year stringdate 1961-01-01 00:00:00 2025-01-01 00:00:00 ⌀ | tier stringclasses 5
values | problem_label stringclasses 119
values | problem_type stringclasses 13
values | exam stringclasses 28
values | problem stringlengths 87 2.77k | solution stringlengths 834 13k | metadata dict | problem_tokens int64 50 903 | solution_tokens int64 500 3.93k |
|---|---|---|---|---|---|---|---|---|---|
1970 | T0 | 10 | null | IMO | 10. (SWE 4) ${ }^{\mathrm{IMO} 3}$ Let $1=a_{0} \leq a_{1} \leq a_{2} \leq \cdots \leq a_{n} \leq \cdots$ be a sequence of real numbers. Consider the sequence $b_{1}, b_{2}, \ldots$ defined by: $$ b_{n}=\sum_{k=1}^{n}\left(1-\frac{a_{k-1}}{a_{k}}\right) \frac{1}{\sqrt{a_{k}}} $$ Prove that: (a) For all natural number... | 10. (a) Since $a_{n-1}<a_{n}$, we have $$ \begin{aligned} \left(1-\frac{a_{k-1}}{a_{k}}\right) \frac{1}{\sqrt{a_{k}}} & =\frac{a_{k}-a_{k-1}}{a_{k}^{3 / 2}} \\ & \leq \frac{2\left(\sqrt{a_{k}}-\sqrt{a_{k-1}}\right) \sqrt{a_{k}}}{a_{k} \sqrt{a_{k-1}}}=2\left(\frac{1}{\sqrt{a_{k-1}}}-\frac{1}{\sqrt{a_{k}}}\right) . \end{... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 225 | 784 |
1970 | T0 | 12 | null | IMO | 12. (USS 4) ${ }^{\mathrm{IMO} 6}$ We are given 100 points in the plane, no three of which are on the same line. Consider all triangles that have all vertices chosen from the 100 given points. Prove that at most $70 \%$ of these triangles are acute angled. | 12. Lemma. Five points are given in the plane such that no three of them are collinear. Then there are at least three triangles with vertices at these points that are not acute-angled. Proof. We consider three cases, according to whether the convex hull of these points is a triangle, quadrilateral, or pentagon. (i) Let... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 76 | 550 |
1970 | T0 | 3 | null | IMO | 3. (BUL 6) ${ }^{\mathrm{IMO}}$ In the tetrahedron $S A B C$ the angle $B S C$ is a right angle, and the projection of the vertex $S$ to the plane $A B C$ is the intersection of the altitudes of the triangle $A B C$. Let $z$ be the radius of the inscribed circle of the triangle $A B C$. Prove that $$ S A^{2}+S B^{2}+S... | 3 . We shall use the following lemma Lemma. If an altitude of a tetrahedron passes through the orthocenter of the opposite side, then each of the other altitudes possesses the same property. Proof. Denote the tetrahedron by $S A B C$ and let $a=B C, b=C A$, $c=A B, m=S A, n=S B, p=S C$. It is enough to prove that an al... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 126 | 930 |
1970 | T0 | 5 | null | IMO | 5. (CZS 3) Let $M$ be an interior point of the tetrahedron $A B C D$. Prove that $$ \begin{aligned} & \overrightarrow{M A} \operatorname{vol}(M B C D)+\overrightarrow{M B} \operatorname{vol}(M A C D) \\ & \quad+\overrightarrow{M C} \operatorname{vol}(M A B D)+\overrightarrow{M D} \operatorname{vol}(M A B C)=0 \end{ali... | 5. Denote respectively by $A_{1}, B_{1}, C_{1}$ and $D_{1}$ the points of intersection of the lines $A M, B M, C M$, and $D M$ with the opposite sides of the tetrahedron. Since $\operatorname{vol}(M B C D)=\operatorname{vol}(A B C D) \overrightarrow{M A_{1}} / \overrightarrow{A A_{1}}$, the relation we have to prove is... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 152 | 838 |
1970 | T0 | 6 | null | IMO | 6. (FRA 1) In the triangle $A B C$ let $B^{\prime}$ and $C^{\prime}$ be the midpoints of the sides $A C$ and $A B$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $A B^{\prime} C^{\prime}, B C^{\prime} H$, and $B^{\prime} C H$ have a common poi... | 6. Let $F$ be the midpoint of $B^{\prime} C^{\prime}, A^{\prime}$ the midpoint of $B C$, and $I$ the intersection point of the line $H F$ and the circle circumscribed about $\triangle B H C^{\prime}$. Denote by $M$ the intersection point of the line $A A^{\prime}$ with the circumscribed circle about the triangle $A B C... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 137 | 650 |
1971 | T0 | 1 | null | IMO | 1. (BUL 2) Consider a sequence of polynomials $P_{0}(x), P_{1}(x), P_{2}(x), \ldots$, $P_{n}(x), \ldots$, where $P_{0}(x)=2, P_{1}(x)=x$ and for every $n \geq 1$ the following equality holds: $$ P_{n+1}(x)+P_{n-1}(x)=x P_{n}(x) $$ Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1$, $$ \... | 1. Assuming that $a, b, c$ in (1) exist, let us find what their values should be. Since $P_{2}(x)=x^{2}-2$, equation (1) for $n=1$ becomes $\left(x^{2}-4\right)^{2}=$ $\left[a\left(x^{2}-2\right)+b x+2 c\right]^{2}$. Therefore, there are two possibilities for $(a, b, c)$ : $(1,0,-1)$ and $(-1,0,1)$. In both cases we mu... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 195 | 585 |
1971 | T0 | 4 | null | IMO | 4. (GBR 3) We are given two mutually tangent circles in the plane, with radii $r_{1}, r_{2}$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_{1}$ and $r_{2}$ and the condition for the solvability of the problem. | 4. In the coordinate system in which the $x$-axis passes through the centers of the circles and the $y$-axis is their common tangent, the circles have equations $$ x^{2}+y^{2}+2 r_{1} x=0, \quad x^{2}+y^{2}-2 r_{2} x=0 . $$ Let $p$ be the desired line with equation $y=a x+b$. The abscissas of points of intersection of ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 78 | 809 |
1971 | T0 | 7 | null | IMO | 7. (NET 1) ${ }^{\mathrm{IMO} /}$ Given a tetrahedron $A B C D$ whose all faces are acuteangled triangles, set $$ \sigma=\measuredangle D A B+\measuredangle B C D-\measuredangle A B C-\measuredangle C D A $$ Consider all closed broken lines $X Y Z T X$ whose vertices $X, Y, Z, T$ lie in the interior of segments $A B,... | 7. (a) Suppose that $X, Y, Z$ are fixed on segments $A B, B C, C D$. It is proven in a standard way that if $\angle A T X \neq \angle Z T D$, then $Z T+T X$ can be reduced. It follows that if there exists a broken line $X Y Z T X$ of minimal length, then the following conditions hold: $$ \begin{aligned} & \angle D A B=... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 212 | 748 |
1972 | T0 | 11 | null | IMO | 11. (NET 6) Consider a sequence of circles $K_{1}, K_{2}, K_{3}, K_{4}, \ldots$ of radii $r_{1}, r_{2}, r_{3}, r_{4}, \ldots$, respectively, situated inside a triangle $A B C$. The circle $K_{1}$ is tangent to $A B$ and $A C ; K_{2}$ is tangent to $K_{1}, B A$, and $B C ; K_{3}$ is tangent to $K_{2}, C A$, and $C B ; K... | 11. Let $\angle A=2 x, \angle B=2 y, \angle C=2 z$. (a) Denote by $M_{i}$ the center of $K_{i}, i=1,2, \ldots$ If $N_{1}, N_{2}$ are the projections of $M_{1}, M_{2}$ onto $A B$, we have $A N_{1}=r_{1} \cot x, N_{2} B=r_{2} \cot y$, and $N_{1} N_{2}=\sqrt{\left(r_{1}+r_{2}\right)^{2}-\left(r_{1}-r_{2}\right)^{2}}=2 \sq... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 328 | 682 |
1972 | T0 | 4 | null | IMO | 4. (GDR 1) Let $n_{1}, n_{2}$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_{1}$ and $M_{2}$ consisting of $2 n_{1}$ and $2 n_{2}$ points, respectively, and such that no three points of the union $M_{1} \cup M_{2}$ are collinear. Prove that there exists a straightline $g$ with the followi... | 4. Choose in $E$ a half-line $s$ beginning at a point $O$. For every $\alpha$ in the interval $\left[0,180^{\circ}\right]$, denote by $s(\alpha)$ the line obtained by rotation of $s$ about $O$ by $\alpha$, and by $g(\alpha)$ the oriented line containing $s(\alpha)$ on which $s(\alpha)$ defines the positive direction. F... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 155 | 590 |
1972 | T0 | 5 | null | IMO | 5. (GDR 2) Prove the following assertion: The four altitudes of a tetrahedron $A B C D$ intersect in a point if and only if $$ A B^{2}+C D^{2}=B C^{2}+A D^{2}=C A^{2}+B D^{2} $$ | 5. Lemma. If $X, Y, Z, T$ are points in space, then the lines $X Z$ and $Y T$ are perpendicular if and only if $X Y^{2}+Z T^{2}=Y Z^{2}+T X^{2}$. Proof. Consider the plane $\pi$ through $X Z$ parallel to $Y T$. If $Y^{\prime}, T^{\prime}$ are the feet of the perpendiculars to $\pi$ from $Y, T$ respectively, then $$ \be... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 74 | 827 |
1972 | T0 | 9 | null | IMO | 9. (NET 2) $)^{\mathrm{IMO} 4}$ Find all solutions in positive real numbers $x_{i}(i=$ $1,2,3,4,5)$ of the following system of inequalities: $$ \begin{aligned} & \left(x_{1}^{2}-x_{3} x_{5}\right)\left(x_{2}^{2}-x_{3} x_{5}\right) \leq 0, \\ & \left(x_{2}^{2}-x_{4} x_{1}\right)\left(x_{3}^{2}-x_{4} x_{1}\right) \leq 0... | 9. Clearly $x_{1}=x_{2}=x_{3}=x_{4}=x_{5}$ is a solution. We shall show that this describes all solutions. Suppose that not all $x_{i}$ are equal. Then among $x_{3}, x_{5}, x_{2}, x_{4}, x_{1}$ two consecutive are distinct: Assume w.l.o.g. that $x_{3} \neq x_{5}$. Moreover, since $\left(1 / x_{1}, \ldots, 1 / x_{5}\rig... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 286 | 830 |
