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 |
|---|---|---|---|---|---|---|---|---|---|
1989 | T0 | 23 | null | IMO | 23. (POL 2) ${ }^{\mathrm{IMO} 6} \mathrm{We}$ consider permutations $\left(x_{1}, \ldots, x_{2 n}\right)$ of the set $\{1, \ldots$, $2 n\}$ such that $\left|x_{i}-x_{i+1}\right|=n$ for at least one $i \in\{1, \ldots, 2 n-1\}$. For every natural number $n$, find out whether permutations with this property are more or l... | 23. Two numbers $x, y \in\{1, \ldots, 2 n\}$ will be called twins if $|x-y|=n$. Then the set $\{1, \ldots, 2 n\}$ splits into $n$ pairs of twins. A permutation $\left(x_{1}, \ldots, x_{2 n}\right)$ of this set is said to be of type $T_{k}$ if $\left|x_{i}-x_{i+1}\right|=n$ holds for exactly $k$ indices $i$ (thus a perm... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 139 | 1,132 |
1989 | T0 | 28 | null | IMO | 28. (ROM 2) Consider in a plane $\Pi$ the points $O, A_{1}, A_{2}, A_{3}, A_{4}$ such that $\sigma\left(O A_{i} A_{j}\right) \geq 1$ for all $i, j=1,2,3,4, i \neq j$. Prove that there is at least one pair $i_{0}, j_{0} \in\{1,2,3,4\}$ such that $\sigma\left(O A_{i_{0}} A_{j_{0}}\right) \geq \sqrt{2}$. (We have denoted ... | 28. Assume w.l.o.g. that the rays $O A_{1}, O A_{2}, O A_{3}, O A_{4}$ are arranged clockwise. Setting $O A_{1}=a, O A_{2}=b, O A_{3}=c, O A_{4}=d$, and $\angle A_{1} O A_{2}=x$, $\angle A_{2} O A_{3}=y, \angle A_{3} O A_{4}=z$, we have $$ \begin{aligned} & S_{1}=\sigma\left(O A_{1} A_{2}\right)=\frac{1}{2} a b|\sin x|... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 178 | 561 |
1989 | T0 | 29 | null | IMO | 29. (ROM 4) A flock of 155 birds sit down on a circle $C$. Two birds $P_{i}, P_{j}$ are mutually visible if $m\left(P_{i} P_{j}\right) \leq 10^{\circ}$. Find the smallest number of mutually visible pairs of birds. (One assumes that a position (point) on $C$ can be occupied simultaneously by several birds.) | 29. Let $P_{i}$, sitting at the place $A$, and $P_{j}$ sitting at $B$, be two birds that can see each other. Let $k$ and $l$ respectively be the number of birds visible from $B$ but not from $A$, and the number of those visible from $A$ but not from $B$. Assume that $k \geq l$. Then if all birds from $B$ fly to $A$, ea... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 96 | 581 |
1989 | T0 | 3 | null | IMO | 3. (AUS 4) Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a di... | 3. Let the carpet have width $x$, length $y$. Let the length of the storerooms be $q$. Let $y / x=k$. Then, as in the previous problem, $(k q-50)^{2}+(50 k-q)^{2}=$ $(k q-38)^{2}+(38 k-q)^{2}$, i.e., $$ k q=22\left(k^{2}+1\right) $$ Also, as before, $x^{2}=\left(\frac{k q-50}{k^{2}-1}\right)^{2}+\left(\frac{50 k-q}{k^{... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 124 | 586 |
1989 | T0 | 31 | null | IMO | 31. (SWE 3) Let $a_{1} \geq a_{2} \geq a_{3}$ be given positive integers and let $N\left(a_{1}, a_{2}, a_{3}\right)$ be the number of solutions $\left(x_{1}, x_{2}, x_{3}\right)$ of the equation $$ \frac{a_{1}}{x_{1}}+\frac{a_{2}}{x_{2}}+\frac{a_{3}}{x_{3}}=1 $$ where $x_{1}, x_{2}$, and $x_{3}$ are positive integers... | 31. Let us denote by $N_{p q r}$ the number of solutions for which $a_{p} / x_{p} \geq a_{q} / x_{q} \geq$ $a_{r} / x_{r}$, where $(p, q, r)$ is one of six permutations of $(1,2,3)$. It is clearly enough to prove that $N_{p q r}+N_{q p r} \leq 2 a_{1} a_{2}\left(3+\ln \left(2 a_{1}\right)\right)$. First, from $$ \frac{... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 192 | 1,297 |
1989 | T0 | 8 | null | IMO | 8. (FRA 2) Let $R$ be a rectangle that is the union of a finite number of rectangles $R_{i}, 1 \leq i \leq n$, satisfying the following conditions: (i) The sides of every rectangle $R_{i}$ are parallel to the sides of $R$. (ii) The interiors of any two different $R_{i}$ are disjoint. (iii) Every $R_{i}$ has at least on... | 8. Let $A, B, C, D$ denote the vertices of $R$. We consider the set $\mathcal{S}$ of all points $E$ of the plane that are vertices of at least one rectangle, and its subset $\mathcal{S}^{\prime}$ consisting of those points in $\mathcal{S}$ that have both coordinates integral in the orthonormal coordinate system with po... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 118 | 608 |
1989 | T0 | 9 | null | IMO | 9. (FRA 4) For all integers $n, n \geq 0$, there exist uniquely determined integers $a_{n}, b_{n}, c_{n}$ such that $$ (1+4 \sqrt[3]{2}-4 \sqrt[3]{4})^{n}=a_{n}+b_{n} \sqrt[3]{2}+c_{n} \sqrt[3]{4} $$ Prove that $c_{n}=0$ implies $n=0$. | 9. From $a_{n+1}+b_{n+1} \sqrt[3]{2}+c_{n+1} \sqrt[3]{4}=\left(a_{n}+b_{n} \sqrt[3]{2}+c_{n} \sqrt[3]{4}\right)(1+4 \sqrt[3]{2}-4 \sqrt[3]{4})$ we obtain $a_{n+1}=a_{n}-8 b_{n}+8 c_{n}$. Since $a_{0}=1, a_{n}$ is odd for all $n$. For an integer $k>0$, we can write $k=2^{l} k^{\prime}, k^{\prime}$ being odd and $l$ a no... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 111 | 943 |
1990 | T0 | 1 | null | IMO | 1. (AUS 3) The integer 9 can be written as a sum of two consecutive integers: $9=4+5$. Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $9=4+5=2+3+4$. Is there an integer that can be written as a sum of 1990 consecutive integers and that can be written as a sum ... | 1. Let $N$ be a number that can be written as a sum of 1990 consecutive integers and as a sum of consecutive positive integers in exactly 1990 ways. The former requirement gives us $N=m+(m+1)+\cdots+(m+1989)=$ $995(2 m+1989)$ for some $m$. Thus $2 \nmid N, 5 \mid N$, and $199 \mid N$. The latter requirement tells us th... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 111 | 658 |
1990 | T0 | 10 | null | IMO | 10. (ICE 2) A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone. | 10. Let $r$ be the radius of the base and $h$ the height of the cone. We may assume w.l.o.g. that $r=1$. Let $A$ be the top of the cone, $B C$ the diameter of the circumference of the base such that the plane touches the circumference at $B, O$ the center of the base, and $H$ the midpoint of $O A$ (also belonging to th... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 59 | 652 |
1990 | T0 | 13 | null | IMO | 13. (IRE 2) An eccentric mathematician has a ladder with $n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $a$ rungs of the ladder, and when he descends, each step he takes covers $b$ rungs of the ladder, where $a$ and $b$ are fixed positive integers. By a se... | 13. We will call the ground the "zeroth" rung. We will prove that the minimum $n$ is $n=a+b-(a, b)$. It is plain that if $(a, b)=k>1$, the scientist can climb only onto the rungs divisible by $k$ and we can just observe these rungs to obtain the situation equivalent to $a^{\prime}=a / k, b^{\prime}=b / k$, and $n^{\pri... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 140 | 574 |
1990 | T0 | 15 | null | IMO | 15. (MEX 2) Determine for which positive integers $k$ the set $$ X=\{1990,1990+1,1990+2, \ldots, 1990+k\} $$ can be partitioned into two disjoint subsets $A$ and $B$ such that the sum of the elements of $A$ is equal to the sum of the elements of $B$. | 15. Let $S(Z)$ denote the sum of all the elements of a set $Z$. We have $S(X)=$ $(k+1) \cdot 1990+\frac{k(k+1)}{2}$. To partition the set into two parts with equal sums, $S(X)$ must be even and hence $\frac{k(k+1)}{2}$ must be even. Hence $k$ is of the form $4 r$ or $4 r+3$, where $r$ is an integer. For $k=4 r+3$ we ca... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 95 | 610 |
1990 | T0 | 16 | null | IMO | 16. (NET 1) ${ }^{\text {IMO6 }}$ Is there a 1990-gon with the following properties (i) and (ii)? (i) All angles are equal; (ii) The lengths of the 1990 sides are a permutation of the numbers $1^{2}, 2^{2}, \ldots, 1989^{2}, 1990^{2}$. | 16. Let $A_{0} A_{1} \ldots A_{1989}$ be the desired 1990-gon. We also define $A_{1990}=A_{0}$. Let $O$ be an arbitrary point. For $1 \leq i \leq 1990$ let $B_{i}$ be a point such that $\overrightarrow{O B_{i}}=\overrightarrow{A_{i-1} A_{i}}$. We define $B_{0}=B_{1990}$. The points $B_{i}$ must satisfy the following pr... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 97 | 753 |
