Navvye/Olympiad-Problems-CSV huggingface dataset:
1992 IMO Longlists
Let $ABC$ be an arbitrary scalene triangle. Define $\sum$ to be the set of all circles $y$ that have the following properties:



(i) $y$ meets each side of  $ABC$ in two (possibly coincident) points;



(ii) if the points of intersection of $y$ with the sides of the triangle are labeled by $P, Q, R, S, T , U$, with the points occurring on the sides in orders $\mathcal B(B,P,Q,C), \mathcal B(C, R, S,A), \mathcal B(A, T,U,B)$, then the following relations of parallelism hold: $TS \parallel BC; PU\parallel CA; RQ\parallel AB$. (In the limiting cases, some of the conditions of parallelism will hold vacuously; e.g., if $A$ lies on the circle $y$, then $T$ , $S$ both coincide with $A$ and the relation $TS \parallel BC$ holds vacuously.)



(a) Under what circumstances is $\sum$ nonempty?



(b) Assuming that Σ is nonempty, show how to construct the locus of centers of the circles in the set $\sum$.



(c) Given that the set $\sum$has just one element, deduce the size of the largest angle of  $ABC.$

(d) Show how to construct the circles in $\sum$ that have, respectively, the largest and the smallest radii.
1992 IMO Longlists
Let $P_n = (19 + 92)(19^2 +92^2) \cdots(19^n +92^n)$ for each positive integer $n$. Determine, with proof, the least positive integer $m$, if it exists, for which $P_m$ is divisible by $33^{33}.$
1992 IMO Longlists
Let $S_n = \{1, 2,\cdots, n\}$ and $f_n : S_n \to S_n$ be defined inductively as follows: $f_1(1) = 1, f_n(2j) = j  \ (j = 1, 2, \cdots , [n/2])$ and



(i) if $n = 2k \  (k \geq  1)$, then $f_n(2j - 1) = f_k(j) + k  \ (j = 1, 2, \cdots, k);$





(ii) if $n = 2k + 1 \  (k \geq 1)$, then $f_n(2k + 1) = k + f_{k+1}(1), f_n(2j - 1) = k + f_{k+1}(j + 1)  \ (j = 1, 2,\cdots , k).$





Prove that $f_n(x) = x$ if and only if $x$ is an integer of the form

$$\frac{(2n + 1)(2^d - 1)}{2^{d+1} - 1}$$

for some positive integer $d.$
1992 IMO Longlists
Let $a, b, c$ be positive real numbers and $p, q, r$ complex numbers. Let $S$ be the set of all solutions $(x, y, z)$ in $\mathbb C$ of the system of simultaneous equations

$$ax + by + cz = p,$$$$ax2 + by2 + cz2 = q,$$$$ax3 + bx3 + cx3 = r.$$

Prove that $S$ has at most six elements.
1992 IMO Longlists
Let $a, b, c$ be integers. Prove that there are integers $p_1, q_1, r_1, p_2, q_2, r_2$ such that

$$a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1, c = p_1q_2 - p_2q_1.$$
1992 IMO Longlists
Find all rational solutions of

$$a^2 + c^2 + 17(b^2 + d^2) = 21,$$$$ab + cd = 2.$$
1992 IMO Longlists
Let the circles $C_1, C_2$, and $C_3$ be orthogonal to the circle $C$ and intersect each other inside $C$ forming acute angles of measures $A, B$, and $C$. Show that $A + B +C < \pi.$
1992 IMO Longlists
Let $n \geq 2$ be an integer. Find the minimum $k$ for which there exists a partition of $\{1, 2, . . . , k\}$ into $n$ subsets $X_1,X_2, \cdots , X_n$ such that the following condition holds:

for any $i, j, 1 \leq i < j \leq n$, there exist $x_i \in X_1, x_j \in X_2$ such that $|x_i - x_j | = 1.$
1992 IMO Longlists
The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?
1992 IMO Longlists
Let $S$ be a set of positive integers $n_1, n_2, \cdots, n_6$ and let $n(f)$ denote the number $n_1n_{f(1)} +n_2n_{f(2)} +\cdots+n_6n_{f(6)}$, where $f$ is a permutation of $\{1, 2, . . . , 6\}$. Let

$$\Omega=\{n(f) | f \text{ is a permutation of } \{1, 2, . . . , 6\} \} $$

Give an example of positive integers $n_1, \cdots, n_6$ such that $\Omega$ contains as many elements as possible and determine the number of elements of $\Omega$.
1992 IMO Longlists
Find the number of positive integers $n$ satisfying $\phi(n) | n$ such that

$$\sum_{m=1}^{\infty} \left( \left[ \frac nm \right] - \left[\frac{n-1}{m} \right]  \right) =  1992$$

What is the largest number among them? As usual, $\phi(n)$ is the number of positive integers less than or equal to $n$ and relatively prime to $n.$
1992 IMO Longlists
Let $n$ be a positive integer. Prove that the number of ways to express $n$ as a sum of distinct positive integers (up to order) and the number of ways to express $n$ as a sum of odd positive integers (up to order) are the same.
1992 IMO Longlists
Prove that the sequence $5, 12, 19, 26, 33,\cdots $ contains no term of the form $2^n -1.$
1992 IMO Longlists
Evaluate

$$\left \lfloor  \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \  \right \rfloor$$
1992 IMO Longlists
Find all the functions $f : \mathbb R^+ \to \mathbb R$ satisfying the identity

$$f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \qquad \forall x,y \in \mathbb R^+$$

Where $\alpha,\beta$ are given real numbers.
1992 IMO Longlists
Given real numbers $x_i  \ (i = 1, 2, \cdots, 4k + 2)$ such that

$$\sum_{i=1}^{4k +2} (-1)^{i+1} x_ix_{i+1} = 4m \qquad ( \ x_1=x_{4k+3} \ )$$

prove that it is possible to choose numbers $x_{k_{1}}, \cdots, x_{k_{6}}$ such that

$$\sum_{i=1}^{6} (-1)^{i} k_i k_{i+1} > m  \qquad ( \ x_{k_{1}} = x_{k_{7}} \ )$$
1992 IMO Longlists
Let $N$ be a point inside the triangle $ABC$. Through the midpoints of the segments $AN, BN$, and $CN$ the lines parallel to the opposite sides of  $\triangle ABC$ are constructed. Let $AN, BN$, and $CN$ be the intersection points of these lines. If $N$ is the orthocenter of the triangle $ABC$, prove that the nine-point circles of  $\triangle ABC$ and  $\triangle A_NB_NC_N$ coincide.



Remark.Remark. The statement of the original problem was that the nine-point circles of the triangles $A_NB_NC_N$ and $A_MB_MC_M$ coincide, where $N$ and $M$ are the orthocenter and the centroid of  $ABC$. This statement is false.
1992 IMO Longlists
Let $n$ be an integer $> 1$. In a circular arrangement of $n$ lamps $L_0, \cdots, L_{n-1}$, each one of which can be either ON or OFF, we start with the situation that all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \cdots$. If $L_{j-1}$ ($j$ is taken mod n) is ON, then $Step_j$ changes the status of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the status of any of the other lamps. If $L_{j-1}$ is OFF, then $Step_j$ does not change anything at all. Show that:



(a) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again.



(b) If $n$ has the form $2^k$, then all lamps are ON after $n^2 - 1$ steps.