1973 | T0 | 1 | null | IMO | 1. (BUL 6) Let a tetrahedron $A B C D$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality $$ \frac{A P}{P A_{1}}+\frac{B P}{P B_{1}}+\frac{C P}{P C_{1}}+\frac{D P}{P D_{1}}=4 $$ holds, where $A_{1}, B_{1}, C_{1}$, and $D_{1}$ are the intersection points of $S$ with... | 1. The condition of the point $P$ can be written in the form $\frac{A P^{2}}{A P \cdot P A_{1}}+\frac{B P^{2}}{B P \cdot P B_{1}}+$ $\frac{C P^{2}}{C P \cdot P C_{1}}+\frac{D P^{2}}{D P \cdot P D_{1}}=4$. All the four denominators are equal to $R^{2}-O P^{2}$, i.e., to the power of $P$ with respect to $S$. Thus the con... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 148 | 502 |
1973 | T0 | 11 | null | IMO | 11. (SWE 4) ${ }^{\mathrm{IMO} 3}$ Determine the minimum of $a^{2}+b^{2}$ if $a$ and $b$ are real numbers for which the equation $$ x^{4}+a x^{3}+b x^{2}+a x+1=0 $$ has at least one real solution. | 11. Putting $x+\frac{1}{x}=t$ we also get $x^{2}+\frac{1}{x^{2}}=t^{2}-2$, and the given equation reduces to $t^{2}+a t+b-2=0$. Since $x=\frac{t \pm \sqrt{t^{2}-4}}{2}, x$ will be real if and only if $|t| \geq 2, t \in \mathbb{R}$. Thus we need the minimum value of $a^{2}+b^{2}$ under the condition $a t+b=-\left(t^{2}-... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 83 | 537 |
1973 | T0 | 16 | null | IMO | 16. (CUB 2) Given $a, \theta \in \mathbb{R}, m \in \mathbb{N}$, and $P(x)=x^{2 m}-2|a|^{m} x^{m} \cos \theta+a^{2 m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials. | 16. First, we have $P(x)=Q(x) R(x)$ for $Q(x)=x^{m}-|a|^{m} e^{i \theta}$ and $R(x)=$ $x^{m}-|a|^{m} e^{-i \theta}$, where $e^{i \varphi}$ means of course $\cos \varphi+i \sin \varphi$. It remains to factor both $Q$ and $R$. Suppose that $Q(x)=\left(x-q_{1}\right) \cdots\left(x-q_{m}\right)$ and $R(x)=\left(x-r_{1}\rig... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 82 | 645 |
1974 | T0 | 10 | null | IMO | 10. II 4 (FIN 3) ${ }^{\mathrm{IMO} 2}$ Let $\triangle A B C$ be a triangle. Prove that there exists a point $D$ on the side $A B$ such that $C D$ is the geometric mean of $A D$ and $B D$ if and only if $\sqrt{\sin A \sin B} \leq \sin \frac{C}{2}$. | 10. If we set $\angle A C D=\gamma_{1}$ and $\angle B C D=\gamma_{2}$ for a point $D$ on the segment $A B$, then by the sine theorem, $$ f(D)=\frac{C D^{2}}{A D \cdot B D}=\frac{C D}{A D} \cdot \frac{C D}{B D}=\frac{\sin \alpha \sin \beta}{\sin \gamma_{1} \sin \gamma_{2}} . $$ The denominator of the last fraction is $$... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 96 | 624 |
1974 | T0 | 6 | null | IMO | 6. I 6 (ROM 4) ${ }^{\mathrm{IMO} 3}$ Does there exist a natural number $n$ for which the number $$ \sum_{k=0}^{n}\binom{2 n+1}{2 k+1} 2^{3 k} $$ is divisible by 5 ? | 6. We set $$ \begin{aligned} & x=\sum_{k=0}^{n}\binom{2 n+1}{2 k+1} 2^{3 k}=\frac{1}{\sqrt{8}} \sum_{k=0}^{n}\binom{2 n+1}{2 k+1} \sqrt{8}^{2 k+1}, \\ & y=\sum_{k=0}^{n}\binom{2 n+1}{2 k} 2^{3 k}=\sum_{k=0}^{n}\binom{2 n+1}{2 k} \sqrt{8}^{2 k} \end{aligned} $$ Both $x$ and $y$ are positive integers. Also, from the bino... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 73 | 708 |
1974 | T0 | 7 | null | IMO | 7. II 1 (POL 2) Let $a_{i}, b_{i}$ be coprime positive integers for $i=1,2, \ldots, k$, and $m$ the least common multiple of $b_{1}, \ldots, b_{k}$. Prove that the greatest common divisor of $a_{1} \frac{m}{b_{1}}, \ldots, a_{k} \frac{m}{b_{k}}$ equals the greatest common divisor of $a_{1}, \ldots, a_{k}$. | 7. Consider an arbitrary prime number $p$. If $p \mid m$, then there exists $b_{i}$ that is divisible by the same power of $p$ as $m$. Then $p$ divides neither $a_{i} \frac{m}{b_{i}}$ nor $a_{i}$, because $\left(a_{i}, b_{i}\right)=1$. If otherwise $p \nmid m$, then $\frac{m}{b_{i}}$ is not divisible by $p$ for any $i$... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 123 | 540 |
1974 | T0 | 8 | null | IMO | 8. II 2 (NET 3) ${ }^{\mathrm{IMO} 5}$ If $a, b, c, d$ are arbitrary positive real numbers, find all possible values of $$ S=\frac{a}{a+b+d}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{a+c+d} . $$ | 8. It is clear that $$ \begin{gathered} \frac{a}{a+b+c+d}+\frac{b}{a+b+c+d}+\frac{c}{a+b+c+d}+\frac{d}{a+b+c+d}<S \\ \text { and } \quad S<\frac{a}{a+b}+\frac{b}{a+b}+\frac{c}{c+d}+\frac{d}{c+d} \end{gathered} $$ or equivalently, $1<S<2$. On the other hand, all values from $(1,2)$ are attained. Since $S=1$ for $(a, b, ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 83 | 525 |
1975 | T0 | 13 | null | IMO | 13. (ROM) Let $A_{0}, A_{1}, \ldots, A_{n}$ be points in a plane such that (i) $A_{0} A_{1} \leq \frac{1}{2} A_{1} A_{2} \leq \cdots \leq \frac{1}{2^{n-1}} A_{n-1} A_{n}$ and (ii) $0<\measuredangle A_{0} A_{1} A_{2}<\measuredangle A_{1} A_{2} A_{3}<\cdots<\measuredangle A_{n-2} A_{n-1} A_{n}<180^{\circ}$, where all the... | 13. Suppose that $A_{k} A_{k+1} \cap A_{m} A_{m+1} \neq \emptyset$ for some $k, m>k+1$. Without loss of generality we may suppose that $k=0, m=n-1$ and that no two segments $A_{k} A_{k+1}$ and $A_{m} A_{m+1}$ intersect for $0 \leq k<m-1<n-1$ except for $k=0, m=n-1$. Also, shortening $A_{0} A_{1}$, we may suppose that $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 229 | 781 |
1975 | T0 | 7 | null | IMO | 7. (GDR) Prove that from $x+y=1(x, y \in \mathbb{R})$ it follows that $$ x^{m+1} \sum_{j=0}^{n}\binom{m+j}{j} y^{j}+y^{n+1} \sum_{i=0}^{m}\binom{n+i}{i} x^{i}=1 \quad(m, n=0,1,2, \ldots) . $$ | 7. We use induction on $m$. Denote by $S_{m}$ the left-hand side of the equality to be proved. First $S_{0}=(1-y)\left(1+y+\cdots+y^{n}\right)+y^{n+1}=1$, since $x=1-y$. Furthermore, $$ \begin{aligned} & S_{m+1}-S_{m} \\ = & \binom{m+n+1}{m+1} x^{m+1} y^{n+1}+x^{m+1} \sum_{j=0}^{n}\left(\binom{m+1+j}{j} x y^{j}-\binom{... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 107 | 525 |
1975 | T0 | 8 | null | IMO | 8. (NET) ${ }^{\mathrm{IMO} 3}$ On the sides of an arbitrary triangle $A B C$, triangles $B P C$, $C Q A$, and $A R B$ are externally erected such that $\measuredangle P B C=\measuredangle C A Q=45^{\circ}$, $\measuredangle B C P=\measuredangle Q C A=30^{\circ}$, $\measuredangle A B R=\measuredangle B A R=15^{\circ}$. ... | 8. Let $K$ and $L$ be the feet of perpendiculars from $P$ and $Q$ to $B C$ and $A C$ respectively. Let $M, N$ be points on $A B$ (ordered $A-N-M-B$ ) such that $R M N$ is a right isosceles triangle with $\angle R=90^{\circ}$. By sine theorem we have $\frac{B M}{B A}=\frac{B M}{B R} \cdot \frac{B R}{B A}=\frac{\sin 15^{... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 140 | 672 |
1976 | T0 | 1 | null | IMO | 1. (BUL 1) Let $A B C$ be a triangle with bisectors $A A_{1}, B B_{1}, C C_{1}\left(A_{1} \in\right.$ $B C$, etc.) and $M$ their common point. Consider the triangles $M B_{1} A$, $M C_{1} A, M C_{1} B, M A_{1} B, M A_{1} C, M B_{1} C$, and their inscribed circles. Prove that if four of these six inscribed circles have ... | 1. Let $r$ denote the common inradius. Some two of the four triangles with the inradii $\rho$ have cross angles at $M$ : Suppose these are $\triangle A M B_{1}$ and $\triangle B M A_{1}$. We shall show that $\triangle A M B_{1} \cong \triangle B M A_{1}$. Indeed, the altitudes of these two triangles are both equal to $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 140 | 877 |
1976 | T0 | 12 | null | IMO | 12. (VIE 2) The polynomial $1976\left(x+x^{2}+\cdots+x^{n}\right)$ is decomposed into a sum of polynomials of the form $a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n}$, where $a_{1}, a_{2}, \cdots, a_{n}$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible. | 12. Suppose the decomposition into $k$ polynomials is possible. The sum of coefficients of each polynomial $a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}$ equals $1+\cdots+$ $n=n(n+1) / 2$ while the sum of coefficients of $1976\left(x+x^{2}+\cdots+x^{n}\right)$ is $1976 n$. Hence we must have $1976 n=k n(n+1) / 2$, which redu... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 113 | 738 |