1990 | T0 | 17 | null | IMO | 17. (NET 3) Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $A$ be the beginning vertex and $B$ be the end vertex. Let there ... | 17. Let us set a coordinate system denoting the vertices of the block. The vertices of the unit cubes of the block can be described as $\{(x, y, z) \mid 0 \leq$ $x \leq p, 0 \leq y \leq q, 0 \leq z \leq r\}$, and we restrict our attention to only these points. Suppose the point $A$ is fixed at $(a, b, c)$. Then for eve... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 161 | 521 |
1990 | T0 | 19 | null | IMO | 19. (POL 1) Let $P$ be a point inside a regular tetrahedron $T$ of unit volume. The four planes passing through $P$ and parallel to the faces of $T$ partition $T$ into 14 pieces. Let $f(P)$ be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not ... | 19. Let $d_{1}, d_{2}, d_{3}, d_{4}$ be the distances of the point $P$ to the tetrahedron. Let $d$ be the height of the regular tetrahedron. Let $x_{i}=d_{i} / d$. Clearly, $x_{1}+$ $x_{2}+x_{3}+x_{4}=1$, and given this condition, the parameters vary freely as we vary $P$ within the tetrahedron. The four tetrahedra hav... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 114 | 1,065 |
1990 | T0 | 21 | null | IMO | 21. (ROM $\mathbf{1}^{\prime}$ ) Let $n$ be a composite natural number and $p$ a proper divisor of $n$. Find the binary representation of the smallest natural number $N$ such that $\frac{\left(1+2^{p}+2^{n-p}\right) N-1}{2^{n}}$ is an integer. | 21. We must solve the congruence $\left(1+2^{p}+2^{n-p}\right) N \equiv 1\left(\bmod 2^{n}\right)$. Since $(1+$ $2^{p}+2^{n-p}$ ) and $2^{n}$ are coprime, there clearly exists a unique $N$ satisfying this equation and $0<N<2^{n}$. Let us assume $n=m p$. Then we have $\left(1+2^{p}\right)\left(\sum_{j=0}^{m-1}(-1)^{j} 2... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 82 | 808 |
1990 | T0 | 22 | null | IMO | 22. (ROM 4) Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules offer round-trip service between the cities they serve. Prove that at least one of the airlines can offer two disjoint round trips eac... | 22. We can assume without loss of generality that each connection is serviced by only one airline and the problem reduces to finding two disjoint monochromatic cycles of the same color and of odd length on a complete graph of 10 points colored by two colors. We use the following two standard lemmas: Lemma 1. Given a co... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 74 | 654 |
1990 | T0 | 24 | null | IMO | 24. (THA 2) Let $a, b, c, d$ be nonnegative real numbers such that $a b+b c+$ $c d+d a=1$. Show that $$ \frac{a^{3}}{b+c+d}+\frac{b^{3}}{a+c+d}+\frac{c^{3}}{a+b+d}+\frac{d^{3}}{a+b+c} \geq \frac{1}{3} $$ ![](https://cdn.mathpix.com/cropped/2024_11_18_8e985d6b9c83aa3e9d0eg-262.jpg?height=62&width=1190&top_left_y=791&t... | 24. Let us denote $A=b+c+d, B=a+c+d, C=a+b+d, D=a+b+c$. Since $a b+b c+c d+d a=1$ the numbers $A, B, C, D$ are all positive. By trivially applying the AM-GM inequality we have: $$ a^{2}+b^{2}+c^{2}+d^{2} \geq a b+b c+c d+d a=1 . $$ We will prove the inequality assuming only that $A, B, C, D$ are positive and $a^{2}+b^{... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 248 | 541 |
1990 | T0 | 26 | null | IMO | 26. (USA 2) Let $P$ be a cubic polynomial with rational coefficients, and let $q_{1}, q_{2}, q_{3}, \ldots$ be a sequence of rational numbers such that $q_{n}=P\left(q_{n+1}\right)$ for all $n \geq 1$. Prove that there exists $k \geq 1$ such that for all $n \geq 1, q_{n+k}=q_{n}$. | 26. We note that $|P(x) / x| \rightarrow \infty$. Hence, there exists an integer number $M$ such that $M>\left|q_{1}\right|$ and $|P(x)| \leq|x| \Rightarrow|x|<M$. It follows that $\left|q_{i}\right|<M$ for all $i \in \mathbb{N}$ because assuming $\left|q_{i}\right| \geq M$ for some $i$ we get $\left|q_{i-1}\right|=$ $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 108 | 741 |
1990 | T0 | 28 | null | IMO | 28. (USS 3) Prove that on the coordinate plane it is impossible to draw a closed broken line such that (i) the coordinates of each vertex are rational; (ii) the length each of its edges is 1 ; (iii) the line has an odd number of vertices. | 28. Let us first prove the following lemma. Lemma. Let $\left(b^{\prime} / a^{\prime}, d^{\prime} / c^{\prime}\right)$ and $\left(b^{\prime \prime} / a^{\prime \prime}, d^{\prime \prime} / c^{\prime \prime}\right)$ be two points with rational coordinates where the fractions given are irreducible. If both $a^{\prime}$ a... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 63 | 697 |
1990 | T0 | 6 | null | IMO | 6. (FRG 2) ${ }^{\mathrm{IMO} 5}$ Two players $A$ and $B$ play a game in which they choose numbers alternately according to the following rule: At the beginning, an initial natural number $n_{0}>1$ is given. Knowing $n_{2 k}$, player $A$ may choose any $n_{2 k+1} \in \mathbb{N}$ such that $$ n_{2 k} \leq n_{2 k+1} \le... | 6. Let $W$ denote the set of all $n_{0}$ for which player $A$ has a winning strategy, $L$ the set of all $n_{0}$ for which player $B$ has a winning strategy, and $T$ the set of all $n_{0}$ for which a tie is ensured. Lemma. Assume $\{m, m+1, \ldots 1990\} \subseteq W$ and that there exists $s \leq 1990$ such that $s / ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 301 | 1,098 |
1990 | T0 | 8 | null | IMO | 8. (HUN 1) For a given positive integer $k$ denote the square of the sum of its digits by $f_{1}(k)$ and let $f_{n+1}(k)=f_{1}\left(f_{n}(k)\right)$. Determine the value of $f_{1991}\left(2^{1990}\right)$. | 8. Since $2^{1990}<8^{700}<10^{700}$, we have $f_{1}\left(2^{1990}\right)<(9 \cdot 700)^{2}<4 \cdot 10^{7}$. We then have $f_{2}\left(2^{1990}\right)<(3+9 \cdot 7)^{2}<4900$ and finally $f_{3}\left(2^{1990}\right)<(3+9 \cdot 3)^{2}=30^{2}$. It is easily shown that $f_{k}(n) \equiv f_{k-1}(n)^{2}(\bmod 9)$. Since $2^{6}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 82 | 577 |
1991 | T0 | 12 | null | IMO | 12. $(\mathbf{C H N} 3)^{\mathrm{IMO} 3}$ Let $S=\{1,2,3, \ldots, 280\}$. Find the minimal natural number $n$ such that in any $n$-element subset of $S$ there are five numbers that are pairwise relatively prime. | 12. Let $A_{m}$ be the set of those elements of $S$ which are divisible by $m$. By the inclusion-exclusion principle, the number of elements divisible by $2,3,5$ or 7 equals $$ \begin{aligned} & \left|A_{2} \cup A_{3} \cup A_{5} \cup A_{7}\right| \\ & =\left|A_{2}\right|+\left|A_{3}\right|+\left|A_{5}\right|+\left|A_{7... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 77 | 991 |
1991 | T0 | 13 | null | IMO | 13. (POL 4) Given any integer $n \geq 2$, assume that the integers $a_{1}, a_{2}, \ldots, a_{n}$ are not divisible by $n$ and, moreover, that $n$ does not divide $a_{1}+a_{2}+$ $\cdots+a_{n}$. Prove that there exist at least $n$ different sequences $\left(e_{1}, e_{2}, \cdots, e_{n}\right)$ consisting of zeros or ones ... | 13. Call a sequence $e_{1}, \ldots, e_{n}$ good if $e_{1} a_{1}+\cdots+e_{n} a_{n}$ is divisible by $n$. Among the sums $s_{0}=0, s_{1}=a_{1}, s_{2}=a_{1}+a_{2}, \ldots, s_{n}=a_{1}+\cdots+a_{n}$, two give the same remainder modulo $n$, and their difference corresponds to a good sequence. To show that, permuting the $a... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 152 | 914 |
1991 | T0 | 14 | null | IMO | 14. (POL 3) Let $a, b, c$ be integers and $p$ an odd prime number. Prove that if $f(x)=a x^{2}+b x+c$ is a perfect square for $2 p-1$ consecutive integer values of $x$, then $p$ divides $b^{2}-4 a c$. | 14. Suppose that $f\left(x_{0}\right), f\left(x_{0}+1\right), \ldots, f\left(x_{0}+2 p-2\right)$ are squares. If $p \mid a$ and $p \nmid b$, then $f(x) \equiv b x+c(\bmod p)$ for $x=x_{0}, \ldots, x_{0}+p-1$ form a complete system of residues modulo $p$. However, a square is always congruent to exactly one of the $\fra... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 77 | 1,095 |
1991 | T0 | 19 | null | IMO | 19. (IRE 5) Let $a$ be a rational number with $0<a<1$ and suppose that $$ \cos 3 \pi a+2 \cos 2 \pi a=0 $$ (Angle measurements are in radians.) Prove that $a=2 / 3$. | 19. Set $x=\cos (\pi a)$. The given equation is equivalent to $4 x^{3}+4 x^{2}-3 x-2=0$, which factorizes as $(2 x+1)\left(2 x^{2}+x-2\right)=0$. The case $2 x+1=0$ yields $\cos (\pi a)=-1 / 2$ and $a=2 / 3$. It remains to show that if $x$ satisfies $2 x^{2}+x-2=0$ then $a$ is not rational. The polynomial equation $2 x... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 66 | 664 |
