment [BG].

1. Give without justification the coordinates of points A, B, G, I and J.
2. a. Determine a parametric representation of the line (AJ). b. Show that a parametric representation of the line (IG) is: $$\left\{ \begin{aligned} x & = \frac { 1 } { 2 } + \frac { 1 } { 2 } t \\ y & = t \\ z & = t \end{aligned} \text { with } t \in \mathbb { R } . \right.$$ c. Prove that the lines (AJ) and (IG) intersect at a point S with coordinates $S \left( \frac { 2 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.
3. a. Show that the vector $\vec { n } ( 0 ; - 1 ; 1 )$ is normal to the plane (ABG). b. Deduce a Cartesian equation of the plane (ABG). c. We admit that a parametric representation of the line (d) with direction vector $\vec { n }$ and passing through the point K with coordinates $\left( \frac { 1 } { 2 } ; 0 ; 1 \right)$ is: $$\left\{ \begin{array} { l } x = \frac { 1 } { 2 } \\ y = - t \quad \text { with } t \in \mathbb { R } . \\ z = 1 + t \end{array} \right.$$ Show that this line (d) intersects the plane (ABG) at a point L with coordinates $L \left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$. d. Show that the point L is equidistant from the points $\mathrm { A } , \mathrm { B }$ and G.
4. Show that the triangle ABG is right-angled at B.
5. a. Identify the center of the circumscribed circle, the centroid and the orthocenter of triangle ABG (no justification is expected). b. Verify by calculation that these three points are indeed collinear.

QUESTION 152
The number of diagonals of a polygon with 8 sides is
(A) 16
(B) 18
(C) 20
(D) 22
(E) 24

Let $f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_0$ be a polynomial with integer coefficients and whose degree is at least 2. Suppose each $a_i$ ($0 \leq i \leq n-1$) is of the form $$a_i = \pm \frac{17!}{r!(17-r)!}$$ with $1 \leq r \leq 16$. Show that $f(m)$ is not equal to zero for any integer $m$.

A polynomial $f ( x )$ has integer coefficients such that $f ( 0 )$ and $f ( 1 )$ are both odd numbers. Prove that $f ( x ) = 0$ has no integer solutions.

Show that there is no infinite arithmetic progression consisting of distinct integers all of which are squares.

If 8 points in a plane are chosen to lie on or inside a circle of diameter 2 cm then show that the distance between some two points will be less than 1 cm.

If $f ( x ) = \frac { x ^ { n } } { n ! } + \frac { x ^ { n - 1 } } { ( n - 1 ) ! } + \cdots + x + 1$, then show that $f ( x ) = 0$ has no repeated roots.

Using the fact that $\sqrt { n }$ is an irrational number whenever $n$ is not a perfect square, show that $\sqrt { 3 } + \sqrt { 7 } + \sqrt { 21 }$ is irrational.

In an isoceles $\triangle \mathrm { ABC }$ with A at the apex the height and the base are both equal to 1 cm. Points $\mathrm { D } , \mathrm { E }$ and F are chosen one from each side such that BDEF is a rhombus. Find the length of the side of this rhombus.

Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { 100 }$ be 100 positive integers. Show that for some $m , n$ with $1 \leq m \leq n \leq 100 , \sum _ { i = m } ^ { n } a _ { i }$ is divisible by 100.

In $\triangle \mathrm { ABC } , \mathrm { BE }$ is a median, and O the mid-point of BE. The line joining A and O meets BC at D. Find the ratio $\overline { \mathrm { AO } } : \overline { \mathrm { OD } }$ (Hint: Draw a line through E parallel to AD.)

(a) Show that the area of a right-angled triangle with all side lengths integers is an integer divisible by 6.
(b) If all the sides and area of a triangle were rational numbers then show that the triangle is got by 'pasting' two right-angled triangles having the same property.

Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ and $b _ { 1 } , b _ { 2 } , \ldots , b _ { n }$ be two arithmetic progressions. Prove that the points $\left( a _ { 1 } , b _ { 1 } \right) , \left( a _ { 2 } , b _ { 2 } \right) , \ldots , \left( a _ { n } , b _ { n } \right)$ are collinear.

In a rectangle ABCD , the length BC is twice the width AB . Pick a point P on side BC such that the lengths of AP and BC are equal. The measure of angle CPD is
(A) $75 ^ { \circ }$
(B) $60 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

A differentiable function $f : \mathbb { R } \rightarrow \mathbb { R }$ satisfies $f ( 1 ) = 2 , f ( 2 ) = 3$ and $f ( 3 ) = 1$. Show that $f ^ { \prime } ( x ) = 0$ for some $x$.