1976 | T0 | 4 | null | IMO | 4. (GBR 1a) ${ }^{\mathrm{IMO} 6}$ For all positive integral $n$, $u_{n+1}=u_{n}\left(u_{n-1}^{2}-2\right)-u_{1}$, $u_{0}=2$, and $u_{1}=5 / 2$. Prove that $$ 3 \log _{2}\left[u_{n}\right]=2^{n}-(-1)^{n} $$ where $[x]$ is the integral part of $x$. | 4. The first few values are easily verified to be $2^{r_{n}}+2^{-r_{n}}$, where $r_{0}=0$, $r_{1}=r_{2}=1, r_{3}=3, r_{4}=5, r_{5}=11, \ldots$. Let us put $u_{n}=2^{r_{n}}+2^{-r_{n}}$ (we will show that $r_{n}$ exists and is integer for each $n$ ). A simple calculation gives us $u_{n}\left(u_{n-1}^{2}-2\right)=2^{r_{n}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 120 | 544 |
1976 | T0 | 6 | null | IMO | 6. (NET 4) ${ }^{\mathrm{IMO} 3} \mathrm{~A}$ rectangular box can be filled completely with unit cubes. If one places cubes with volume 2 in the box such that their edges are parallel to the edges of the box, one can fill exactly $40 \%$ of the box. Determine all possible (interior) sizes of the box. | 6. Suppose $a_{1} \leq a_{2} \leq a_{3}$ are the dimensions of the box. If we set $b_{i}=$ $\left[a_{i} / \sqrt[3]{2}\right]$, the condition of the problem is equivalent to $\frac{a_{1}}{b_{1}} \cdot \frac{a_{2}}{b_{2}} \cdot \frac{a_{3}}{b_{3}}=5$. We list some values of $a, b=[a / \sqrt[3]{2}]$ and $a / b$ : | $a$ | ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 83 | 664 |
1977 | T0 | 26 | null | IMO | 26. (NET 1) Let $p$ be a prime number greater than 5 . Let $V$ be the collection of all positive integers $n$ that can be written in the form $n=k p+1$ or $n=k p-1(k=1,2, \ldots)$. A number $n \in V$ is called indecomposable in $V$ if it is impossible to find $k, l \in V$ such that $n=k l$. Prove that there exists a nu... | 26. The result is an immediate consequence (for $G=\{-1,1\}$ ) of the following generalization. (1) Let $G$ be a proper subgroup of $\mathbb{Z}_{n}^{*}$ (the multiplicative group of residue classes modulo $n$ coprime to $n$ ), and let $V$ be the union of elements of $G$. A number $m \in V$ is called indecomposable in $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 138 | 606 |
1977 | T0 | 30 | null | IMO | 30. (NET 5) A triangle $A B C$ with $\angle A=30^{\circ}$ and $\angle C=54^{\circ}$ is given. On $B C$ a point $D$ is chosen such that $\angle C A D=12^{\circ}$. On $A B$ a point $E$ is chosen such that $\angle A C E=6^{\circ}$. Let $S$ be the point of intersection of $A D$ and $C E$. Prove that $B S=B C$. | 30. Suppose $\angle S B A=x$. By the trigonometric form of Ceva's theorem we have $$ \frac{\sin \left(96^{\circ}-x\right)}{\sin x} \frac{\sin 18^{\circ}}{\sin 12^{\circ}} \frac{\sin 6^{\circ}}{\sin 48^{\circ}}=1 $$ We claim that $x=12^{\circ}$ is a solution of this equation. To prove this, it is enough to show that $\s... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 121 | 717 |
1977 | T0 | 46 | null | IMO | 46. (FIN 3) Let $f$ be a strictly increasing function defined on the set of real numbers. For $x$ real and $t$ positive, set $$ g(x, t)=\frac{f(x+t)-f(x)}{f(x)-f(x-t)} $$ Assume that the inequalities $$ 2^{-1}<g(x, t)<2 $$ hold for all positive $t$ if $x=0$, and for all $t \leq|x|$ otherwise. Show that $$ 14^{-1}<... | 46. We need to consider only the case $t>|x|$. There is no loss of generality in assuming $x>0$. To obtain the estimate from below, set $$ \begin{aligned} a_{1} & =f\left(-\frac{x+t}{2}\right)-f(-(x+t)), & a_{2}=f(0)-f\left(-\frac{x+t}{2}\right), \\ a_{3} & =f\left(\frac{x+t}{2}\right)-f(0), & a_{4}=f(x+t)-f\left(\frac... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 138 | 571 |
1977 | T0 | 7 | null | IMO | 7. (CZS 3) Prove the following assertion: If $c_{1}, c_{2}, \ldots, c_{n}(n \geq 2)$ are real numbers such that $$ (n-1)\left(c_{1}^{2}+c_{2}^{2}+\cdots+c_{n}^{2}\right)=\left(c_{1}+c_{2}+\cdots+c_{n}\right)^{2} $$ then either all these numbers are nonnegative or all these numbers are nonpositive. | 7. Let us suppose that $c_{1} \leq c_{2} \leq \cdots \leq c_{n}$ and that $c_{1}<0<c_{n}$. There exists $k, 1 \leq k<n$, such that $c_{k} \leq 0<c_{k+1}$. Then we have $$ \begin{aligned} (n-1)\left(c_{1}^{2}+c_{2}^{2}+\cdots+c_{n}^{2}\right) \geq & k\left(c_{1}^{2}+\cdots+c_{k}^{2}\right)+(n-k)\left(c_{k+1}^{2}+\cdots+... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 121 | 553 |
1978 | T0 | 13 | null | IMO | 13. (USA 6) ${ }^{\mathrm{IMO} 2}$ Given any point $P$ in the interior of a sphere with radius $R$, three mutually perpendicular segments $P A, P B, P C$ are drawn terminating on the sphere and having one common vertex in $P$. Consider the rectangular parallelepiped of which $P A, P B, P C$ are coterminal edges. Find t... | 13. Lemma. If $M N P Q$ is a rectangle and $O$ any point in space, then $O M^{2}+$ $O P^{2}=O N^{2}+O Q^{2}$. Proof. Let $O_{1}$ be the projection of $O$ onto $M N P Q$, and $m, n, p, q$ denote the distances of $O_{1}$ from $M N, N P, P Q, Q M$, respectively. Then $O M^{2}=O O_{1}^{2}+q^{2}+m^{2}, O N^{2}=O O_{1}^{2}+m... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 121 | 674 |
1978 | T0 | 14 | null | IMO | 14. (VIE 2) Prove that it is possible to place $2 n(2 n+1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times(n+1)$ in a cubic box with edge $2 n+1$ if and only if $n$ is even or $n=1$. Remark. It is assumed that the edges of the pieces of soap are parallel to the edges of the box. | 14. We label the cells of the cube by $\left(a_{1}, a_{2}, a_{3}\right), a_{i} \in\{1,2, \ldots, 2 n+1\}$, in a natural way: for example, as Cartesian coordinates of centers of the cells $\left((1,1,1)\right.$ is one corner, etc.). Notice that there should be $(2 n+1)^{3}-$ $2 n(2 n+1) \cdot 2(n+1)=2 n+1$ void cells, i... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 100 | 1,277 |
1978 | T0 | 17 | null | IMO | 17. (FRA 3) Prove that for any positive integers $x, y, z$ with $x y-z^{2}=1$ one can find nonnegative integers $a, b, c, d$ such that $x=a^{2}+b^{2}, y=c^{2}+d^{2}$, $z=a c+b d$. Set $z=(2 q)$ ! to deduce that for any prime number $p=4 q+1, p$ can be represented as the sum of squares of two integers. | 17. Let $z_{0} \geq 1$ be a positive integer. Supposing that the statement is true for all triples $(x, y, z)$ with $z<z_{0}$, we shall prove that it is true for $z=z_{0}$ too. If $z_{0}=1$, verification is trivial, while $x_{0}=y_{0}$ is obviously impossible. So let there be given a triple $\left(x_{0}, y_{0}, z_{0}\r... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 117 | 1,061 |
1978 | T0 | 5 | null | IMO | 5. (GDR 2) For every integer $d \geq 1$, let $M_{d}$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A=M_{1}, B=M_{2} \backslash\{2\}, C=M_{3}$. Prove that every $c \in C$ may be wri... | 5. We first explicitly describe the elements of the sets $M_{1}, M_{2}$. $x \notin M_{1}$ is equivalent to $x=a+(a+1)+\cdots+(a+n-1)=n(2 a+n-1) / 2$ for some natural numbers $n, a, n \geq 2$. Among $n$ and $2 a+n-1$, one is odd and the other even, and both are greater than 1 ; so $x$ has an odd factor $\geq 3$. On the ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 120 | 507 |
1978 | T0 | 9 | null | IMO | 9. $\mathbf{( G B R} \mathbf{5})^{\mathrm{IMO} 3}$ Let $\{f(n)\}$ be a strictly increasing sequence of positive integers: $0<f(1)<f(2)<f(3)<\cdots$. Of the positive integers not belonging to the sequence, the $n$th in order of magnitude is $f(f(n))+1$. Determine $f(240)$. | 9. Since the $n$th missing number (gap) is $f(f(n))+1$ and $f(f(n))$ is a member of the sequence, there are exactly $n-1$ gaps less than $f(f(n))$. This leads to $$ f(f(n))=f(n)+n-1 $$ Since 1 is not a gap, we have $f(1)=1$. The first gap is $f(f(1))+1=2$. Two consecutive integers cannot both be gaps (the predecessor o... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 95 | 951 |
1979 | T0 | 12 | null | IMO | 12. (GDR 3) Let $R$ be a set of exactly 6 elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions: (i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$; (ii) For any three sets $X, Y, Z$ in $F, X \cup Y \cup Z \neq R$, (iii) $\bigcup_{X \in F} X=... | 12. The first criterion ensures that all sets in an $S$-family are distinct. Since the number of different families of subsets is finite, $h$ has to exist. In fact, we will show that $h=11$. First of all, if there exists $X \in F$ such that $|X| \geq 5$, then by (3) there exists $Y \in F$ such that $X \cup Y=R$. In thi... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 187 | 584 |
1979 | T0 | 26 | null | IMO | 26. (YUG 4) Prove that the functional equations $$ \begin{aligned} f(x+y) & =f(x)+f(y) \\ \text { and } \quad f(x+y+x y) & =f(x)+f(y)+f(x y) \quad(x, y \in \mathbb{R}) \end{aligned} $$ are equivalent. | 26. Let us assume that $f(x+y)=f(x)+f(y)$ for all reals. In this case we trivially apply the equation to get $f(x+y+x y)=f(x+y)+f(x y)=$ $f(x)+f(y)+f(x y)$. Hence the equivalence is proved in the first direction. Now let us assume that $f(x+y+x y)=f(x)+f(y)+f(x y)$ for all reals. Plugging in $x=y=0$ we get $f(0)=0$. Pl... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 83 | 527 |