1991 | T0 | 2 | null | IMO | 2. (JAP 5) For an acute triangle $A B C, M$ is the midpoint of the segment $B C, P$ is a point on the segment $A M$ such that $P M=B M, H$ is the foot of the perpendicular line from $P$ to $B C, Q$ is the point of intersection of segment $A B$ and the line passing through $H$ that is perpendicular to $P B$, and finally... | 2. Let $H Q$ meet $P B$ at $Q^{\prime}$ and $H R$ meet $P C$ at $R^{\prime}$. From $M P=M B=M C$ we have $\angle B P C=90^{\circ}$. So $P R^{\prime} H Q^{\prime}$ is a rectangle. Since $P H$ is perpendicular to $B C$, it follows that the circle with diameter $P H$, through $P, R^{\prime}, H, Q^{\prime}$, is tangent to ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 156 | 651 |
1991 | T0 | 22 | null | IMO | 22. (USA 4) Real constants $a, b, c$ are such that there is exactly one square all of whose vertices lie on the cubic curve $y=x^{3}+a x^{2}+b x+c$. Prove that the square has sides of length $\sqrt[4]{72}$. | 22. Suppose w.l.o.g. that the center of the square is at the origin $O(0,0)$. We denote the curve $y=f(x)=x^{3}+a x^{2}+b x+c$ by $\gamma$ and the vertices of the square by $A, B, C, D$ in this order. At first, the symmetry with respect to the point $O$ maps $\gamma$ into the curve $\bar{\gamma}\left(y=f(-x)=x^{3}-a x^... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 69 | 690 |
1991 | T0 | 24 | null | IMO | 24. (IND 1) An odd integer $n \geq 3$ is said to be "nice" if there is at least one permutation $a_{1}, a_{2}, \ldots, a_{n}$ of $1,2, \ldots, n$ such that the $n$ sums $a_{1}-a_{2}+$ $a_{3}-\cdots-a_{n-1}+a_{n}, a_{2}-a_{3}+a_{4}-\cdots-a_{n}+a_{1}, a_{3}-a_{4}+a_{5}-\cdots-a_{1}+$ $a_{2}, \ldots, a_{n}-a_{1}+a_{2}-\c... | 24. Let $y_{k}=a_{k}-a_{k+1}+a_{k+2}-\cdots+a_{k+n-1}$ for $k=1,2, \ldots, n$, where we define $x_{i+n}=x_{i}$ for $1 \leq i \leq n$. We then have $y_{1}+y_{2}=2 a_{1}, y_{2}+y_{3}=$ $2 a_{2}, \ldots, y_{n}+y_{1}=2 a_{n}$. (i) Let $n=4 k-1$ for some integer $k>0$. Then for each $i=1,2, \ldots, n$ we have that $y_{i}=\l... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 196 | 529 |
1991 | T0 | 26 | null | IMO | 26. (CZS 1) Let $n \geq 2$ be a natural number and let the real numbers $p, a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ satisfy $1 / 2 \leq p \leq 1,0 \leq a_{i}, 0 \leq b_{i} \leq p$, $i=1, \ldots, n$, and $\sum_{i=1}^{n} a_{i}=\sum_{i=1}^{n} b_{i}=1$. Prove the inequality $$ \sum_{i=1}^{n} b_{i} \prod_... | 26. Without loss of generality we can assume $b_{1} \geq b_{2} \geq \cdots \geq b_{n}$. We denote by $A_{i}$ the product $a_{1} a_{2} \ldots a_{i-1} a_{i+1} \ldots a_{n}$. If for some $i<j$ holds $A_{i}<A_{j}$, then $b_{i} A_{i}+b_{j} A_{j} \leq b_{i} A_{j}+b_{j} A_{i}$ (or equivalently $\left(b_{i}-b_{j}\right)\left(A... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 205 | 510 |
1991 | T0 | 27 | null | IMO | 27. (POL 2) Determine the maximum value of the sum $$ \sum_{i<j} x_{i} x_{j}\left(x_{i}+x_{j}\right) $$ over all $n$-tuples $\left(x_{1}, \ldots, x_{n}\right)$, satisfying $x_{i} \geq 0$ and $\sum_{i=1}^{n} x_{i}=1$. | 27. Write $F\left(x_{1}, \ldots, x_{n}\right)=\sum_{i<j} x_{i} x_{j}\left(x_{i}+x_{j}\right)$. Choose an $n$-tuple $\left(x_{1}, \ldots, x_{n}\right)$, $\sum_{i=1}^{n} x_{i}=1, x_{i} \geq 0$ with at least three nonzero components, and assume w.l.o.g. that $x_{1} \geq \cdots \geq x_{k-1} \geq x_{k} \geq x_{k+1}=\cdots=x... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 99 | 602 |
1991 | T0 | 28 | null | IMO | 28. (NET 1) ${ }^{\mathrm{IMO}}$ Given a real number $a>1$, construct an infinite and bounded sequence $x_{0}, x_{1}, x_{2}, \ldots$ such that for all natural numbers $i$ and $j, i \neq j$, the following inequality holds: $$ \left|x_{i}-x_{j}\right||i-j|^{a} \geq 1 $$ | 28. Let $x_{n}=c(n \sqrt{2}-[n \sqrt{2}])$ for some constant $c>0$. For $i>j$, putting $p=[i \sqrt{2}]-[j \sqrt{2}]$, we have $\left|x_{i}-x_{j}\right|=c|(i-j) \sqrt{2}-p|=\frac{\left|2(i-j)^{2}-p^{2}\right| c}{(i-j) \sqrt{2}+p} \geq \frac{c}{(i-j) \sqrt{2}+p} \geq \frac{c}{4(i-j)}$, because $p<(i-j) \sqrt{2}+1$. Takin... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 99 | 519 |
1991 | T0 | 29 | null | IMO | 29. (FIN 2) We call a set $S$ on the real line $\mathbb{R}$ superinvariant if for any stretching $A$ of the set by the transformation taking $x$ to $A(x)=x_{0}+$ $a\left(x-x_{0}\right)$ there exists a translation $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 a $y \... | 29. One easily observes that the following sets are super-invariant: one-point set, its complement, closed and open half-lines or their complements, and the whole real line. To show that these are the only possibilities, we first observe that $S$ is super-invariant if and only if for each $a>0$ there is a $b$ such that... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 172 | 618 |
1991 | T0 | 3 | null | IMO | 3. (PRK 1) Let $S$ be any point on the circumscribed circle of $\triangle P Q R$. Then the feet of the perpendiculars from $S$ to the three sides of the triangle lie on the same straight line. Denote this line by $l(S, P Q R)$. Suppose that the hexagon $A B C D E F$ is inscribed in a circle. Show that the four lines $l... | 3. Consider the problem with the unit circle on the complex plane. For convenience, we use the same letter for a point in the plane and its corresponding complex number. Lemma 1. Line $l(S, P Q R)$ contains the point $Z=\frac{P+Q+R+S}{2}$. Proof. Suppose $P^{\prime}, Q^{\prime}, R^{\prime}$ are the feet of perpendicula... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 142 | 1,021 |
1991 | T0 | 4 | null | IMO | 4. (FRA 2) ${ }^{\mathrm{IMO5}}$ Let $A B C$ be a triangle and $M$ an interior point in $A B C$. Show that at least one of the angles $\measuredangle M A B, \measuredangle M B C$, and $\measuredangle M C A$ is less than or equal to $30^{\circ}$. | 4. Assume the contrary, that $\angle M A B, \angle M B C, \angle M C A$ are all greater than $30^{\circ}$. By the sine Ceva theorem, it holds that $$ \begin{aligned} & \sin \angle M A C \sin \angle M B A \sin \angle M C B \\ = & \sin \angle M A B \sin \angle M B C \sin \angle M C A>\sin ^{3} 30^{\circ}=\frac{1}{8} . \e... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 87 | 603 |
1992 | T0 | 13 | null | IMO | 13. (NZL 1) ${ }^{\mathrm{IMO1}}$ Find all integer triples $(p, q, r)$ such that $1<p<q<r$ and $(p-1)(q-1)(r-1)$ is a divisor of $(p q r-1)$. | 13. Let us define $$ \begin{aligned} F(p, q, r)= & \frac{(p q r-1)}{(p-1)(q-1)(r-1)} \\ = & 1+\frac{1}{p-1}+\frac{1}{q-1}+\frac{1}{r-1} \\ & +\frac{1}{(p-1)(q-1)}+\frac{1}{(q-1)(r-1)}+\frac{1}{(r-1)(p-1)} \end{aligned} $$ Obviously $F$ is a decreasing function of $p, q, r$. Suppose that $1<p<q<r$ are integers for which... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 65 | 514 |
1992 | T0 | 17 | null | IMO | 17. (SWE 1) Let $\alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $n$. Prove that: (a) the inequality $\alpha\left(n^{2}\right) \leq \frac{1}{2} \alpha(n)(\alpha(n)+1)$ holds; (b) the above inequality is an equality for infinitely many positive integers; (c) there exist... | 17. (a) Let $n=\sum_{i=1}^{k} 2^{a_{i}}$, so that $\alpha(n)=k$. Then $$ n^{2}=\sum_{i} 2^{2 a_{i}}+\sum_{i<j} 2^{a_{i}+a_{j}+1} $$ has at most $k+\binom{k}{2}=\frac{k(k+1)}{2}$ binary ones. (b) The above inequality is an equality for all numbers $n_{k}=2^{k}$. (c) Put $n_{m}=2^{2^{m}-1}-\sum_{j=1}^{m} 2^{2^{m}-2^{j}}$... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 247 | 1,078 |
1992 | T0 | 18 | null | IMO | 18. (USA 2) Let $[x]$ denote the greatest integer less than or equal to $x$. Pick any $x_{1}$ in $[0,1)$ and define the sequence $x_{1}, x_{2}, x_{3}, \ldots$ by $x_{n+1}=0$ if $x_{n}=0$ and $x_{n+1}=1 / x_{n}-\left[1 / x_{n}\right]$ otherwise. Prove that $$ x_{1}+x_{2}+\cdots+x_{n}<\frac{F_{1}}{F_{2}}+\frac{F_{2}}{F_... | 18. Let us define inductively $f^{1}(x)=f(x)=\frac{1}{x+1}$ and $f^{n}(x)=f\left(f^{n-1}(x)\right)$, and let $g_{n}(x)=x+f(x)+f^{2}(x)+\cdots+f^{n}(x)$. We shall prove first the following statement. Lemma. The function $g_{n}(x)$ is strictly increasing on $[0,1]$, and $g_{n-1}(1)=$ $F_{1} / F_{2}+F_{2} / F_{3}+\cdots+F... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 209 | 1,032 |