(c) If $n$ has the form $2^k +1$, then all lamps are ON after $n^2 -n+1$ steps.
1992 IMO Longlists
Suppose that $n > m \geq 1$ are integers such that the string of digits $143$ occurs somewhere in the decimal representation of the fraction $\frac{m}{n}$. Prove that $n > 125.$
1992 IMO Longlists
A directed graph (any two distinct vertices joined by at most one directed line) has the following property: If $x, u,$ and $v$ are three distinct vertices such that $x \to u$ and $x \to v$, then $u \to w$ and $v \to w$ for some vertex $w$. Suppose that $x \to u \to y \to\cdots \to z$ is a path of length $n$, that cannot be extended to the right (no arrow goes away from $z$). Prove that every path beginning at $x$ arrives after $n$ steps at $z.$
1992 IMO Longlists
For positive numbers $a, b, c$ define $A = \frac{(a + b + c)}{3}$, $G = \sqrt[3]{abc}$, $H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.$ Prove that

$$ \left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot  \frac AH.$$
1992 IMO Longlists
Let a regular $7$-gon $A_0A_1A_2A_3A_4A_5A_6$ be inscribed in a circle. Prove that for any two points $P, Q$ on the arc $A_0A_6$ the following equality holds:

$$\sum_{i=0}^6 (-1)^{i} PA_i = \sum_{i=0}^6 (-1)^{i} QA_i .$$
1992 IMO Longlists
There are a board with $2n \cdot 2n \ (= 4n^2)$ squares and $4n^2-1$ cards numbered with different natural numbers. These cards are put one by one on each of the squares. One square is empty. We can move a card to an empty square from one of the adjacent squares (two squares are adjacent if they have a common edge). Is it possible to exchange two cards on two adjacent squares of a column (or a row) in a finite number of movements?
1992 IMO Longlists
Let $c_1, \cdots, c_n  \ (n \geq  2)$ be real numbers such that $0 \leq \sum c_i \leq n$. Prove that there exist integers $k_1, \cdots , k_n$ such that $\sum k_i=0$ and $1-n \leq c_i + nk_i \leq n$ for every $i = 1, \cdots , n.$
1992 IMO Longlists
Let $a$ and $b$ be integers. Prove that $\frac{2a^2-1}{b^2+2}$ is not an integer.
1992 IMO Longlists
For any positive integer $n$ consider all representations $n = a_1 + \cdots+ a_k$, where $a_1 > a_2 > \cdots > a_k > 0$ are integers such that for all $i \in \{1, 2, \cdots , k - 1\}$, the number $a_i$ is divisible by $a_{i+1}$. Find the longest such representation of the number $1992.$
1992 IMO Longlists
If $A, B, C$, and $D$ are four distinct points in space, prove that there is a plane $P$ on which the orthogonal projections of $A, B, C$, and $D$ form a parallelogram (possibly degenerate).
1992 IMO Longlists
A circle of radius $\rho$ is tangent to the sides $AB$ and $AC$ of the triangle $ABC$, and its center $K$ is at a distance $p$ from $BC$.



(a) Prove that $a(p - \rho) = 2s(r - \rho)$, where $r$ is the inradius and $2s$ the perimeter of $ABC$.

(b) Prove that if the circle intersect $BC$ at $D$ and $E$, then

$$DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}$$



where $r_1$ is the exradius corresponding to the vertex $A.$
1992 IMO Longlists
In a triangle, a symmedian is a line through a vertex that is symmetric to the median with the respect to the internal bisector (all relative to the same vertex). In the triangle $ABC$, the median $m_a$ meets $BC$ at $A'$ and the circumcircle again at $A_1$. The symmedian $s_a$ meets $BC$ at $M$ and the circumcircle again at $A_2$. Given that the line $A_1A_2$ contains the circumcenter $O$ of the triangle, prove that:



(a) $\frac{AA'}{AM} = \frac{b^2+c^2}{2bc} ;$



(b) $1+4b^2c^2 = a^2(b^2+c^2)$
1992 IMO Longlists
Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$, where $r$ is a positive integer less than $15$ and relatively prime to $15$, satisfy

$$x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.$$
1992 IMO Longlists
Let two circles $A$ and $B$ with unequal radii $r$ and $R$, respectively, be tangent internally at the point $A_0$. If there exists a sequence of distinct circles $(C_n)$ such that each circle is tangent to both $A$ and $B$, and each circle $C_{n+1}$ touches circle $C_{n}$ at the point $A_n$, prove that

$$\sum_{n=1}^{\infty} |A_{n+1}A_n| < \frac{4 \pi Rr}{R+r}.$$
1992 IMO Longlists
Let $P_1(x, y)$ and $P_2(x, y)$ be two relatively prime polynomials with complex coefficients. Let $Q(x, y)$ and $R(x, y)$ be polynomials with complex coefficients and each of degree not exceeding $d$. Prove that there exist two integers $A_1, A_2$ not simultaneously zero with $|A_i| \leq d + 1 \  (i = 1, 2)$ and such that the polynomial $A_1P_1(x, y) + A_2P_2(x, y)$ is coprime to $Q(x, y)$ and $R(x, y).$
1992 IMO Longlists
In a school six different courses are taught: mathematics, physics, biology, music, history, geography. The students were required to rank these courses according to their preferences, where equal preferences were allowed. It turned out that:



(i) mathematics was ranked among the most preferred courses by all students;





(ii) no student ranked music among the least preferred ones;





(iii) all students preferred history to geography and physics to biology; and





(iv) no two rankings were the same.





Find the greatest possible value for the number of students in this school.
1992 IMO Longlists
Let $\{A_n | n = 1, 2, \cdots \} $ be a set of points in the plane such that for each $n$, the disk with center $A_n$ and radius $2^n$ contains no other point $A_j$ . For any given positive real numbers $a < b$ and $R$, show that there is a subset $G$ of the plane satisfying:



(i) the area of $G$ is greater than or equal to $R$;



(ii) for each point $P$ in $G$, $a < \sum_{n=1}^{\infty} \frac{1}{|A_nP|} <b.$
1992 IMO Longlists
Let $S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.$ Show that every real number $x \geq 0$ is an accumulation point of $S.$
1992 IMO Longlists
A sequence $\{an\}$ of positive integers is defined by

$$a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N$$

Determine the positive integers that occur in the sequence.
1992 IMO Longlists
Given any triangle $ABC$ and any positive integer $n$, we say that $n$ is a decomposable number for triangle $ABC$ if there exists a decomposition of the triangle $ABC$ into $n$ subtriangles with each subtriangle similar to  $\triangle ABC$. Determine the positive integers that are decomposable numbers for every triangle.
1992 IMO Longlists
Show that if $994$ integers are chosen from $1, 2,\cdots , 1992$ and one of the chosen integers is less than $64$, then there exist two among the chosen integers such that one of them is a factor of the other.
1992 IMO Longlists
Let $F_n$ be the nth Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$. Let $A_0, A_1, A_2,\cdots$ be a sequence of points on a circle of radius $1$ such that the minor arc from $A_{k-1}$ to $A_k$ runs clockwise and such that

$$\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+1}^2+1}$$

for $k \geq 1$, where $\mu(XY )$ denotes the radian measure of the arc $XY$ in the clockwise direction. What is the limit of the radian measure of arc $A_0A_n$ as $n$ approaches infinity?
1992 IMO Longlists
Given a graph with $n$ vertices and a positive integer $m$ that is less than $ n$, prove that the graph contains a set of $m+1$ vertices in which the difference between the largest degree of any vertex in the set and the smallest degree of any vertex in the set is at most $m-1.$
1992 IMO Longlists
Suppose that points $X, Y,Z$ are located on sides $BC, CA$, and $AB$, respectively, of triangle $ABC$ in such a way that triangle $XY Z$ is similar to triangle $ABC$. Prove that the orthocenter of triangle $XY Z$ is the circumcenter of triangle $ABC.$
1992 IMO Longlists
Let $f(x) = x^m + a_1x^{m-1} + \cdots+ a_{m-1}x + a_m$ and $g(x) = x^n + b_1x^{n-1} + \cdots + b_{n-1}x + b_n$ be two polynomials with real coefficients such that for each real number $x, f(x)$ is the square of an integer if and only if so is $g(x)$. Prove that if $n +m > 0$, then there exists a polynomial $h(x)$ with real coefficients such that $f(x) \cdot g(x) = (h(x))^2.$