1979 | T0 | 3 | null | IMO | 3. (BUL 1) Find all polynomials $f(x)$ with real coefficients for which $$ f(x) f\left(2 x^{2}\right)=f\left(2 x^{3}+x\right) $$ | 3. An obvious solution is $f(x)=0$. We now look for nonzero solutions. We note that plugging in $x=0$ we get $f(0)^{2}=f(0)$; hence $f(0)=0$ or $f(0)=1$. If $f(0)=0$, then $f$ is of the form $f(x)=x^{k} g(x)$, where $g(0) \neq 0$. Plugging this formula into $f(x) f\left(2 x^{2}\right)=f\left(2 x^{3}+x\right)$ we get $2... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 52 | 520 |
1979 | T0 | 4 | null | IMO | 4. (BUL 3) ${ }^{\mathrm{IMO} 2} \mathrm{~A}$ pentagonal prism $A_{1} A_{2} \ldots A_{5} B_{1} B_{2} \ldots B_{5}$ is given. The edges, the diagonals of the lateral walls and the internal diagonals of the prism are each colored either red or green in such a way that no triangle whose vertices are vertices of the prism ... | 4. Let us prove first that the edges $A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{5} A_{1}$ are of the same color. Assume the contrary, and let w.l.o.g. $A_{1} A_{2}$ be red and $A_{2} A_{3}$ be green. Three of the segments $A_{2} B_{l}(l=1,2,3,4,5)$, say $A_{2} B_{i}, A_{2} B_{j}, A_{2} B_{k}$, have to be of the same color, ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 124 | 594 |
1979 | T0 | 8 | null | IMO | 8. (FRG 3) For all rational $x$ satisfying $0 \leq x<1, f$ is defined by $$ f(x)= \begin{cases}f(2 x) / 4, & \text { for } 0 \leq x<1 / 2 \\ 3 / 4+f(2 x-1) / 4, & \text { for } 1 / 2 \leq x<1\end{cases} $$ Given that $x=0 . b_{1} b_{2} b_{3} \ldots$ is the binary representation of $x$, find $f(x)$. | 8. By the definition of $f$, it holds that $f\left(0 . b_{1} b_{2} \ldots\right)=3 b_{1} / 4+f\left(0 . b_{2} b_{3} \ldots\right) / 4$ $=0 . b_{1} b_{1}+f\left(0 . b_{2} b_{3} \ldots\right) / 4$. Continuing this argument we obtain $$ f\left(0 . b_{1} b_{2} b_{3} \ldots\right)=0 . b_{1} b_{1} \ldots b_{n} b_{n}+\frac{1}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 143 | 580 |
1981 | T0 | 1 | null | IMO | 1. (BEL) ${ }^{\mathrm{IMO} 4}$ (a) For which values of $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers? (b) For which values of $n>2$ is there a unique set having the stated property? | 1. Assume that the set $\{a-n+1, a-n+2, \ldots, a\}$ of $n$ consecutive numbers satisfies the condition $a \mid \operatorname{lcm}[a-n+1, \ldots, a-1]$. Let $a=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{r}^{\alpha_{r}}$ be the canonic representation of $a$, where $p_{1}<p_{2}<\cdots<p_{r}$ are primes and $\alpha_{... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 87 | 598 |
1981 | T0 | 16 | null | IMO | 16. (GBR) A sequence of real numbers $u_{1}, u_{2}, u_{3}, \ldots$ is determined by $u_{1}$ and the following recurrence relation for $n \geq 1$ : $$ 4 u_{n+1}=\sqrt[3]{64 u_{n}+15} $$ Describe, with proof, the behavior of $u_{n}$ as $n \rightarrow \infty$. | 16. The sequence $\left\{u_{n}\right\}$ is bounded, whatever $u_{1}$ is. Indeed, assume the opposite, and let $u_{m}$ be the first member of the sequence such that $\left|u_{m}\right|>\max \left\{2,\left|u_{1}\right|\right\}$. Then $\left|u_{m-1}\right|=\left|u_{m}^{3}-15 / 64\right|>\left|u_{m}\right|$, which is impos... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 101 | 751 |
1981 | T0 | 2 | null | IMO | 2. (BUL) A sphere $S$ is tangent to the edges $A B, B C, C D, D A$ of a tetrahedron $A B C D$ at the points $E, F, G, H$ respectively. The points $E, F, G, H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $A C$, then it is also tangent to the edge $B D$. | 2. Lemma. Let $E, F, G, H, I$, and $K$ be points on edges $A B, B C, C D, D A$, $A C$, and $B D$ of a tetrahedron. Then there is a sphere that touches the edges at these points if and only if $$ \begin{aligned} & A E=A H=A I, \quad B E=B F=B K, \\ & C F=C G=C I, \quad D G=D H=D K . \end{aligned} $$ Proof. The "only if"... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 100 | 1,865 |
1981 | T0 | 4 | null | IMO | 4. (CAN) Let $\left\{f_{n}\right\}$ be the Fibonacci sequence $\{1,1,2,3,5, \ldots\}$. (a) Find all pairs $(a, b)$ of real numbers such that for each $n, a f_{n}+b f_{n+1}$ is a member of the sequence. (b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $u f_{n}^{2}+v f_{n+1}^{2}$ is a member o... | 4. We shall use the known formula for the Fibonacci sequence $$ f_{n}=\frac{1}{\sqrt{5}}\left(\alpha^{n}-(-1)^{n} \alpha^{-n}\right), \quad \text { where } \alpha=\frac{1+\sqrt{5}}{2} . $$ (a) Suppose that $a f_{n}+b f_{n+1}=f_{k_{n}}$ for all $n$, where $k_{n}>0$ is an integer depending on $n$. By (1), this is equival... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 129 | 749 |
1982 | T0 | 13 | Combinatorics | IMO | 13. C1 (NET 1) ${ }^{\mathrm{IMO} 2} \mathrm{~A}$ scalene triangle $A_{1} A_{2} A_{3}$ is given with sides $a_{1}, a_{2}, a_{3}$ ( $a_{i}$ is the side opposite to $A_{i}$ ). For all $i=1,2,3, M_{i}$ is the midpoint of side $a_{i}, T_{i}$ is the point where the incircle touches side $a_{i}$, and the reflection of $T_{i}... | 13. Points $S_{1}, S_{2}, S_{3}$ clearly lie on the inscribed circle. Let $\widehat{X Y}$ denote the oriented arc $X Y$. The $\operatorname{arcs} \widehat{T_{2} S_{1}}$ and $\widehat{T_{1} T_{3}}$ are equal, since they are symmetric with respect to the bisector of $\angle A_{1}$. Similarly, $\widehat{T_{3} T_{2}}=\wide... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 186 | 680 |
1982 | T0 | 3 | Algebra | IMO | 3. A3 (USS 4) ${ }^{\mathrm{IMO} 3}$ Consider the infinite sequences $\left\{x_{n}\right\}$ of positive real numbers with the following properties: $$ x_{0}=1 \quad \text { and for all } i \geq 0, x_{i+1} \leq x_{i} $$ (a) Prove that for every such sequence there is an $n \geq 1$ such that $\frac{x_{0}^{2}}{x_{1}}+$ ... | 3. (a) By the Cauchy-Schwarz inequality we have $\left(x_{0}^{2} / x_{1}+\cdots+x_{n-1}^{2} / x_{n}\right)$. $\left(x_{1}+\cdots+x_{n}\right) \geq\left(x_{0}+\cdots+x_{n-1}\right)^{2}$. Let us set $X_{n-1}=x_{1}+x_{2}+$ $\cdots+x_{n-1}$. Using $x_{0}=1$, the last inequality can be rewritten as $$ \frac{x_{0}^{2}}{x_{1}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 228 | 978 |
1982 | T0 | 6 | Algebra | IMO | 6. A6 (VIE 1) ${ }^{\text {IMO6 }}$ Let $S$ be a square with sides of length 100 and let $L$ be a path within $S$ that does not meet itself and that is composed of linear segments $A_{0} A_{1}, A_{1} A_{2}, \ldots, A_{n-1} A_{n}$ with $A_{0} \neq A_{n}$. Suppose that for every point $P$ of the boundary of $S$ there is ... | 6. Denote by $d(U, V)$ the distance between points or sets of points $U$ and $V$. For $P, Q \in L$ we shall denote by $L_{P Q}$ the part of $L$ between points $P$ and $Q$ and by $l_{P Q}$ the length of this part. Let us denote by $S_{i}(i=1,2,3,4)$ the vertices of $S$ and by $T_{i}$ points of $L$ such that $S_{i} T_{i}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 210 | 648 |
1983 | T0 | 1 | null | IMO | 1. (AUS 1) The localities $P_{1}, P_{2}, \ldots, P_{1983}$ are served by ten international airlines $A_{1}, A_{2}, \ldots, A_{10}$. It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules offer round-trip flights. Prove that at least one of the airli... | 1. Suppose that there are $n$ airlines $A_{1}, \ldots, A_{n}$ and $N>2^{n}$ cities. We shall prove that there is a round trip by at least one $A_{i}$ containing an odd number of stops. For $n=1$ the statement is trivial, since one airline serves at least 3 cities and hence $P_{1} P_{2} P_{3} P_{1}$ is a round trip with... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 108 | 616 |
1983 | T0 | 11 | null | IMO | 11. (FIN $\mathbf{2}^{\prime}$ ) Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous and satisfy: $$ \begin{aligned} b f(2 x) & =f(x), & & 0 \leq x \leq 1 / 2 \\ f(x) & =b+(1-b) f(2 x-1), & & 1 / 2 \leq x \leq 1 \end{aligned} $$ where $b=\frac{1+c}{2+c}, c>0$. Show that $0<f(x)-x<c$ for every $x, 0<x<1$. | 11. First suppose that the binary representation of $x$ is finite: $x=0, a_{1} a_{2} \ldots a_{n}$ $=\sum_{j=1}^{n} a_{j} 2^{-j}, a_{i} \in\{0,1\}$. We shall prove by induction on $n$ that $$ f(x)=\sum_{j=1}^{n} b_{0} \ldots b_{j-1} a_{j}, \quad \text { where } b_{k}=\left\{\begin{array}{lr} -b & \text { if } a_{k}=0 \... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 149 | 805 |