1992 | T0 | 19 | null | IMO | 19. (IRE 1) Let $f(x)=x^{8}+4 x^{6}+2 x^{4}+28 x^{2}+1$. Let $p>3$ be a prime and suppose there exists an integer $z$ such that $p$ divides $f(z)$. Prove that there exist integers $z_{1}, z_{2}, \ldots, z_{8}$ such that if $$ g(x)=\left(x-z_{1}\right)\left(x-z_{2}\right) \cdots\left(x-z_{8}\right) $$ then all coeffic... | 19. Observe that $f(x)=\left(x^{4}+2 x^{2}+3\right)^{2}-8\left(x^{2}-1\right)^{2}=\left[x^{4}+2(1-\sqrt{2}) x^{2}+\right.$ $3+2 \sqrt{2}]\left[x^{4}+2(1+\sqrt{2}) x^{2}+3-2 \sqrt{2}\right]$. Now it is easy to find that the roots of $f$ are $$ x_{1,2,3,4}= \pm i(i \sqrt[4]{2} \pm 1) \quad \text { and } \quad x_{5,6,7,8}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 147 | 700 |
1992 | T0 | 21 | null | IMO | 21. (GBR 2) ${ }^{\mathrm{IMO} 6}$ For each positive integer $n$, denote by $s(n)$ the greatest integer such that for all positive integers $k \leq s(n), n^{2}$ can be expressed as a sum of squares of $k$ positive integers. (a) Prove that $s(n) \leq n^{2}-14$ for all $n \geq 4$. (b) Find a number $n$ such that $s(n)=\o... | 21. (a) Representing $n^{2}$ as a sum of $n^{2}-13$ squares is equivalent to representing 13 as a sum of numbers of the form $x^{2}-1, x \in \mathbb{N}$, such as $0,3,8,15, \ldots$ But it is easy to check that this is impossible, and hence $s(n) \leq n^{2}-14$. (b) Let us prove that $s(13)=13^{2}-14=155$. Observe that ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 153 | 1,027 |
1992 | T0 | 4 | null | IMO | 4. $(\mathbf{C H N} 3)^{\mathrm{IMO}}$ Given nine points in space, no four of which are coplanar, find the minimal natural number $n$ such that for any coloring with red or blue of $n$ edges drawn between these nine points there always exists a triangle having all edges of the same color. | 4. There are 36 possible edges in total. If not more than 3 edges are left undrawn, then we can choose 6 of the given 9 points no two of which are connected by an undrawn edge. These 6 points together with the edges between them form a two-colored complete graph, and thus by a wellknown result there exists at least one... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 75 | 754 |
1992 | T0 | 8 | null | IMO | 8. (IND 5) Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions: (i) its side lengths are $1,2,3, \ldots, 1992$ in some order; (ii) the polygon is circumscribable about a circle. Alternative formulation. Does there exist a 1992-gon with side lengths $1,2,3, \ldots, 199... | 8. For simplicity, we shall write $n$ instead of 1992. Lemma. There exists a tangent $n$-gon $A_{1} A_{2} \ldots A_{n}$ with sides $A_{1} A_{2}=a_{1}$, $A_{2} A_{3}=a_{2}, \ldots, A_{n} A_{1}=a_{n}$ if and only if the system $$ x_{1}+x_{2}=a_{1}, x_{2}+x_{3}=a_{2}, \ldots, x_{n}+x_{1}=a_{n} $$ has a solution $\left(x_{... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 127 | 927 |
1993 | T0 | 13 | null | IMO | 13. (IRE 3) Let $S$ be the set of all pairs $(m, n)$ of relatively prime positive integers $m, n$ with $n$ even and $m<n$. For $s=(m, n) \in S$ write $n=2^{k} n_{0}$, where $k, n_{0}$ are positive integers with $n_{0}$ odd and define $f(s)=\left(n_{0}, m+\right.$ $n-n_{0}$ ). Prove that $f$ is a function from $S$ to $S... | 13. For an odd integer $N>1$, let $S_{N}=\{(m, n) \in S \mid m+n=N\}$. If $f(m, n)=\left(m_{1}, n_{1}\right)$, then $m_{1}+n_{1}=m+n$ with $m_{1}$ odd and $m_{1} \leq \frac{n}{2}<$ $\frac{N}{2}<n_{1}$, so $f$ maps $S_{N}$ to $S_{N}$. Also $f$ is bijective, since if $f(m, n)=$ $\left(m_{1}, n_{1}\right)$, then $n$ is un... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 297 | 951 |
1993 | T0 | 14 | null | IMO | 14. (ISR 1) The vertices $D, E, F$ of an equilateral triangle lie on the sides $B C, C A, A B$ respectively of a triangle $A B C$. If $a, b, c$ are the respective lengths of these sides, and $S$ the area of $A B C$, prove that $$ D E \geq \frac{2 \sqrt{2} S}{\sqrt{a^{2}+b^{2}+c^{2}+4 \sqrt{3} S}} $$ | 14. Consider any point $T$ inside the triangle $A B C$ or on its boundary. Since $$ \begin{aligned} 2 S & =2\left(S_{A E T F}+S_{B F T D}+S_{C D T E}\right) \\ & \leq A T \cdot E F+B T \cdot F D+C T \cdot D E=(A T+B T+C T) D E \end{aligned} $$ it suffices to find a point $T$ such that $$ (A T+B T+C T)^{2} \geq \frac{a^... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 120 | 634 |
1993 | T0 | 17 | null | IMO | 17. (NET 2) ${ }^{\mathrm{IMO} 6}$ Let $n$ be an integer greater than 1 . In a circular arrangement of $n$ lamps $L_{0}, \ldots, L_{n-1}$, each one of that can be either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $S_{t e p_{0}}, S t e p_{1}, \ldots$. If $L_{j-... | 17. We introduce the rotation operation Rot to the left by one, so that $S_{t e p_{j}}=$ Rot $^{-j} \circ$ Step $_{0} \circ$ Rot $^{j}$. Now writing Step ${ }^{*}={\text { Rot } \circ \text { Step }_{0} \text {, the problem is }}^{-20}$ transformed into the question whether there is an $M(n)$ such that all lamps are $O... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 297 | 811 |
1993 | T0 | 19 | null | IMO | 19. (ROM 2) Let $a, b, n$ be positive integers, $b>1$ and $b^{n}-1 \mid a$. Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero. | 19. Let $s$ be the minimum number of nonzero digits that can appear in the $b$ adic representation of any number divisible by $b^{n}-1$. Among all numbers divisible by $b^{n}-1$ and having $s$ nonzero digits in base $b$, we choose the number $A$ with the minimum sum of digits. Let $A=a_{1} b^{n_{1}}+\cdots+a_{s} b^{n_{... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 63 | 526 |
1993 | T0 | 21 | null | IMO | 21. (GBR 1) A circle $S$ is said to cut a circle $\Sigma$ diametrally if their common chord is a diameter of $\Sigma$. Let $S_{A}, S_{B}, S_{C}$ be three circles with distinct centers $A, B, C$ respectively. Prove that $A, B, C$ are collinear if and only if there is no unique circle $S$ that cuts each of $S_{A}, S_{B},... | 21. Assume that $S$ is a circle with center $O$ that cuts $S_{i}$ diametrically in points $P_{i}, Q_{i}, i \in\{A, B, C\}$, and denote by $r_{i}, r$ the radii of $S_{i}$ and $S$ respectively. Since $O A$ is perpendicular to $P_{A} Q_{A}$, it follows by Pythagoras's theorem that $O A^{2}+A P_{A}^{2}=O P_{A}^{2}$, i.e., ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 179 | 609 |
1993 | T0 | 22 | null | IMO | 22. (GBR 2) ${ }^{\mathrm{IMO} 2} A, B, C, D$ are four points in the plane, with $C, D$ on the same side of the line $A B$, such that $A C \cdot B D=A D \cdot B C$ and $\measuredangle A D B=$ $90^{\circ}+\measuredangle A C B$. Find the ratio $$ \frac{A B \cdot C D}{A C \cdot B D} $$ and prove that circles $A C D, B C... | 22. Let $M$ be the point inside $\angle A D B$ that satisfies $D M=D B$ and $D M \perp$ $D B$. Then $\angle A D M=\angle A C B$ and $A D / D M=A C / C B$. It follows that the triangles $A D M, A C B$ are similar; hence $\angle C A D=\angle B A M$ (because $\angle C A B=\angle D A M$ ) and $A B / A M=A C / A D$. Consequ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 153 | 594 |
1993 | T0 | 23 | null | IMO | 23. (GBR 3) A finite set of (distinct) positive integers is called a " $D S$-set" if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some $D S$-set. | 23. Let the given numbers be $a_{1}, \ldots, a_{n}$. Put $s=a_{1}+\cdots+a_{n}$ and $m=$ $\operatorname{lcm}\left(a_{1}, \ldots, a_{n}\right)$ and write $m=2^{k} r$ with $k \geq 0$ and $r$ odd. Let the binary expansion of $r$ be $r=2^{k_{0}}+2^{k_{1}}+\cdots+2^{k_{t}}$, with $0=k_{0}<\cdots<k_{t}$. Adjoin to the set $\... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 62 | 644 |
1993 | T0 | 24 | null | IMO | 24. (USA 3) Prove that $$ \frac{a}{b+2 c+3 d}+\frac{b}{c+2 d+3 a}+\frac{c}{d+2 a+3 b}+\frac{d}{a+2 b+3 c} \geq \frac{2}{3} $$ for all positive real numbers $a, b, c, d$. | 24. By the Cauchy-Schwarz inequality, if $x_{1}, x_{2}, \ldots, x_{n}$ and $y_{1}, y_{2}, \ldots, y_{n}$ are positive numbers, then $$ \left(\sum_{i=1}^{n} \frac{x_{i}}{y_{i}}\right)\left(\sum_{i=1}^{n} x_{i} y_{i}\right) \geq\left(\sum_{i=1}^{n} x_{i}\right)^{2} $$ Applying this to the numbers $a, b, c, d$ and $b+2 c+... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 91 | 527 |