Remark.Remark. The original problem stated $g(x) = x^n + b_1x^{n-1} + \cdots + {\color{red}{ b_{n-1}}} + b_n$, but I think the right form of the problem is what I wrote.
1990 IMO Longlists
In triangle $ABC, O$ is the circumcenter, $H$ is the orthocenter. Construct the circumcircles of triangles $CHB, CHA$ and $AHB$, and let their centers be $A_1, B_1, C_1$, respectively. Prove that triangles $ABC$ and $A_1B_1C_1$ are congruent, and their nine-point circles coincide.
1990 IMO Longlists
Given the condition that there exist exactly $1990$ triangles $ABC$ with integral side-lengths satisfying the following conditions:

(i) $\angle ABC =\frac 12 \angle BAC;$

(ii) $AC = b.$

Find the minimal value of $b.$
1990 IMO Longlists
Let function $f : \mathbb Z_{\geq 0}^0 \to \mathbb N$ satisfy the following conditions:



(i) $ f(0, 0, 0) = 1;$



(ii)  $f(x, y, z) = f(x - 1, y, z) + f(x, y - 1, z) + f(x, y, z - 1);$



(iii) when applying above relation iteratively, if any of $x', y', z$' is negative, then $f(x', y', z') = 0.$



Prove that if $x, y, z$ are the side lengths of a triangle, then  $\frac{\left(f(x,y,z) \right) ^k}{ f(mx ,my, mz)}$ is not an integer for any integers $k, m > 1.$
1990 IMO Longlists
$A$ and $B$ are two points in the plane $\alpha$, and line $r$ passes through points $A, B$. There are $n$ distinct points $P_1, P_2, \ldots, P_n$ in one of the half-plane divided by line $r$. Prove that there are at least $\sqrt n$ distinct values among the distances $AP_1, AP_2, \ldots, AP_n, BP_1, BP_2, \ldots, BP_n.$
1990 IMO Longlists
Let $p, k$ and $x$ be positive integers such that $p \geq k$ and $x < \left[ \frac{p(p-k+1)}{2(k-1)} \right]$, where $[q]$ is the largest integer no larger than $q$. Prove that when $x$ balls are put into $p$ boxes arbitrarily, there exist $k$ boxes with the same number of balls.
1990 IMO Longlists
In a group of mathematicians, every mathematician has some friends (the relation of friend is reciprocal). Prove that there exists a mathematician, such that the average of the number of friends of all his friends is no less than the average of the number of friends of all these mathematicians.
1990 IMO Longlists
For any permutation $p$ of set $\{1, 2, \ldots, n\}$, define $d(p) = |p(1) - 1| + |p(2) - 2| + \ldots + |p(n) - n|$. Denoted by $i(p)$ the number of integer pairs $(i, j)$ in permutation $p$ such that $1 \leqq < j \leq n$ and $p(i) > p(j)$. Find all the real numbers $c$, such that the inequality $i(p) \leq c \cdot d(p)$ holds for any positive integer $n$ and any permutation $p.$
1990 IMO Longlists
Six cities $A, B, C, D, E$, and $F$ are located on the vertices of a regular hexagon in that order. $G$ is the center of the hexagon. The sides of the hexagon are the roads connecting these cities. Further more, there are roads connecting cities $B, C, E, F$ and $G$, respectively. Because of raining, one or more roads maybe destroyed. The probability of the road keeping undestroyed between two consecutive cities is $p$. Determine the probability of the road between cities $A$ and $D$ is undestroyed.
1990 IMO Longlists
We call a set $S$ on the real line $R$ "superinvariant", if for any stretching $A$ of the set $S$ by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0)$, where $a > 0$, there exists a transformation $B, B(x) = x + b$, such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$, there is $y \in S$ such that $A(x) = B(y)$, and for any $t \in S$, there is a $u  \in S$ such that $B(t) = A(u).$ Determine all superinvariant sets.
1990 IMO Longlists
We call an integer $k \geq 1$ having property $P$, if there exists at least one integer $m \geq 1$ which cannot be expressed in the form $m = \varepsilon_1 z_1^k  + \varepsilon_2 z_2^k + \cdots + \varepsilon_{2k} z_{2k}^k $ , where $z_i$  are nonnegative integer and $\varepsilon _i = 1$ or $-1$, $i = 1, 2, \ldots, 2k$. Prove that there are infinitely many integers $k$ having the property $P.$
1990 IMO Longlists
1990 mathematicians attend a meeting, every mathematician has at least 1327 friends (the relation of friend is reciprocal). Prove that there exist four mathematicians among them such that any two of them are friends.
1990 IMO Longlists
Find, with proof, the least positive integer $n$ having the following property: in the binary representation of $\frac 1n$, all the binary representations of $1, 2, \ldots, 1990$ (each consist of consecutive digits) are appeared after the decimal point.
1990 IMO Longlists
Could the three-dimensional space be expressed as the union of disjoint circumferences?
1990 IMO Longlists
Point $O$ is interior to triangle $ABC$. Through $O$, draw three lines $DE \parallel BC, FG \parallel CA$, and $HI \parallel AB$, where $D, G$ are on $AB$, $I, F$ are on $BC$ and $E, H$ are on $CA$. Denote by $S_1$ the area of hexagon $DGHEFI$, and $S_2$ the area of triangle $ABC$. Prove that $S_1 \geq \frac 23 S_2.$
1990 IMO Longlists
Find the real number $t$, such that the following system of equations has a unique real solution $(x, y, z, v)$:



$$ \left\{\begin{array}{cc}x+y+z+v=0\\ (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. $$
1990 IMO Longlists
Prove that there exist infinitely many positive integers $n$ such that the number $\frac{1^2+2^2+\cdots+n^2}{n}$ is a perfect square. Obviously, $1$ is the least integer having this property. Find the next two least integers having this property.
1990 IMO Longlists
Let $ABC$ be an arbitrary acute triangle. Circle $\Gamma$ satisfies the following conditions:

(i) Circle $\Gamma$ intersects all three sides of triangle $ABC.$

(ii) In the convex hexagon formed by above six intersections, the three pairs of opposite sides are parallel respectively. (The hexagon maybe degenerate, that is, two or more vertices are coincide. In this case, "opposite sides are parallel" is defined through limit opinion.)

Find the locus of the center of circle $\Gamma$, and explain how to construct the locus.
1990 IMO Longlists
Function $f(n), n \in \mathbb N$, is defined as follows:

Let $\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}$ , where $A(n), B(n)$ are coprime positive integers; if $B(n) = 1$, then $f(n) = 1$; if $B(n) \neq 1$, then $f(n)$ is the largest prime factor of $B(n)$. Prove that the values of $f(n)$ are finite, and find the maximum value of $f(n).$
1990 IMO Longlists
Let $S = \{1, 2, \ldots, 1990\}$. A $31$-element subset of $S$ is called "good" if the sum of its elements is divisible by $5$. Find the number of good subsets of $S.$
1990 IMO Longlists
Using following five figures, can a parallelepiped be constructed, whose side lengths are all integers larger than $1$ and has volume $1990$ ? (In the figure, every square represents a unit cube.)