1983 | T0 | 15 | null | IMO | 15. (POL 3) Decide whether there exists a set $M$ of natural numbers satisfying the following conditions: (i) For any natural number $m>1$ there are $a, b \in M$ such that $a+b=m$. (ii) If $a, b, c, d \in M, a, b, c, d>10$ and $a+b=c+d$, then $a=c$ or $a=d$. | 15. There is no such set. Suppose $M$ satisfies $(a)$ and $(b)$ and let $q_{n}=$ $|\{a \in M: a \leq n\}|$. Consider the differences $b-a$, where $a, b \in M$ and $10<a<b \leq k$. They are all positive and less than $k$, and (b) implies that they are $\left(\frac{q_{k}}{-} \frac{q_{10}}{2}\right)$ different integers. H... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 98 | 505 |
1983 | T0 | 18 | null | IMO | 18. (FRG 3) ${ }^{\mathrm{IMO} 3}$ Let $a, b, c$ be positive integers satisfying $(a, b)=(b, c)=$ $(c, a)=1$. Show that $2 a b c-a b-b c-c a$ is the largest integer not representable as $$ x b c+y c a+z a b $$ with nonnegative integers $x, y, z$. | 18. Let $\left(x_{0}, y_{0}, z_{0}\right)$ be one solution of $b c x+c a y+a b z=n$ (not necessarily nonnegative). By subtracting $b c x_{0}+c a y_{0}+a b z_{0}=n$ we get $$ b c\left(x-x_{0}\right)+c a\left(y-y_{0}\right)+a b\left(z-z_{0}\right)=0 . $$ Since $(a, b)=(a, c)=1$, we must have $a \mid x-x_{0}$ or $x-x_{0}=... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 95 | 625 |
1983 | T0 | 22 | null | IMO | 22. (SWE 4) Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_{1}, a_{2}, \ldots, a_{n}$ of the integers $1,2, \ldots, n$ such that $$ \sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_{k}}{n}=0 $$ | 22. Decompose $n$ into $n=s t$, where the greatest common divisor of $s$ and $t$ is 1 and where $s>1$ and $t>1$. For $1 \leq k \leq n$ put $k=v s+u$, where $0 \leq v \leq t-1$ and $1 \leq u \leq s$, and let $a_{k}=a_{v s+u}$ be the unique integer in the set $\{1,2,3, \ldots, n\}$ such that $v s+u t-a_{v s+u}$ is a mult... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 96 | 756 |
1983 | T0 | 6 | null | IMO | 6. (CAN 2) Suppose that $\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}$ are positive integers for which $x_{1}+x_{2}+\cdots+x_{n}=2(n+1)$. Show that there exists an integer $r$ with $0 \leq r \leq n-1$ for which the following $n-1$ inequalities hold: $$ \begin{aligned} x_{r+1}+\cdots+x_{r+i} & \leq 2 i+1 & & \forall i, 1... | 6. The existence of $r$ : Let $S=\left\{x_{1}+x_{2}+\cdots+x_{i}-2 i \mid i=1,2, \ldots, n\right\}$. Let $\max S$ be attained for the first time at $r^{\prime}$. If $r^{\prime}=n$, then $x_{1}+x_{2}+\cdots+x_{i}-2 i<2$ for $1 \leq i \leq n-1$, so one can take $r=r^{\prime}$. Suppose that $r^{\prime}<n$. Then for $l<n-r... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 233 | 1,210 |
1984 | T0 | 16 | null | IMO | 16. (POL 1) ${ }^{\mathrm{IMO} 6}$ Let $a, b, c, d$ be odd positive integers such that $a<b<c<$ $d$, $a d=b c$, and $a+d=2^{k}, b+c=2^{m}$ for some integers $k$ and $m$. Prove that $a=1$. | 16. From the first two conditions we can easily conclude that $a+d>b+c$ (indeed, $(d+a)^{2}-(d-a)^{2}=(c+b)^{2}-(c-b)^{2}=4 a d=4 b c$ and $d-a>c-b>0)$. Thus $k>m$. From $d=2^{k}-a$ and $c=2^{m}-b$ we get $a\left(2^{k}-a\right)=b\left(2^{m}-b\right)$, or equivalently, $$ (b+a)(b-a)=2^{m}\left(b-2^{k-m} a\right) $$ Sinc... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 83 | 648 |
1984 | T0 | 17 | null | IMO | 17. (FRG 3) In a permutation $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ of the set $1,2, \ldots, n$ we call a pair $\left(x_{i}, x_{j}\right)$ discordant if $i<j$ and $x_{i}>x_{j}$. Let $d(n, k)$ be the number of such permutations with exactly $k$ discordant pairs. Find $d(n, 2)$ and $d(n, 3)$. | 17. For any $m=0,1, \ldots, n-1$, we shall find the number of permutations $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ with exactly $k$ discordant pairs such that $x_{n}=n-m$. This $x_{n}$ is a member of exactly $m$ discordant pairs, and hence the permutation $\left(x_{1}, \ldots, x_{n-1}\right.$ of the set $\{1,2, \ldo... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 119 | 532 |
1984 | T0 | 18 | null | IMO | 18. (USA 5) Inside triangle $A B C$ there are three circles $k_{1}, k_{2}, k_{3}$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_{1}, k_{2}, k_{3}$ are 1, 4, and 9 . Determine the radius of $k$. | 18. Suppose that circles $k_{1}\left(O_{1}, r_{1}\right), k_{2}\left(O_{2}, r_{2}\right)$, and $k_{3}\left(O_{3}, r_{3}\right)$ touch the edges of the angles $\angle B A C, \angle A B C$, and $\angle A C B$, respectively. Denote also by $O$ and $r$ the center and radius of the incircle. Let $P$ be the point of tangency... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 86 | 673 |
1984 | T0 | 3 | null | IMO | 3. (USS 3) Find all positive integers $n$ such that $$ n=d_{6}^{2}+d_{7}^{2}-1 $$ where $1=d_{1}<d_{2}<\cdots<d_{k}=n$ are all positive divisors of the number $n$. | 3. From the equality $n=d_{6}^{2}+d_{7}^{2}-1$ we see that $d_{6}$ and $d_{7}$ are relatively prime and $d_{7}\left|d_{6}^{2}-1=\left(d_{6}-1\right)\left(d_{6}+1\right), d_{6}\right| d_{7}^{2}-1=\left(d_{7}-1\right)\left(d_{7}+1\right)$. Suppose that $d_{6}=a b, d_{7}=c d$ with $1<a<b, 1<c<d$. Then $n$ has 7 divisors s... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 69 | 787 |
1985 | T0 | 12 | null | IMO | 12. 3b.(GBR 4) A sequence of polynomials $P_{m}(x, y, z), m=0,1,2, \ldots$, in $x, y$, and $z$ is defined by $P_{0}(x, y, z)=1$ and by $$ P_{m}(x, y, z)=(x+z)(y+z) P_{m-1}(x, y, z+1)-z^{2} P_{m-1}(x, y, z) $$ for $m>0$. Prove that each $P_{m}(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of ... | 12. We shall prove by induction on $m$ that $P_{m}(x, y, z)$ is symmetric and that $$ (x+y) P_{m}(x, z, y+1)-(x+z) P_{m}(x, y, z+1)=(y-z) P_{m}(x, y, z) $$ holds for all $x, y, z$. This is trivial for $m=0$. Assume now that it holds for $m=n-1$. Since obviously $P_{n}(x, y, z)=P_{n}(y, x, z)$, the symmetry of $P_{n}$ w... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 161 | 587 |
1985 | T0 | 14 | null | IMO | 14. 4b.(IRE 4) A set of 1985 points is distributed around the circumference of a circle and each of the points is marked with 1 or -1 . A point is called "good" if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positiv... | 14. It suffices to prove the existence of a good point in the case of exactly 661 -1 's. We prove by induction on $k$ that in any arrangement with $3 k+2$ points $k$ of which are -1 's a good point exists. For $k=1$ this is clear by inspection. Assume that the assertion holds for all arrangements of $3 n+2$ points and ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 106 | 674 |
1985 | T0 | 15 | null | IMO | 15. 5a.(FRA 3) Let $K$ and $K^{\prime}$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_{1}, T_{2}, \ldots, T_{p}$ with mutually disjoint interiors and find translations $t_{1}, t_{2}, \ldots, t_{p}$ such that $$ K^{\prime}=\bigcup_{i=... | 15. There is no loss of generality if we assume $K=A B C D, K^{\prime}=$ $A B^{\prime} C^{\prime} D^{\prime}$, and that $K^{\prime}$ is obtained from $K$ bya clockwise rotation around $A$ by $\phi, 0 \leq \phi \leq \pi / 4$. Let $C^{\prime} D^{\prime}, B^{\prime} C^{\prime}$, and the parallel to $A B$ through $D^{\prim... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 126 | 917 |
1985 | T0 | 2 | null | IMO | 2. (BRA 3) A polyhedron has 12 faces and is such that: (i) all faces are isosceles triangles, (ii) all edges have length either $x$ or $y$, (iii) at each vertex either 3 or 6 edges meet, and (iv) all dihedral angles are equal. Find the ratio $x / y$. | 2. The polyhedron has $3 \cdot 12 / 2=18$ edges, and by Euler's formula, 8 vertices. Let $v_{1}$ and $v_{2}$ be the numbers of vertices at which respectively 3 and 6 edges meet. Then $v_{1}+v_{2}=8$ and $3 v_{1}+6 v_{2}=2 \cdot 18$, implying that $v_{1}=4$. Let $A, B, C, D$ be the vertices at which three edges meet. Si... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 85 | 534 |
1986 | T0 | 11 | null | IMO | 11. (BUL 1) Let $f(n)$ be the least number of distinct points in the plane such that for each $k=1,2, \ldots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n)$. Simplified version. Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right]$ ( $[x]$ ... | 11. Let $X$ be a finite set in the plane and $l_{k}$ a line containing exactly $k$ points of $X(k=1, \ldots, n)$. Then $l_{n}$ contains $n$ points, $l_{n-1}$ contains at least $n-2$ points not lying on $l_{n}, l_{n-2}$ contains at least $n-4$ points not lying on $l_{n}$ or $l_{n-1}$, etc. It follows that $$ |X| \geq g(... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 114 | 557 |