1993 | T0 | 25 | null | IMO | 25. (VIE 1) Solve the following system of equations, in which $a$ is a given number satisfying $|a|>1$ : $$ \begin{aligned} x_{1}^{2} & =a x_{2}+1, \\ x_{2}^{2} & =a x_{3}+1 \\ \cdots & \cdots \\ x_{999}^{2} & =a x_{1000}+1 \\ x_{1000}^{2} & =a x_{1}+1 \end{aligned} $$ | 25. We need only consider the case $a>1$ (since the case $a<-1$ is reduced to $a>1$ by taking $a^{\prime}=-a, x_{i}^{\prime}=-x_{i}$ ). Since the left sides of the equations are nonnegative, we have $x_{i} \geq-\frac{1}{a}>-1, i=1, \ldots, 1000$. Suppose w.l.o.g. that $x_{1}=\max \left\{x_{i}\right\}$. In particular, $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 131 | 873 |
1993 | T0 | 26 | null | IMO | 26. (VIE 2) Let $a, b, c, d$ be four nonnegative numbers satisfying $a+b+c+d=$ 1. Prove the inequality $$ a b c+b c d+c d a+d a b \leq \frac{1}{27}+\frac{176}{27} a b c d $$ | 26. Set $$ \begin{aligned} f(a, b, c, d) & =a b c+b c d+c d a+d a b-\frac{176}{27} a b c d \\ & =a b(c+d)+c d\left(a+b-\frac{176}{27} a b\right) . \end{aligned} $$ If $a+b-\frac{176}{a} b \leq 0$, by the arithmetic-geometric inequality we have $f(a, b, c, d) \leq a b(c+d) \leq \frac{1}{27}$. On the other hand, if $a+b-... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 78 | 621 |
1993 | T0 | 3 | null | IMO | 3. (SPA 1) Consider the triangle $A B C$, its circumcircle $k$ with center $O$ and radius $R$, and its incircle with center $I$ and radius $r$. Another circle $k_{c}$ is tangent to the sides $C A, C B$ at $D, E$, respectively, and it is internally tangent to $k$. Show that the incenter $I$ is the midpoint of $D E$. | 3. Let $O_{1}$ and $\rho$ be the center and radius of $k_{c}$. It is clear that $C, I, O_{1}$ are collinear and $C I / C O_{1}=r / \rho$. By Stewart's theorem applied to $\triangle O C O_{1}$, $$ O I^{2}=\frac{r}{\rho} O O_{1}^{2}+\left(1-\frac{r}{\rho}\right) O C^{2}-C I \cdot I O_{1} . $$ Since $O O_{1}=R-\rho, O C=R... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 97 | 1,730 |
1993 | T0 | 8 | null | IMO | 8. (IND 1) Define a sequence $\langle f(n)\rangle_{n=1}^{\infty}$ of positive integers by $f(1)=1$ and $$ f(n)= \begin{cases}f(n-1)-n, & \text { if } f(n-1)>n ; \\ f(n-1)+n, & \text { if } f(n-1) \leq n,\end{cases} $$ for $n \geq 2$. Let $S=\{n \in \mathbb{N} \mid f(n)=1993\}$. (a) Prove that $S$ is an infinite set. ... | 8. Suppose that $f(n)=1$ for some $n>0$. Then $f(n+1)=n+2, f(n+$ $2)=2 n+4, f(n+3)=n+1, f(n+4)=2 n+5, f(n+5)=n$, and so by induction $f(n+2 k)=2 n+3+k, f(n+2 k-1)=n+3-k$ for $k=1,2, \ldots, n+2$. Particularly, $n^{\prime}=3 n+3$ is the smallest value greater than $n$ for which $f\left(n^{\prime}\right)=1$. It follows t... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 227 | 586 |
1993 | T0 | 9 | null | IMO | 9. (IND 4) (a) Show that the set $\mathbb{Q}^{+}$of all positive rational numbers can be partitioned into three disjoint subsets $A, B, C$ satisfying the following conditions: $$ B A=B, \quad B^{2}=C, \quad B C=A, $$ where $H K$ stands for the set $\{h k \mid h \in H, k \in K\}$ for any two subsets $H, K$ of $\mathbb... | 9. We shall first complete the "multiplication table" for the sets $A, B, C$. It is clear that this multiplication is commutative and associative, so that we have the following relations: $$ \begin{aligned} & A C=(A B) B=B B=C \\ & A^{2}=A A=(A B) C=B C=A \\ & C^{2}=C C=B(B C)=B A=B \end{aligned} $$ (a) Now put 1 in $A... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 249 | 826 |
1994 | T0 | 12 | Combinatorics | IMO | 12. C7 (BRA) Prove that for any integer $n \geq 2$, there exists a set of $2^{n-1}$ points in the plane such that no 3 lie on a line and no $2 n$ are the vertices of a convex $2 n$-gon. | 12. Define $S_{n}$ recursively as follows: Let $S_{2}=\{(0,0),(1,1)\}$ and $S_{n+1}=$ $S_{n} \cup T_{n}$, where $T_{n}=\left\{\left(x+2^{n-1}, y+M_{n}\right) \mid(x, y) \in S_{n}\right\}$, with $M_{n}$ chosen large enough so that the entire set $T_{n}$ lies above every line passing through two points of $S_{n}$. By def... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 67 | 608 |
1994 | T0 | 15 | Geometry | IMO | 15. G3 (RUS) A circle $\omega$ is tangent to two parallel lines $l_{1}$ and $l_{2}$. A second circle $\omega_{1}$ is tangent to $l_{1}$ at $A$ and to $\omega$ externally at $C$. A third circle $\omega_{2}$ is tangent to $l_{2}$ at $B$, to $\omega$ externally at $D$, and to $\omega_{1}$ externally at $E$. $A D$ intersec... | 15. We shall prove that $A D$ is a common tangent of $\omega$ and $\omega_{2}$. Denote by $K, L$ the points of tangency of $\omega$ with $l_{1}$ and $l_{2}$ respectively. Let $r, r_{1}, r_{2}$ be the radii of $\omega, \omega_{1}, \omega_{2}$ respectively, and set $K A=x, L B=y$. It will be enough if we show that $x y=2... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 133 | 674 |
1994 | T0 | 21 | Number Theory | IMO | 21. N4 (FRA) For any positive integer $x_{0}$, three sequences $\left\{x_{n}\right\},\left\{y_{n}\right\}$, and $\left\{z_{n}\right\}$ are defined as follows: (i) $y_{0}=4$ and $z_{0}=1$; (ii) if $x_{n}$ is even for $n \geq 0, x_{n+1}=\frac{x_{n}}{2}, y_{n+1}=2 y_{n}$, and $z_{n+1}=z_{n}$; (iii) if $x_{n}$ is odd for $... | 21. Note first that $y_{n}=2^{k}(k \geq 2)$ and $z_{k} \equiv 1(\bmod 4)$ for all $n$, so if $x_{n}$ is odd, $x_{n+1}$ will be even. Further, it is shown by induction on $n$ that $y_{n}>z_{n}$ when $x_{n-1}$ is even and $2 y_{n}>z_{n}>y_{n}$ when $x_{n-1}$ is odd. In fact, $n=1$ is the trivial case, while if it holds f... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 262 | 953 |
1994 | T0 | 23 | Number Theory | IMO | 23. N6 (LAT) Let $x_{1}$ and $x_{2}$ be relatively prime positive integers. For $n \geq 2$, define $x_{n+1}=x_{n} x_{n-1}+1$. (a) Prove that for every $i>1$, there exists $j>i$ such that $x_{i}^{i}$ divides $x_{j}^{j}$. (b) Is it true that $x_{1}$ must divide $x_{j}^{j}$ for some $j>1$ ? | 23. (a) Let $p$ be a prime divisor of $x_{i}, i>1$, and let $x_{j} \equiv u_{j}(\bmod p)$ where $0 \leq u_{j} \leq p-1$ (particularly $u_{i} \equiv 0$ ). Then $u_{j+1} \equiv u_{j} u_{j-1}+$ $1(\bmod p)$. The number of possible pairs $\left(u_{j}, u_{j+1}\right)$ is finite, so $u_{j}$ is eventually periodic. We claim t... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 125 | 663 |
1994 | T0 | 24 | Number Theory | IMO | 24. N7 (GBR) A wobbly number is a positive integer whose digits in base 10 are alternately nonzero and zero, the units digit being nonzero. Determine all positive integers that do not divide any wobbly number. | 24. A multiple of 10 does not divide any wobbly number. Also, if $25 \mid n$, then every multiple of $n$ ends with $25,50,75$, or 00 ; hence it is not wobbly. We now show that every other number $n$ divides some wobbly number. (i) Let $n$ be odd and not divisible by 5 . For any $k \geq 1$ there exists $l$ such that $\l... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 52 | 518 |
1994 | T0 | 5 | Algebra | IMO | 5. A5 (POL) Let $f(x)=\frac{x^{2}+1}{2 x}$ for $x \neq 0$. Define $f^{(0)}(x)=x$ and $f^{(n)}(x)=$ $f\left(f^{(n-1)}(x)\right)$ for all positive integers $n$ and $x \neq 0$. Prove that for all nonnegative integers $n$ and $x \neq-1,0$, or 1 , $$ \frac{f^{(n)}(x)}{f^{(n+1)}(x)}=1+\frac{1}{f\left(\left(\frac{x+1}{x-1}\r... | 5. If $f^{(n)}(x)=\frac{p_{n}(x)}{q_{n}(x)}$ for some positive integer $n$ and polynomials $p_{n}, q_{n}$, then $$ f^{(n+1)}(x)=f\left(\frac{p_{n}(x)}{q_{n}(x)}\right)=\frac{p_{n}(x)^{2}+q_{n}(x)^{2}}{2 p_{n}(x) q_{n}(x)} $$ Note that $f^{(0)}(x)=x / 1$. Thus $f^{(n)}(x)=\frac{p_{n}(x)}{q_{n}(x)}$, where the sequence o... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 171 | 706 |
1994 | T0 | 9 | Combinatorics | IMO | 9. C4 (EST) There are $n+1$ fixed positions in a row, labeled 0 to $n$ in increasing order from right to left. Cards numbered 0 to $n$ are shuffled and dealt, one in each position. The object of the game is to have card $i$ in the $i$ th position for $0 \leq i \leq n$. If this has not been achieved, the following move ... | 9. (a) For $i=1, \ldots, n$, let $d_{i}$ be 0 if the card $i$ is in the $i$ th position, and 1 otherwise. Define $b=d_{1}+2 d_{2}+2^{2} d_{3}+\cdots+2^{n-1} d_{n}$, so that $0 \leq b \leq$ $2^{n}-1$, and $b=0$ if and only if the game is over. After each move some digit $d_{l}$ changes from 1 to 0 while $d_{l+1}, d_{l+2... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 210 | 511 |