$$\text{Squares are the same and all are } \Huge{1 \times 1}$$

[asy]

import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1);

draw((2,4)--(0,4),linewidth(2pt)); draw((0,4)--(0,0),linewidth(2pt)); draw((0,0)--(2,0),linewidth(2pt)); draw((2,0)--(2,1),linewidth(2pt)); draw((2,1)--(0,1),linewidth(2pt)); draw((1,0)--(1,4),linewidth(2pt)); draw((2,4)--(2,3),linewidth(2pt)); draw((2,3)--(0,3),linewidth(2pt)); draw((0,2)--(1,2),linewidth(2pt));

label("(1)", (0.56,-1.54), SE*lsf); draw((4,2)--(4,1),linewidth(2pt)); draw((7,2)--(7,1),linewidth(2pt)); draw((4,2)--(7,2),linewidth(2pt)); draw((4,1)--(7,1),linewidth(2pt)); draw((6,0)--(6,3),linewidth(2pt)); draw((5,3)--(5,0),linewidth(2pt)); draw((5,0)--(6,0),linewidth(2pt)); draw((5,3)--(6,3),linewidth(2pt)); label("(2)", (5.13,-1.46), SE*lsf); draw((9,0)--(9,3),linewidth(2pt)); draw((10,3)--(10,0),linewidth(2pt)); draw((12,3)--(12,0),linewidth(2pt)); draw((11,0)--(11,3),linewidth(2pt)); draw((9,2)--(12,2),linewidth(2pt)); draw((12,1)--(9,1),linewidth(2pt)); draw((9,3)--(10,3),linewidth(2pt)); draw((11,3)--(12,3),linewidth(2pt)); draw((12,0)--(11,0),linewidth(2pt)); draw((9,0)--(10,0),linewidth(2pt)); label("(3)", (10.08,-1.48), SE*lsf); draw((14,1)--(17,1),linewidth(2pt)); draw((15,2)--(17,2),linewidth(2pt)); draw((15,2)--(15,0),linewidth(2pt)); draw((15,0)--(14,0)); draw((14,1)--(14,0),linewidth(2pt)); draw((16,2)--(16,0),linewidth(2pt)); label("(4)", (15.22,-1.5), SE*lsf); draw((14,0)--(16,0),linewidth(2pt)); draw((17,2)--(17,1),linewidth(2pt)); draw((19,3)--(19,0),linewidth(2pt)); draw((20,3)--(20,0),linewidth(2pt)); draw((20,3)--(19,3),linewidth(2pt)); draw((19,2)--(20,2),linewidth(2pt)); draw((19,1)--(20,1),linewidth(2pt)); draw((20,0)--(19,0),linewidth(2pt)); label("(5)", (19.11,-1.5), SE*lsf); dot((0,0),ds); dot((0,1),ds); dot((0,2),ds); dot((0,3),ds); dot((0,4),ds); dot((1,4),ds); dot((2,4),ds); dot((2,3),ds); dot((1,3),ds); dot((1,2),ds); dot((1,1),ds); dot((2,1),ds); dot((2,0),ds); dot((1,0),ds); dot((5,0),ds); dot((6,0),ds); dot((5,1),ds); dot((6,1),ds); dot((5,2),ds); dot((6,2),ds); dot((5,3),ds); dot((6,3),ds); dot((7,2),ds); dot((7,1),ds); dot((4,1),ds); dot((4,2),ds); dot((9,0),ds); dot((9,1),ds); dot((9,2),ds); dot((9,3),ds); dot((10,0),ds); dot((11,0),ds); dot((12,0),ds); dot((10,1),ds); dot((10,2),ds); dot((10,3),ds); dot((11,1),ds); dot((11,2),ds); dot((11,3),ds); dot((12,1),ds); dot((12,2),ds); dot((12,3),ds); dot((14,0),ds); dot((15,0),ds); dot((16,0),ds); dot((15,1),ds); dot((14,1),ds); dot((16,1),ds); dot((15,2),ds); dot((16,2),ds); dot((17,2),ds); dot((17,1),ds); dot((19,0),ds); dot((20,0),ds); dot((19,1),ds); dot((20,1),ds); dot((19,2),ds); dot((20,2),ds); dot((19,3),ds); dot((20,3),ds); clip((-0.41,-10.15)--(-0.41,8.08)--(21.25,8.08)--(21.25,-10.15)--cycle);

[/asy]