1986 | T0 | 12 | null | IMO | 12. (GDR 3) ${ }^{\mathrm{IMO} 3}$ To each vertex $P_{i}(i=1, \ldots, 5)$ of a pentagon an integer $x_{i}$ is assigned, the sum $s=\sum x_{i}$ being positive. The following operation is allowed, provided at least one of the $x_{i}$ 's is negative: Choose a negative $x_{i}$, replace it by $-x_{i}$, and add the former va... | 12. We define $f\left(x_{1}, \ldots, x_{5}\right)=\sum_{i=1}^{5}\left(x_{i+1}-x_{i-1}\right)^{2}\left(x_{0}=x_{5}, x_{6}=x_{1}\right)$. Assuming that $x_{3}<0$, according to the rules the lattice vector $X=$ $\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)$ changes into $Y=\left(x_{1}, x_{2}+x_{3},-x_{3}, x_{4}+x_{3}, x... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 163 | 503 |
1986 | T0 | 17 | null | IMO | 17. ( $\mathbf{C H N} 3)^{\mathrm{IMO} 2}$ Let $A, B, C$ be fixed points in the plane. A man starts from a certain point $P_{0}$ and walks directly to $A$. At $A$ he turns his direction by $60^{\circ}$ to the left and walks to $P_{1}$ such that $P_{0} A=A P_{1}$. After he does the same action 1986 times successively ar... | 17. We use complex numbers to represent the position of a point in the plane. For convenience, let $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, \ldots$ be $A, B, C, A, B, \ldots$ respectively, and let $P_{0}$ be the origin. After the $k$ th step, the position of $P_{k}$ will be $P_{k}=A_{k}+\left(P_{k-1}-A_{k}\right) u, k=1,2,3... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 169 | 827 |
1986 | T0 | 19 | null | IMO | 19. (BUL 2) A tetrahedron $A B C D$ is given such that $A D=B C=a$; $A C=B D=b ; A B \cdot C D=c^{2}$. Let $f(P)=A P+B P+C P+D P$, where $P$ is an arbitrary point in space. Compute the least value of $f(P)$. | 19. Let $M$ and $N$ be the midpoints of segments $A B$ and $C D$, respectively. The given conditions imply that $\triangle A B D \cong \triangle B A C$ and $\triangle C D A \cong \triangle D C B$; hence $M C=M D$ and $N A=N B$. It follows that $M$ and $N$ both lie on the perpendicular bisectors of $A B$ and $C D$, and ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 86 | 668 |
1986 | T0 | 4 | null | IMO | 4. (CZS 3) Let $n$ be a positive integer and let $p$ be a prime number, $p>3$. Find at least $3(n+1)$ [easier version: $2(n+1)]$ sequences of positive integers $x, y, z$ satisfying $$ x y z=p^{n}(x+y+z) $$ that do not differ only by permutation. | 4. Let $x=p^{\alpha} x^{\prime}, y=p^{\beta} y^{\prime}, z=p^{\gamma} z^{\prime}$ with $p \nmid x^{\prime} y^{\prime} z^{\prime}$ and $\alpha \geq \beta \geq \gamma$. From the given equation it follows that $p^{n}(x+y)=z\left(x y-p^{n}\right)$ and consequently $z^{\prime} \mid x+y$. Since also $p^{\gamma} \mid x+y$, we... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 88 | 638 |
1986 | T0 | 9 | null | IMO | 9. (GDR 1) ${ }^{\mathrm{IMO} 6}$ Prove or disprove: Given a finite set of points with integer coordinates in the plane, it is possible to color some of these points red and the remaining ones white in such a way that for any straight line $L$ parallel to one of the coordinate axes, the number of red colored points and... | 9. We shall use induction on the number $n$ of points. The case $n=1$ is trivial. Let us suppose that the statement is true for all $1,2, \ldots, n-1$, and that we are given a set $T$ of $n$ points. If there exists a point $P \in T$ and a line $l$ that is parallel to an axis and contains $P$ and no other points of $T$,... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 97 | 531 |
1987 | T0 | 10 | null | IMO | 10. (ICE 3) Let $S_{1}$ and $S_{2}$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of... | 10. Denote by $r, R$ (take w.l.o.g. $r<R$ ) the radii and by $A, B$ the centers of the spheres $S_{1}, S_{2}$ respectively. Let $s$ be the common radius of the spheres in the ring, $C$ the center of one of them, say $S$, and $D$ the foot of the perpendicular from $C$ to $A B$. The centers of the spheres in the ring for... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 116 | 596 |
1987 | T0 | 12 | null | IMO | 12. (POL 5) Given a nonequilateral triangle $A B C$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A^{\prime} B^{\prime} C^{\prime}$ (the vertices listed counterclockwise) for which the triples of points $A, B^{\prime}, C^{\prime} ; A^{\prime}, B, C^{\prime}$; and $... | 12. Here all angles will be oriented and measured counterclockwise. Note that $\measuredangle C A^{\prime} B=\measuredangle A B^{\prime} C=$ $\measuredangle B C^{\prime} A=\pi / 3$. Let $a^{\prime}, b^{\prime}, c^{\prime}$ denote respectively the inner bisectors of angles $A^{\prime}, B^{\prime}, C^{\prime}$ in triangl... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 114 | 719 |
1987 | T0 | 16 | null | IMO | 16. (FRG 3) ${ }^{\mathrm{IMO1}}$ Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_{n}(k)$. Prove: (a) $\sum_{k=0}^{n} k p_{n}(k)=n$ !; (b) $\sum_{k=0}^{n}(k-1)^{2} p_{n}(k)=n$ !. | 16. We assume that $S=\{1,2, \ldots, n\}$, and use the obvious fact $$ \sum_{k=0}^{n} p_{n}(k)=n! $$ (a) To each permutation $\pi$ of $S$ we assign an $n$-vector $\left(e_{1}, e_{2}, \ldots, e_{n}\right)$, where $e_{i}$ is 1 if $i$ is a fixed point of $\pi$, and 0 otherwise. Since exactly $p_{n}(k)$ of the assigned vec... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 117 | 710 |
1987 | T0 | 2 | null | IMO | 2. (USA 3) At a party attended by $n$ married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques $C_{1}, C_{2}, \ldots, C_{k}$ with the following property: no couple are members of the same clique, but for every other pair of persons t... | 2. Let $d_{i}$ denote the number of cliques of which person $i$ is a member. Clearly $d_{i} \geq 2$. We now distinguish two cases: (i) For some $i, d_{i}=2$. Suppose that $i$ is a member of two cliques, $C_{p}$ and $C_{q}$. Then $\left|C_{p}\right|=\left|C_{q}\right|=n$, since for each couple other than $i$ and his/her... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 113 | 758 |
1987 | T0 | 4 | null | IMO | 4. (FRA 5) Let $A B C D E F G H$ be a parallelepiped with $A E\|B F\| C G \| D H$. Prove the inequality $$ A F+A H+A C \leq A B+A D+A E+A G $$ In what cases does equality hold? | 4. Setting $x_{1}=\overrightarrow{A B}, x_{2}=\overrightarrow{A D}, x_{3}=\overrightarrow{A E}$, we have to prove that $$ \left\|x_{1}+x_{2}\right\|+\left\|x_{2}+x_{3}\right\|+\left\|x_{3}+x_{1}\right\| \leq\left\|x_{1}\right\|+\left\|x_{2}\right\|+\left\|x_{3}\right\|+\left\|x_{1}+x_{2}+x_{3}\right\| . $$ We have $$ \... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 72 | 1,764 |
1987 | T0 | 7 | null | IMO | 7. (NET 1) Given five real numbers $u_{0}, u_{1}, u_{2}, u_{3}, u_{4}$, prove that it is always possible to find five real numbers $v_{0}, v_{1}, v_{2}, v_{3}, v_{4}$ that satisfy the following conditions: (i) $u_{i}-v_{i} \in \mathbb{N}$. (ii) $\sum_{0 \leq i<j \leq 4}\left(v_{i}-v_{j}\right)^{2}<4$. | 7. For all real numbers $v$ the following inequality holds: $$ \sum_{0 \leq i<j \leq 4}\left(v_{i}-v_{j}\right)^{2} \leq 5 \sum_{i=0}^{4}\left(v_{i}-v\right)^{2} $$ Indeed, $$ \begin{aligned} \sum_{0 \leq i<j \leq 4}\left(v_{i}-v_{j}\right)^{2} & =\sum_{0 \leq i<j \leq 4}\left[\left(v_{i}-v\right)-\left(v_{j}-v\right)\... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 126 | 611 |
1987 | T0 | 8 | null | IMO | 8. (HUN 1) (a) Let $(m, k)=1$. Prove that there exist integers $a_{1}, a_{2}, \ldots, a_{m}$ and $b_{1}, b_{2}, \ldots, b_{k}$ such that each product $a_{i} b_{j}(i=1,2, \ldots, m ; j=$ $1,2, \ldots, k)$ gives a different residue when divided by $m k$. (b) Let $(m, k)>1$. Prove that for any integers $a_{1}, a_{2}, \ldo... | 8. (a) Consider $$ a_{i}=i k+1, \quad i=1,2, \ldots, m ; \quad b_{j}=j m+1, \quad j=1,2, \ldots, k $$ Assume that $m k \mid a_{i} b_{j}-a_{s} b_{t}=(i k+1)(j m+1)-(s k+1)(t m+1)=$ $k m(i j-s t)+m(j-t)+k(i-s)$. Since $m$ divides this sum, we get that $m \mid k(i-s)$, or, together with $\operatorname{gcd}(k, m)=1$, that ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 211 | 622 |