1995 | T0 | 10 | Geometry | IMO | 10. G4 (UKR) An acute triangle $A B C$ is given. Points $A_{1}$ and $A_{2}$ are taken on the side $B C$ (with $A_{2}$ between $A_{1}$ and $C$ ), $B_{1}$ and $B_{2}$ on the side $A C$ (with $B_{2}$ between $B_{1}$ and $A$ ), and $C_{1}$ and $C_{2}$ on the side $A B$ (with $C_{2}$ between $C_{1}$ and $B$ ) such that $$ ... | 10. Let the two triangles be $X_{1} Y_{1} Z_{1}, X_{2} Y_{2} Z_{2}$, with $X_{1}=B B_{1} \cap C C_{1}, Y_{1}=$ $C C_{1} \cap A A_{1}, Z_{1}=A A_{1} \cap B B_{1}$, $X_{2}=B B_{2} \cap C C_{2}, Y_{2}=C C_{2} \cap$ $A A_{2}, Z_{2}=A A_{2} \cap B B_{2}$. First, we observe that $\angle A B B_{2}=\angle A C C_{1}$ and $\angl... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 273 | 992 |
1995 | T0 | 12 | Geometry | IMO | 12. G6 (USA) Let $A_{1} A_{2} A_{3} A_{4}$ be a tetrahedron, $G$ its centroid, and $A_{1}^{\prime}, A_{2}^{\prime}, A_{3}^{\prime}$, and $A_{4}^{\prime}$ the points where the circumsphere of $A_{1} A_{2} A_{3} A_{4}$ intersects $G A_{1}, G A_{2}, G A_{3}$, and $G A_{4}$, respectively. Prove that $$ G A_{1} \cdot G A_{... | 12. Let $O$ be the circumcenter and $R$ the circumradius of $\underset{\longrightarrow}{A_{1} A_{2} A_{3} A_{4}}$. We have $O A_{i}^{2}=\left(\overrightarrow{O G}+\left(\overrightarrow{O A_{i}}-\overrightarrow{O G}\right)\right)^{2}=O G^{2}+G A_{i}^{2}+2 \overrightarrow{O G} \cdot \overrightarrow{G A_{i}}$. Summing up ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 310 | 709 |
1995 | T0 | 13 | Geometry | IMO | 13. G7 (LAT) $O$ is a point inside a convex quadrilateral $A B C D$ of area $S . K, L, M$, and $N$ are interior points of the sides $A B, B C, C D$, and $D A$ respectively. If $O K B L$ and $O M D N$ are parallelograms, prove that $\sqrt{S} \geq \sqrt{S_{1}}+\sqrt{S_{2}}$, where $S_{1}$ and $S_{2}$ are the areas of $O ... | 13. If $O$ lies on $A C$, then $A B C D, A K O N$, and $O L C M$ are similar; hence $A C=A O+O C$ implies $\sqrt{S}=\sqrt{S_{1}}+\sqrt{S_{2}}$. Assume that $O$ does not lie on $A C$ and that w.l.o.g. it lies inside triangle $A D C$. Let us denote by $T_{1}, T_{2}$ the areas of parallelograms $K B L O, N O M D$ respecti... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 138 | 1,186 |
1995 | T0 | 14 | Geometry | IMO | 14. G8 (COL) Let $A B C$ be a triangle. A circle passing through $B$ and $C$ intersects the sides $A B$ and $A C$ again at $C^{\prime}$ and $B^{\prime}$, respectively. Prove that $B B^{\prime}, C C^{\prime}$, and $H H^{\prime}$ are concurrent, where $H$ and $H^{\prime}$ are the orthocenters of triangles $A B C$ and $A ... | 14. Let $B B^{\prime}$ cut $C C^{\prime}$ at $P$. Since $\angle B^{\prime} B C^{\prime}=\angle B^{\prime} C C^{\prime}$, it follows that $\angle P B H=\angle P C H$. Let $D$ and $E$ be points such that $B P C D$ and $H P C E$ are parallelograms (consequently, so is $B H E D$ ). Triangles $B A C$ and $C^{\prime} A B^{\p... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 126 | 675 |
1995 | T0 | 17 | Number Theory | IMO | 17. N3 $(\mathbf{C Z E})^{\mathrm{IMO} 3}$ Determine all integers $n>3$ such that there are $n$ points $A_{1}, A_{2}, \ldots, A_{n}$ in the plane that satisfy the following two conditions simultaneously: (a) No three lie on the same line. (b) There exist real numbers $p_{1}, p_{2}, \ldots, p_{n}$ such that the area of ... | 17. For $n=4$, the vertices of a unit square $A_{1} A_{2} A_{3} A_{4}$ and $p_{1}=p_{2}=p_{3}=$ $p_{4}=\frac{1}{6}$ satisfy the conditions. We claim that there are no solutions for $n=5$ (and thus for any $n \geq 5$ ). Suppose to the contrary that points $A_{i}$ and $p_{i}, i=1, \ldots, 5$, satisfy the conditions. Deno... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 154 | 712 |
1995 | T0 | 18 | Number Theory | IMO | 18. N4 (BUL) Find all positive integers $x$ and $y$ such that $x+y^{2}+z^{3}=x y z$, where $z$ is the greatest common divisor of $x$ and $y$. | 18. Let $x=z a$ and $y=z b$, where $a$ and $b$ are relatively prime. The given Diophantine equation becomes $a+z b^{2}+z^{2}=z^{2} a b$, so $a=z c$ for some $c \in \mathbb{Z}$. We obtain $c+b^{2}+z=z^{2} c b$, or $c=\frac{b^{2}+z}{z^{2} b-1}$. (i) If $z=1$, then $c=\frac{b^{2}+1}{b-1}=b+1+\frac{2}{b-1}$, so $b=2$ or $b... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 54 | 516 |
1995 | T0 | 19 | Number Theory | IMO | 19. N5 (IRE) At a meeting of $12 k$ people, each person exchanges greetings with exactly $3 k+6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting? | 19. For each two people let $n$ be the number of people exchanging greetings with both of them. To determine $n$ in terms of $k$, we shall count in two ways the number of triples $(A, B, C)$ of people such that $A$ exchanged greetings with both $B$ and $C$, but $B$ and $C$ mutually did not. There are $12 k$ possibiliti... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 57 | 532 |
1995 | T0 | 20 | Number Theory | IMO | 20. N6 (POL) ${ }^{\mathrm{IMO} 06}$ Let $p$ be an odd prime. Find the number of $p$-element subsets $A$ of $\{1,2, \ldots, 2 p\}$ such that the sum of all elements of $A$ is divisible by $p$. | 20. We shall consider the set $M=\{0,1, \ldots, 2 p-1\}$ instead. Let $M_{1}=$ $\{0,1, \ldots, p-1\}$ and $M_{2}=\{p, p+1, \ldots, 2 p-1\}$. We shall denote by $|A|$ and $\sigma(A)$ the number of elements and the sum of elements of the set $A$; also, let $C_{p}$ be the family of all $p$-element subsets of $M$. Define t... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 75 | 1,337 |
1995 | T0 | 21 | Number Theory | IMO | 21. N7 (BLR) Does there exist an integer $n>1$ that satisfies the following condition? The set of positive integers can be partitioned into $n$ nonempty subsets such that an arbitrary sum of $n-1$ integers, one taken from each of any $n-1$ of the subsets, lies in the remaining subset. | 21. We shall show that there is no such $n$. Certainly, $n=2$ does not work, so suppose $n \geq 3$. Let $a, b$ be distinct elements of $A_{1}$, and $c$ any integer greater than $-a$ and $-b$. We claim that $a+c, b+c$ belong to the same subsets. Suppose to the contrary that $a+c \in A_{1}$ and $b+c \in A_{2}$, and take ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 75 | 651 |
1995 | T0 | 24 | null | IMO | 24. S2 (POL) ${ }^{\text {IMO4 }}$ The positive real numbers $x_{0}, x_{1}, \ldots, x_{1995}$ satisfy $x_{0}=$ $x_{1995}$ and $$ x_{i-1}+\frac{2}{x_{i-1}}=2 x_{i}+\frac{1}{x_{i}} $$ for $i=1,2, \ldots, 1995$. Find the maximum value that $x_{0}$ can have. | 24. The given condition is equivalent to $\left(2 x_{i}-x_{i-1}\right)\left(x_{i} x_{i-1}-1\right)=0$, so either $x_{i}=\frac{1}{2} x_{i-1}$ or $x_{i}=\frac{1}{x_{i-1}}$. We shall show by induction on $n$ that for any $n \geq 0, x_{n}=2^{k_{n}} x_{0}^{e_{n}}$ for some integer $k_{n}$, where $\left|k_{n}\right| \leq n$ ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 125 | 789 |
1995 | T0 | 27 | null | IMO | 27. S5 (FIN) For positive integers $n$, the numbers $f(n)$ are defined inductively as follows: $f(1)=1$, and for every positive integer $n, f(n+1)$ is the greatest integer $m$ such that there is an arithmetic progression of positive integers $a_{1}<a_{2}<\cdots<a_{m}=n$ for which $$ f\left(a_{1}\right)=f\left(a_{2}\ri... | 27. Computing the first few values of $f(n)$, we observe the following pattern: $$ \begin{aligned} f(4 k) & =k, k \geq 3, & f(8) & =3 ; \\ f(4 k+1) & =1, k \geq 4, & f(5) & =f(13)=2 ; \\ f(4 k+2) & =k-3, k \geq 7, & f(2) & =1, f(6)=f(10)=2, \\ & & f(14) & =f(18)=3, f(26)=4 ; \\ f(4 k+3) & =2 . & & \end{aligned} $$ We s... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 152 | 569 |
1995 | T0 | 28 | null | IMO | 28. S6 (IND) Let $\mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying $$ f(m+f(n))=n+f(m+95) $$ for all $m$ and $n$ in $\mathbb{N}$. What is the value of $\sum_{k=1}^{19} f(k)$ ? | 28. Let $F(x)=f(x)-95$ for $x \geq 1$. Writing $k$ for $m+95$, the given condition becomes $$ F(k+F(n))=F(k)+n, \quad k \geq 96, n \geq 1 $$ Thus for $x, z \geq 96$ and an arbitrary $y$ we have $F(x+y)+z=F(x+$ $y+F(z))=F(x+F(F(y)+z))=F(x)+F(y)+z$, and consequently $F(x+y)=F(x)+F(y)$ whenever $x \geq 96$. Moreover, sinc... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 103 | 605 |
1995 | T0 | 4 | Algebra | IMO | 4. A4 (USA) Let $a, b$, and $c$ be given positive real numbers. Determine all positive real numbers $x, y$, and $z$ such that $$ x+y+z=a+b+c $$ and $$ 4 x y z-\left(a^{2} x+b^{2} y+c^{2} z\right)=a b c $$ | 4. The second equation is equivalent to $\frac{a^{2}}{y z}+\frac{b^{2}}{z x}+\frac{c^{2}}{x y}+\frac{a b c}{x y z}=4$. Let $x_{1}=$ $\frac{a}{\sqrt{y z}}, y_{1}=\frac{b}{\sqrt{z x}}, z_{1}=\frac{c}{\sqrt{x y}}$. Then $x_{1}^{2}+y_{1}^{2}+z_{1}^{2}+x_{1} y_{1} z_{1}=4$, where $0<x_{1}, y_{1}, z_{1}<2$. Regarding this as... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 81 | 1,039 |
1995 | T0 | 5 | Algebra | IMO | 5. A5 (UKR) Let $\mathbb{R}$ be the set of real numbers. Does there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ that simultaneously satisfies the following three conditions? (a) There is a positive number $M$ such that $-M \leq f(x) \leq M$ for all $x$. (b) $f(1)=1$. (c) If $x \neq 0$, then $$ f\left(x+\fr... | 5. Suppose that a function $f$ satisfies the condition, and let $c$ be the least upper bound of $\{f(x) \mid x \in \mathbb{R}\}$. We have $c \geq 2$, since $f(2)=f(1+$ $\left.1 / 1^{2}\right)=f(1)+f(1)^{2}=2$. Also, since $c$ is the least upper bound, for each $k=1,2, \ldots$ there is an $x_{k} \in \mathbb{R}$ such tha... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 148 | 563 |
1995 | T0 | 6 | Algebra | IMO | 6. A6 (JAP) Let $n$ be an integer, $n \geq 3$. Let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers such that $x_{i}<x_{i+1}$ for $1 \leq i \leq n-1$. Prove that $$ \frac{n(n-1)}{2} \sum_{i<j} x_{i} x_{j}>\left(\sum_{i=1}^{n-1}(n-i) x_{i}\right)\left(\sum_{j=2}^{n}(j-1) x_{j}\right) . $$ | 6. Let $y_{i}=x_{i+1}+\cdots+x_{n}, Y=\sum_{j=2}^{n}(j-1) x_{j}$, and $z_{i}=\frac{n(n-1)}{2} y_{i}-(n-$ $i) Y$. Then $\frac{n(n-1)}{2} \sum_{i<j} x_{i} x_{j}-\left(\sum_{i=1}^{n-1}(n-i) x_{i}\right) Y=\frac{n(n-1)}{2} \sum_{i=1}^{n-1} x_{i} y_{i}-$ $\sum_{i=1}^{n-1}(n-i) x_{i} Y=\sum_{i=1}^{n-1} x_{i} z_{i}$, so it re... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 147 | 1,097 |
1996 | T0 | 11 | Geometry | IMO | 11. G2 (CAN) ${ }^{\mathrm{IMO} 2}$ Let $P$ be a point inside $\triangle A B C$ such that $$ \angle A P B-\angle C=\angle A P C-\angle B . $$ Let $D, E$ be the incenters of $\triangle A P B, \triangle A P C$ respectively. Show that $A P, B D$ and $C E$ meet in a point. | 11. Let $X, Y, Z$ respectively be the feet of the perpendiculars from $P$ to $B C$, $C A, A B$. Examining the cyclic quadrilaterals $A Z P Y, B X P Z, C Y P X$, one can easily see that $\angle X Z Y=\angle A P B-\angle C$ and $X Y=P C \sin \angle C$. The first relation gives that $X Y Z$ is isosceles with $X Y=X Z$, so... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 100 | 513 |
1996 | T0 | 12 | Geometry | IMO | 12. G3 (GBR) Let $A B C$ be an acute-angled triangle with $B C>C A$. Let $O$ be the circumcenter, $H$ its orthocenter, and $F$ the foot of its altitude $C H$. Let the perpendicular to $O F$ at $F$ meet the side $C A$ at $P$. Prove that $\angle F H P=\angle B A C$. Possible second part: What happens if $|B C| \leq|C A|$... | 12. It is easy to see that $P$ lies on the segment $A C$. Let $E$ be the foot of the altitude $B H$ and $Y, Z$ the midpoints of $A C, A B$ respectively. Draw the perpendicular $H R$ to $F P(R \in F P)$. Since $Y$ is the circumcenter of $\triangle F C A$, we have $\angle F Y A=180^{\circ}-2 \angle A$. Also, $O F P Y$ is... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 123 | 887 |
1996 | T0 | 14 | Geometry | IMO | 14. G5 (ARM) ${ }^{\mathrm{IMO5}}$ Let $A B C D E F$ be a convex hexagon such that $A B$ is parallel to $D E, B C$ is parallel to $E F$, and $C D$ is parallel to $A F$. Let $R_{A}, R_{C}, R_{E}$ be the circumradii of triangles $F A B, B C D, D E F$ respectively, and let $P$ denote the perimeter of the hexagon. Prove th... | 14. Let $a, b, c, d, e$, and $f$ denote the lengths of the sides $A B, B C, C D, D E$, $E F$, and $F A$ respectively. Note that $\angle A=\angle D, \angle B=\angle E$, and $\angle C=\angle F$. Draw the lines $P Q$ and $R S$ through $A$ and $D$ perpendicular to $B C$ and $E F$ respectively $(P, R \in B C, Q, S \in E F)$... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 146 | 2,262 |
1996 | T0 | 16 | Geometry | IMO | 16. G7 (GBR) Let $A B C$ be an acute-angled triangle with circumcenter $O$ and circumradius $R$. Let $A O$ meet the circle $B O C$ again in $A^{\prime}$, let $B O$ meet the circle $C O A$ again in $B^{\prime}$, and let $C O$ meet the circle $A O B$ again in $C^{\prime}$. Prove that $$ O A^{\prime} \cdot O B^{\prime} \... | 16. Let $A_{1}$ be the point of intersection of $O A^{\prime}$ and $B C$; similarly define $B_{1}$ and $C_{1}$. From the similarity of triangles $O B A_{1}$ and $O A^{\prime} B$ we obtain $O A_{1}$. $O A^{\prime}=R^{2}$. Now it is enough to show that $8 O A_{1} \cdot O B^{\prime} \cdot O C^{\prime} \leq R^{3}$. Thus we... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 143 | 681 |
1996 | T0 | 18 | Geometry | IMO | 18. G9 (UKR) In the plane are given a point $O$ and a polygon $\mathcal{F}$ (not necessarily convex). Let $P$ denote the perimeter of $\mathcal{F}, D$ the sum of the distances from $O$ to the vertices of $\mathcal{F}$, and $H$ the sum of the distances from $O$ to the lines containing the sides of $\mathcal{F}$. Prove t... | 18. We first prove the result in the simplest case. Given a 2 -gon $A B A$ and a point $O$, let $a, b, c, h$ denote $O A, O B, A B$, and the distance of $O$ from $A B$. Then $D=a+b, P=2 c$, and $H=2 h$, so we should show that $$ (a+b)^{2} \geq 4 h^{2}+c^{2} $$ Indeed, let $l$ be the line through $O$ parallel to $A B$, ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 123 | 626 |
1996 | T0 | 20 | Number Theory | IMO | 20. N2 (RUS) ${ }^{\mathrm{IMO} 4}$ The positive integers $a$ and $b$ are such that the numbers $15 a+16 b$ and $16 a-15 b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 20. Let $15 a+16 b=x^{2}$ and $16 a-15 b=y^{2}$, where $x, y \in \mathbb{N}$. Then we obtain $x^{4}+y^{4}=(15 a+16 b)^{2}+(16 a-15 b)^{2}=\left(15^{2}+16^{2}\right)\left(a^{2}+b^{2}\right)=481\left(a^{2}+b^{2}\right)$. In particular, $481=13 \cdot 37 \mid x^{4}+y^{4}$. We have the following lemma. Lemma. Suppose that $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 80 | 539 |
1996 | T0 | 21 | Number Theory | IMO | 21. N3 (BUL) A finite sequence of integers $a_{0}, a_{1}, \ldots, a_{n}$ is called quadratic if for each $i \in\{1,2, \ldots, n\}$ we have the equality $\left|a_{i}-a_{i-1}\right|=i^{2}$. (a) Prove that for any two integers $b$ and $c$, there exist a natural number $n$ and a quadratic sequence with $a_{0}=b$ and $a_{n}... | 21. (a) It clearly suffices to show that for every integer $c$ there exists a quadratic sequence with $a_{0}=0$ and $a_{n}=c$, i.e., that $c$ can be expressed as $\pm 1^{2} \pm 2^{2} \pm \cdots \pm n^{2}$. Since $$ (n+1)^{2}-(n+2)^{2}-(n+3)^{2}+(n+4)^{2}=4 $$ we observe that if our claim is true for $c$, then it is als... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 160 | 537 |
1996 | T0 | 22 | Number Theory | IMO | 22. N4 (BUL) Find all positive integers $a$ and $b$ for which $$ \left[\frac{a^{2}}{b}\right]+\left[\frac{b^{2}}{a}\right]=\left[\frac{a^{2}+b^{2}}{a b}\right]+a b $$ where as usual, $[t]$ refers to greatest integer that is less than or equal to $t$. | 22. Let $a, b \in \mathbb{N}$ satisfy the given equation. It is not possible that $a=b$ (since it leads to $a^{2}+2=2 a$ ), so we assume w.l.o.g. that $a>b$. Next, for $a>b=1$ the equation becomes $a^{2}=2 a$, and one obtains a solution $(a, b)=(2,1)$. Let $b>1$. If $\left[\frac{a^{2}}{b}\right]=\alpha$ and $\left[\fra... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 97 | 675 |
1996 | T0 | 23 | Number Theory | IMO | 23. N5 (ROM) Let $\mathbb{N}_{0}$ denote the set of nonnegative integers. Find a bijective function $f$ from $\mathbb{N}_{0}$ into $\mathbb{N}_{0}$ such that for all $m, n \in \mathbb{N}_{0}$, $$ f(3 m n+m+n)=4 f(m) f(n)+f(m)+f(n) . $$ | 23. We first observe that the given functional equation is equivalent to $$ 4 f\left(\frac{(3 m+1)(3 n+1)-1}{3}\right)+1=(4 f(m)+1)(4 f(n)+1) . $$ This gives us the idea of introducing a function $g: 3 \mathbb{N}_{0}+1 \rightarrow 4 \mathbb{N}_{0}+1$ defined as $g(x)=4 f\left(\frac{x-1}{3}\right)+1$. By the above equal... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 95 | 575 |