1988 | T0 | 1 | null | IMO | 1. (BUL 1) An integer sequence is defined by $$ a_{n}=2 a_{n-1}+a_{n-2} \quad(n>1), \quad a_{0}=0, \quad a_{1}=1 $$ Prove that $2^{k}$ divides $a_{n}$ if and only if $2^{k}$ divides $n$. | 1. Assume that $p$ and $q$ are real and $b_{0}, b_{1}, b_{2}, \ldots$ is a sequence such that $b_{n}=p b_{n-1}+q b_{n-2}$ for all $n>1$. From the equalities $b_{n}=p b_{n-1}+q b_{n-2}$, $b_{n+1}=p b_{n}+q b_{n-1}, b_{n+2}=p b_{n+1}+q b_{n}$, eliminating $b_{n+1}$ and $b_{n-1}$ we obtain that $b_{n+2}=\left(p^{2}+2 q\ri... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 84 | 1,086 |
1988 | T0 | 10 | null | IMO | 10. (GDR 1) Let $N=\{1,2, \ldots, n\}, n \geq 2$. A collection $F=\left\{A_{1}, \ldots, A_{t}\right\}$ of subsets $A_{i} \subseteq N, i=1, \ldots, t$, is said to be separating if for every pair $\{x, y\} \subseteq N$, there is a set $A_{i} \in F$ such that $A_{i} \cap\{x, y\}$ contains just one element. A collection $F... | 10. We claim that if the family $\left\{A_{1}, \ldots, A_{t}\right\}$ separates the $n$-set $N$, then $2^{t} \geq n$. The proof goes by induction. The case $t=1$ is clear, so suppose that the claim holds for $t-1$. Since $A_{t}$ does not separate elements of its own or its complement, it follows that $\left\{A_{1}, \ld... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 210 | 927 |
1988 | T0 | 11 | null | IMO | 11. (GDR 3) The lock on a safe consists of three wheels, each of which may be set in eight different positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations that must be tried if one is to guarantee being... | 11. The answer is 32 . Write the combinations as triples $k=(x, y, z), 0 \leq$ $x, y, z \leq 7$. Define the sets $K_{1}=\{(1,0,0),(0,1,0),(0,0,1),(1,1,1)\}$, $K_{2}=\{(2,0,0),(0,2,0),(0,0,2),(2,2,2)\}, K_{3}=\{(0,0,0),(4,4,4)\}$, and $K=\left\{k=k_{1}+k_{2}+k_{3} \mid k_{i} \in K_{i}, i=1,2,3\right\}$. There are 32 com... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 89 | 938 |
1988 | T0 | 12 | null | IMO | 12. (GRE 2) In a triangle $A B C$, choose any points $K \in B C, L \in A C$, $M \in A B, N \in L M, R \in M K$, and $F \in K L$. If $E_{1}, E_{2}, E_{3}, E_{4}, E_{5}$, $E_{6}$, and $E$ denote the areas of the triangles $A M R, C K R, B K F, A L F$, $B N M, C L N$, and $A B C$ respectively, show that $$ E \geq 8 \sqrt... | 12. Let $E(X Y Z)$ stand for the area of a triangle $X Y Z$. We have $$ \begin{gathered} \frac{E_{1}}{E}=\frac{E(A M R)}{E(A M K)} \cdot \frac{E(A M K)}{E(A B K)} \cdot \frac{E(A B K)}{E(A B C)}=\frac{M R}{M K} \cdot \frac{A M}{A B} \cdot \frac{B K}{B C} \Rightarrow \\ \left(\frac{E_{1}}{E}\right)^{1 / 3} \leq \frac{1}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 214 | 617 |
1988 | T0 | 13 | null | IMO | 13. (GRE 3) ${ }^{\mathrm{IMO5}}$ In a right-angled triangle $A B C$, let $A D$ be the altitude drawn to the hypotenuse and let the straight line joining the incenters of the triangles $A B D, A C D$ intersect the sides $A B, A C$ at the points $K, L$ respectively. If $E$ and $E_{1}$ denote the areas of the triangles $... | 13. Let $A B=c, A C=b, \angle C B A=\beta, B C=a$, and $A D=h$. Let $r_{1}$ and $r_{2}$ be the inradii of $A B D$ and $A D C$ respectively and $O_{1}$ and $O_{2}$ the centers of the respective incircles. We obviously have $r_{1} / r_{2}=$ $c / b$. We also have $D O_{1}=\sqrt{2} r_{1}$, $D O_{2}=\sqrt{2} r_{2}$, and $\a... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 130 | 594 |
1988 | T0 | 14 | null | IMO | 14. (HUN 1) For what values of $n$ does there exist an $n \times n$ array of entries $-1,0$, or 1 such that the $2 n$ sums obtained by summing the elements of the rows and the columns are all different? | 14. Consider an array $\left[a_{i j}\right]$ of the given property and denote the sums of the rows and the columns by $r_{i}$ and $c_{j}$ respectively. Among the $r_{i}$ 's and $c_{j}$ 's, one element of $[-n, n]$ is missing, so that there are at least $n$ nonnegative and $n$ nonpositive sums. By permuting rows and col... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 62 | 804 |
1988 | T0 | 17 | null | IMO | 17. (ISR 2) In the convex pentagon $A B C D E$, the sides $B C, C D, D E$ have the same length. Moreover, each diagonal of the pentagon is parallel to a side ( $A C$ is parallel to $D E, B D$ is parallel to $A E$, etc.). Prove that $A B C D E$ is a regular pentagon. | 17. Let $A C$ and $A D$ meet $B E$ in $R, S$, respectively. Then by the conditions of the problem, $$ \begin{aligned} & \angle A E B=\angle E B D=\angle B D C=\angle D B C=\angle A D B=\angle E A D=\alpha, \\ & \angle A B E=\angle B E C=\angle E C D=\angle C E D=\angle A C E=\angle B A C=\beta \\ & \angle B C A=\angle ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 90 | 644 |
1988 | T0 | 2 | null | IMO | 2. (BUL 3) Let $n$ be a positive integer. Find the number of odd coefficients of the polynomial $$ u_{n}(x)=\left(x^{2}+x+1\right)^{n} $$ | 2. For polynomials $f(x), g(x)$ with integer coefficients, we use the notation $f(x) \sim g(x)$ if all the coefficients of $f-g$ are even. Let $n=2^{s}$. It is immediately shown by induction that $\left(x^{2}+x+1\right)^{2^{s}} \sim x^{2^{s+1}}+x^{2^{s}}+1$, and the required number for $n=2^{s}$ is 3 . Let $n=2^{s}-1$.... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 51 | 1,173 |
1988 | T0 | 20 | null | IMO | 20. (MON 4) Find the least natural number $n$ such that if the set $\{1,2, \ldots, n\}$ is arbitrarily divided into two nonintersecting subsets, then one of the subsets contains three distinct numbers such that the product of two of them equals the third. | 20. Suppose that $A_{n}=\{1,2, \ldots, n\}$ is partitioned into $B_{n}$ and $C_{n}$, and that neither $B_{n}$ nor $C_{n}$ contains 3 distinct numbers one of which is equal to the product of the other two. If $n \geq 96$, then the divisors of 96 must be split up. Let w.l.o.g. $2 \in B_{n}$. There are four cases. (i) $3 ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 65 | 620 |
1988 | T0 | 21 | null | IMO | 21. (POL 4) Forty-nine students solve a set of three problems. The score for each problem is a whole number of points from 0 to 7 . Prove that there exist two students $A$ and $B$ such that for each problem, $A$ will score at least as many points as $B$. | 21. Let $X$ be the set of all ordered triples $a=\left(a_{1}, a_{2}, a_{3}\right)$ for $a_{i} \in\{0,1, \ldots, 7\}$. Write $a \prec b$ if $a_{i} \leq b_{i}$ for $i=1,2,3$ and $a \neq b$. Call a subset $Y \subset X$ independent if there are no $a, b \in Y$ with $a \prec b$. We shall prove that an independent set contai... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 70 | 679 |
1988 | T0 | 22 | null | IMO | 22. (KOR 2) Let $p$ be the product of two consecutive integers greater than 2. Show that there are no integers $x_{1}, x_{2}, \ldots, x_{p}$ satisfying the equation $$ \sum_{i=1}^{p} x_{i}^{2}-\frac{4}{4 p+1}\left(\sum_{i=1}^{p} x_{i}\right)^{2}=1 $$ Alternative formulation. Show that there are only two values of $p$... | 22. Set $X=\sum_{i=1}^{p} x_{i}$ and w.l.o.g. assume that $X \geq 0$ (if $\left(x_{1}, \ldots, x_{p}\right)$ is a solution, then $\left(-x_{1}, \ldots,-x_{p}\right)$ is a solution too). Since $x^{2} \geq x$ for all integers $x$, it follows that $\sum_{i=1}^{p} x_{i}^{2} \geq X$. If the last inequality is an equality, t... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 144 | 634 |
1988 | T0 | 23 | null | IMO | 23. (SIN 2) Let $Q$ be the center of the inscribed circle of a triangle $A B C$. Prove that for any point $P$, $a(P A)^{2}+b(P B)^{2}+c(P C)^{2}=a(Q A)^{2}+b(Q B)^{2}+c(Q C)^{2}+(a+b+c)(Q P)^{2}$, where $a=B C, b=C A$, and $c=A B$. | 23. Denote by $R$ the intersection point of lines $A Q$ and $B C$. We know that $B R: R C=c: b$ and $A Q: Q R=(b+c): a$. By applying Stewart's theorem to $\triangle P B C$ and $\triangle P A R$ we obtain $$ \begin{aligned} a \cdot A P^{2} & +b \cdot B P^{2}+c \cdot C P^{2}=a P A^{2}+(b+c) P R^{2}+(b+c) R B \cdot R C \\... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 113 | 506 |
1988 | T0 | 26 | null | IMO | 26. (GBR 2) ${ }^{\mathrm{IMO} 3} \mathrm{~A}$ function $f$ defined on the positive integers (and taking positive integer values) is given by $$ \begin{aligned} f(1) & =1, \quad f(3)=3 \\ f(2 n) & =f(n) \\ f(4 n+1) & =2 f(2 n+1)-f(n) \\ f(4 n+3) & =3 f(2 n+1)-2 f(n) \end{aligned} $$ for all positive integers $n$. Det... | 26. The overline in this problem will exclusively denote binary representation. We will show by induction that if $n=\overline{c_{k} c_{k-1} \ldots c_{0}}=\sum_{i=0}^{k} c_{i} 2^{i}$ is the binary representation of $n\left(c_{i} \in\{0,1\}\right)$, then $f(n)=\overline{c_{0} c_{1} \ldots c_{k}}=$ $\sum_{i=0}^{k} c_{i} ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 161 | 821 |
1988 | T0 | 27 | null | IMO | 27. (GBR 4) The triangle $A B C$ is acute-angled. Let $L$ be any line in the plane of the triangle and let $u, v, w$ be the lengths of the perpendiculars from $A, B, C$ respectively to $L$. Prove that $$ u^{2} \tan A+v^{2} \tan B+w^{2} \tan C \geq 2 \Delta $$ where $\Delta$ is the area of the triangle, and determine ... | 27. Consider a Cartesian system with the $x$-axis on the line $B C$ and origin at the foot of the perpendicular from $A$ to $B C$, so that $A$ lies on the $y$-axis. Let $A$ be $(0, \alpha), B(-\beta, 0), C(\gamma, 0)$, where $\alpha, \beta, \gamma>0$ (because $A B C$ is acute-angled). Then $\tan B=\frac{\alpha}{\beta},... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 121 | 628 |
1988 | T0 | 29 | null | IMO | 29. (USA 3) A number of signal lights are equally spaced along a one-way railroad track, labeled in order $1,2, \ldots, N(N \geq 2)$. As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the nu... | 29. Let the first train start from Signal 1 at time 0, and let $t_{j}$ be the time it takes for the $j$ th train in the series to travel from one signal to the next. By induction on $k$, we show that Train $k$ arrives at signal $n$ at time $s_{k}+(n-2) m_{k}$, where $s_{k}=t_{1}+\cdots+t_{k}$ and $m_{k}=\max _{j=1, \ld... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 214 | 561 |
1988 | T0 | 3 | null | IMO | 3. (CAN 1) The triangle $A B C$ is inscribed in a circle. The interior bisectors of the angles $A, B$, and $C$ meet the circle again at $A^{\prime}, B^{\prime}$, and $C^{\prime}$ respectively. Prove that the area of triangle $A^{\prime} B^{\prime} C^{\prime}$ is greater than or equal to the area of triangle $A B C$. | 3. Let $R$ be the circumradius, $r$ the inradius, $s$ the semiperimeter, $\Delta$ the area of $A B C$ and $\Delta^{\prime}$ the area of $A^{\prime} B^{\prime} C^{\prime}$. The angles of triangle $A^{\prime} B^{\prime} C^{\prime}$ are $A^{\prime}=90^{\circ}-A / 2, B^{\prime}=90^{\circ}-B / 2$, and $C^{\prime}=90^{\circ}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 101 | 746 |
1988 | T0 | 30 | null | IMO | 30. (USS 1) A point $M$ is chosen on the side $A C$ of the triangle $A B C$ in such a way that the radii of the circles inscribed in the triangles $A B M$ and $B M C$ are equal. Prove that $$ B M^{2}=\Delta \cot \frac{B}{2} $$ where $\Delta$ is the area of the triangle $A B C$. | 30. Let $\Delta_{1}, s_{1}, r^{\prime}$ denote the area, semiperimeter, and inradius of triangle $A B M, \Delta_{2}, s_{2}, r^{\prime}$ the same quantities for triangle $M B C$, and $\Delta, s, r$ those for $\triangle A B C$. Also, let $P^{\prime}$ and $Q^{\prime}$ be the points of tangency of the incircle of $\triangl... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 99 | 535 |
1988 | T0 | 5 | null | IMO | 5. (CZS 2) ${ }^{\mathrm{IMO} 2}$ Let $n$ be an even positive integer. Let $A_{1}, A_{2}, \ldots, A_{n+1}$ be sets having $n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $n$ can one assign to every ele... | 5. Let $n=2 k$ and let $A=\left\{A_{1}, \ldots, A_{2 k+1}\right\}$ denote the family of sets with the desired properties. Since every element of their union $B$ belongs to at least two sets of $A$, it follows that $A_{j}=\bigcup_{i \neq j} A_{i} \cap A_{j}$ holds for every $1 \leq j \leq 2 k+1$. Since each intersection... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 128 | 583 |
1988 | T0 | 6 | null | IMO | 6. (CZS 3) In a given tetrahedron $A B C D$ let $K$ and $L$ be the centers of edges $A B$ and $C D$ respectively. Prove that every plane that contains the line $K L$ divides the tetrahedron into two parts of equal volume. | 6. Let $\omega$ be the plane through $A B$, parallel to $C D$. Define the point transformation $f: X \mapsto X^{\prime}$ in space as follows. If $X \in K L$, then $X^{\prime}=X$; otherwise, let $\omega_{X}$ be the plane through $X$ parallel to $\omega$ : then $X^{\prime}$ is the point symmetric to $X$ with respect to t... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 72 | 861 |
1988 | T0 | 7 | null | IMO | 7. (FRA 2) Let $a$ be the greatest positive root of the equation $x^{3}-3 x^{2}+1=$ 0 . Show that $\left[a^{1788}\right]$ and $\left[a^{1988}\right]$ are both divisible by 17 . ( $[x]$ denotes the integer part of $x$.) | 7. The algebraic equation $x^{3}-3 x^{2}+1=0$ admits three real roots $\beta, \gamma, a$, with $$ -0.6<\beta<-0.5, \quad 0.6<\gamma<0.7, \quad \sqrt{8}<a<3 . $$ Define, for all integers $n$, $$ u_{n}=\beta^{n}+\gamma^{n}+a^{n} . $$ It holds that $u_{n+3}=3 u_{n+2}-u_{n}$. Obviously, $0<\beta^{n}+\gamma^{n}<1$ for all $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 83 | 611 |
1988 | T0 | 8 | null | IMO | 8. (FRA 3) Let $u_{1}, u_{2}, \ldots, u_{m}$ be $m$ vectors in the plane, each of length less than or equal to 1 , which add up to zero. Show that one can rearrange $u_{1}, u_{2}, \ldots, u_{m}$ as a sequence $v_{1}, v_{2}, \ldots, v_{m}$ such that each partial sum $v_{1}, v_{1}+v_{2}, v_{1}+v_{2}+v_{3}, \ldots, v_{1}+... | 8. Consider first the case that the vectors are on the same line. Then if $e$ is a unit vector, we can write $u_{1}=x_{1} e, \ldots, u_{n}=x_{n} e$ for scalars $x_{i},\left|x_{i}\right| \leq 1$, with zero sum. It is now easy to permute $x_{1}, x_{2}, \ldots, x_{n}$ into $z_{1}, z_{2}, \ldots z_{n}$ so that $\left|z_{1}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 161 | 1,056 |
1989 | T0 | 1 | null | IMO | 1. (AUS 2) ${ }^{\mathrm{IMO} 2}$ Let $A B C$ be a triangle. The bisector of angle $A$ meets the circumcircle of triangle $A B C$ in $A_{1}$. Points $B_{1}$ and $C_{1}$ are defined similarly. Let $A A_{1}$ meet the lines that bisect the two external angles at $B$ and $C$ in point $A^{0}$. Define $B^{0}$ and $C^{0}$ sim... | 1. Let $I$ denote the intersection of the three internal bisectors. Then $I A_{1}=A_{1} A^{0}$. One way proving this is to realize that the circumcircle of $A B C$ is the nine-point circle of $A^{0} B^{0} C^{0}$, hence it bisects $I A^{0}$, since $I$ is the orthocenter of $A^{0} B^{0} C^{0}$. Another way is through not... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 211 | 532 |
1989 | T0 | 14 | null | IMO | 14. (IND 2) A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals. | 14. Lemma 1. In a quadrilateral $A B C D$ circumscribed about a circle, with points of tangency $P, Q, R, S$ on $D A, A B, B C, C D$ respectively, the lines $A C, B D, P R, Q S$ concur. Proof. Follows immediately, for example, from Brianchon's theorem. Lemma 2. Let a variable chord $X Y$ of a circle $C(I, r)$ subtend a... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 59 | 804 |
1989 | T0 | 15 | null | IMO | 15. (IRE 1) Let $a, b, c, d, m, n$ be positive integers such that $a^{2}+b^{2}+c^{2}+d^{2}=$ 1989, $a+b+c+d=m^{2}$, and the largest of $a, b, c, d$ is $n^{2}$. Determine, with proof, the values of $m$ and $n$. | 15. By Cauchy's inequality, $44<\sqrt{1989}<a+b+c+d \leq \sqrt{2 \cdot 1989}<90$. Since $m^{2}=a+b+c+d$ is of the same parity as $a^{2}+b^{2}+c^{2}+d^{2}=1989$, $m^{2}$ is either 49 or 81 . Let $d=\max \{a, b, c, d\}$. Suppose that $m^{2}=49$. Then $(49-d)^{2}=(a+b+c)^{2}>a^{2}+b^{2}+c^{2}=$ $1989-d^{2}$, and so $d^{2}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 99 | 500 |
1989 | T0 | 21 | null | IMO | 21. (NET 2) Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $120^{\circ}$. | 21. In order to obtain a triangle as the intersection we must have three points $P, Q, R$ on three sides of the tetrahedron passing through one vertex, say $T$. It is clear that we may suppose w.l.o.g. that $P$ is a vertex, and $Q$ and $R$ lie on the edges $T P_{1}$ and $T P_{2}\left(P_{1}, P_{2}\right.$ are vertices) ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 54 | 604 |
1989 | T0 | 22 | null | IMO | 22. (PHI 1) ${ }^{\text {IMO1 }}$ Prove that the set $\{1,2, \ldots, 1989\}$ can be expressed as the disjoint union of 17 subsets $A_{1}, A_{2}, \ldots, A_{17}$ such that: (i) each $A_{i}$ contains the same number of elements; (ii) the sum of all elements of each $A_{i}$ is the same for $i=1,2, \ldots, 17$. | 22. The statement remains valid if 17 is replaced by any divisor $k$ of $1989=3^{2}$. $13 \cdot 17,1<k<1989$, so let $k$ be one such divisor. The set $\{1,2, \ldots, 1989\}$ can be partitioned as $\{1,2, \ldots, 3 k\} \cup \bigcup_{j=1}^{L}\{(2 j+1) k+1,(2 j+1) k+$ $2, \ldots,(2 j+1) k+2 k\}=X \cup Y_{1} \cup \cdots \c... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 124 | 548 |
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