1996 | T0 | 24 | Combinatorics | IMO | 24. C1 (FIN) ${ }^{\mathrm{IMO} 1}$ We are given a positive integer $r$ and a rectangular board $A B C D$ with dimensions $|A B|=20,|B C|=12$. The rectangle is divided into a grid of $20 \times 12$ unit squares. The following moves are permitted on the board: One can move from one square to another only if the distance... | 24. We shall work on the array of lattice points defined by $\mathcal{A}=\left\{(x, y) \in \mathbb{Z}^{2} \mid\right.$ $0 \leq x \leq 19,0 \leq y \leq 11\}$. Our task is to move from $(0,0)$ to $(19,0)$ via the points of $\mathcal{A}$ so that each move has the form $(x, y) \rightarrow(x+a, y+b)$, where $a, b \in \mathb... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 193 | 703 |
1996 | T0 | 26 | Combinatorics | IMO | 26. C3 (USA) Let $k, m, n$ be integers such that $1<n \leq m-1 \leq k$. Determine the maximum size of a subset $S$ of the set $\{1,2,3, \ldots, k\}$ such that no $n$ distinct elements of $S$ add up to $m$. | 26. Denote the required maximum size by $M_{k}(m, n)$. If $m<\frac{n(n+1)}{2}$, then trivially $M=k$, so from now on we assume that $m \geq \frac{n(n+1)}{2}$. First we give a lower bound for $M$. Let $r=r_{k}(m, n)$ be the largest integer such that $r+(r+1)+\cdots+(r+n-1) \leq m$. This is equivalent to $n r \leq m-\fra... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 80 | 732 |
1996 | T0 | 27 | Combinatorics | IMO | 27. C4 (FIN) Determine whether or not there exist two disjoint infinite sets $\mathcal{A}$ and $\mathcal{B}$ of points in the plane satisfying the following conditions: (i) No three points in $\mathcal{A} \cup \mathcal{B}$ are collinear, and the distance between any two points in $\mathcal{A} \cup \mathcal{B}$ is at le... | 27. Suppose that such sets of points $\mathcal{A}, \mathcal{B}$ exist. First, we observe that there exist five points $A, B, C, D, E$ in $\mathcal{A}$ such that their convex hull does not contain any other point of $\mathcal{A}$. Indeed, take any point $A \in \mathcal{A}$. Since any two points of $\mathcal{A}$ are at d... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 148 | 510 |
1996 | T0 | 28 | Combinatorics | IMO | 28. C5 (FRA) ${ }^{\mathrm{IMO} 6}$ Let $p, q, n$ be three positive integers with $p+q<n$. Let $\left(x_{0}, x_{1}, \ldots, x_{n}\right)$ be an $(n+1)$-tuple of integers satisfying the following conditions: (i) $x_{0}=x_{n}=0$. (ii) For each $i$ with $1 \leq i \leq n$, either $x_{i}-x_{i-1}=p$ or $x_{i}-x_{i-1}=-q$. Sh... | 28. Note that w.l.o.g., we can assume that $p$ and $q$ are coprime. Indeed, otherwise it suffices to consider the problem in which all $x_{i}$ 's and $p, q$ are divided by $\operatorname{gcd}(p, q)$. Let $k, l$ be the number of indices $i$ with $x_{i+1}-x_{i}=p$ and the number of those $i$ with $x_{i+1}-x_{i}=-q(0 \leq... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 179 | 1,070 |
1996 | T0 | 29 | Combinatorics | IMO | 29. C6 (CAN) A finite number of beans are placed on an infinite row of squares. A sequence of moves is performed as follows: At each stage a square containing more than one bean is chosen. Two beans are taken from this square; one of them is placed on the square immediately to the left, and the other is placed on the s... | 29. Let the squares be indexed serially by the integers: ..., $-1,0,1,2, \ldots$. When a bean is moved from $i$ to $i+1$ or from $i+1$ to $i$ for the first time, we may assign the index $i$ to it. Thereafter, whenever some bean is moved in the opposite direction, we shall assume that it is exactly the one marked by $i$... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 126 | 931 |
1996 | T0 | 3 | Algebra | IMO | 3. A3 (GRE) Let $a>2$ be given, and define recursively $$ a_{0}=1, \quad a_{1}=a, \quad a_{n+1}=\left(\frac{a_{n}^{2}}{a_{n-1}^{2}}-2\right) a_{n} $$ Show that for all $k \in \mathbb{N}$, we have $$ \frac{1}{a_{0}}+\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{k}}<\frac{1}{2}\left(2+a-\sqrt{a^{2}-4}\right) . $$ | 3. Since $a_{1}>2$, it can be written as $a_{1}=b+b^{-1}$ for some $b>0$. Furthermore, $a_{1}^{2}-2=b^{2}+b^{-2}$ and hence $a_{2}=\left(b^{2}+b^{-2}\right)\left(b+b^{-1}\right)$. We prove that $$ a_{n}=\left(b+b^{-1}\right)\left(b^{2}+b^{-2}\right)\left(b^{4}+b^{-4}\right) \cdots\left(b^{2^{n-1}}+b^{-2^{n-1}}\right) $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 159 | 652 |
1996 | T0 | 30 | Combinatorics | IMO | 30. C7 (IRE) Let $U$ be a finite set and let $f, g$ be bijective functions from $U$ onto itself. Let $S=\{w \in U: f(f(w))=g(g(w))\}, \quad T=\{w \in U: f(g(w))=g(f(w))\}$, and suppose that $U=S \cup T$. Prove that for $w \in U, f(w) \in S$ if and only if $g(w) \in S$. | 30. For convenience, we shall write $f^{2}, f g, \ldots$ for the functions $f \circ f, f \circ g, \ldots$ We need two lemmas. Lemma 1. If $f(x) \in S$ and $g(x) \in T$, then $x \in S \cap T$. Proof. The given condition means that $f^{3}(x)=g^{2} f(x)$ and $g f g(x)=$ $f g^{2}(x)$. Since $x \in S \cup T=U$, we have two ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 116 | 622 |
1996 | T0 | 6 | Algebra | IMO | 6. A6 (IRE) Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that $$ k=f(x)(x+1)^{n}+g(x)\left(x^{n}+1\right) $$ for some polynomials $f(x), g(x)$ having integer coefficients. If $k_{0}$ denotes the least such $k$, determine $k_{0}$ as a function of $n$. A6 ${ }^{\prime}$ Let ... | 6. Let $f(x), g(x)$ be polynomials with integer coefficients such that $$ f(x)(x+1)^{n}+g(x)\left(x^{n}+1\right)=k_{0} $$ Write $n=2^{r} m$ for $m$ odd and note that $x^{n}+1=\left(x^{2^{r}}+1\right) B(x)$, where $B(x)=x^{2^{r}(m-1)}-x^{2^{r}(m-2)}+\cdots-x^{2^{r}}+1$. Moreover, $B(-1)=1$; hence $B(x)-1=(x+1) c(x)$ and... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 291 | 973 |
1996 | T0 | 9 | Algebra | IMO | 9. A9 (POL) Let the sequence $a(n), n=1,2,3, \ldots$, be generated as follows: $a(1)=0$, and for $n>1$, $$ a(n)=a([n / 2])+(-1)^{\frac{n(n+1)}{2}} . \quad(\text { Here }[t]=\text { the greatest integer } \leq t .) $$ (a) Determine the maximum and minimum value of $a(n)$ over $n \leq 1996$ and find all $n \leq 1996$ f... | 9. From the definition of $a(n)$ we obtain $$ a(n)-a([n / 2])=\left\{\begin{array}{r} 1 \text { if } n \equiv 0 \text { or } n \equiv 3(\bmod 4) \\ -1 \text { if } n \equiv 1 \text { or } n \equiv 2(\bmod 4) . \end{array}\right. $$ Let $n=\overline{b_{k} b_{k-1} \ldots b_{1} b_{0}}$ be the binary representation of $n$,... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 170 | 690 |
1997 | T0 | 1 | null | IMO | 1. (BLR) $)^{\mathrm{IMO}} \mathrm{An}$ infinite square grid is colored in the chessboard pattern. For any pair of positive integers $m, n$ consider a right-angled triangle whose vertices are grid points and whose legs, of lengths $m$ and $n$, run along the lines of the grid. Let $S_{b}$ be the total area of the black ... | 1. Let $A B C$ be the given triangle, with $\angle B=90^{\circ}$ and $A B=m, B C=n$. For an arbitrary polygon $\mathcal{P}$ we denote by $w(\mathcal{P})$ and $b(\mathcal{P})$ respectively the total areas of the white and black parts of $\mathcal{P}$. (a) Let $D$ be the fourth vertex of the rectangle $A B C D$. When $m$... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 196 | 696 |
1997 | T0 | 11 | null | IMO | 11. (NET) Let $P(x)$ be a polynomial with real coefficients such that $P(x)>$ 0 for all $x \geq 0$. Prove that there exists a positive integer $n$ such that $(1+x)^{n} P(x)$ is a polynomial with nonnegative coefficients. | 11. All real roots of $P(x)$ (if any) are negative: say $-a_{1},-a_{2}, \ldots,-a_{k}$. Then $P(x)$ can be factored as $$ P(x)=C\left(x+a_{1}\right) \cdots\left(x+a_{k}\right)\left(x^{2}-b_{1} x+c_{1}\right) \cdots\left(x^{2}-b_{m} x+c_{m}\right) $$ where $x^{2}-b_{i} x+c_{i}$ are quadratic polynomials without real roo... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 67 | 570 |
1997 | T0 | 12 | null | IMO | 12. (ITA) Let $p$ be a prime number and let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that: (i) $f(0)=0, f(1)=1$; (ii) for every positive integer $n$, the remainder of the division of $f(n)$ by $p$ is either 0 or 1. Prove that $d \geq p-1$. | 12. Lemma. For any polynomial $P$ of degree at most $n$, the following equality holds: $$ \sum_{i=0}^{n+1}(-1)^{i}\binom{n+1}{i} P(i)=0 $$ Proof. See (SL81-13). Suppose to the contrary that the degree of $f$ is at most $p-2$. Then it follows from the lemma that $$ 0=\sum_{i=0}^{p-1}(-1)^{i}\binom{p-1}{i} f(i) \equiv \s... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 95 | 510 |
1997 | T0 | 14 | null | IMO | 14. (IND) Let $b, m, n$ be positive integers such that $b>1$ and $m \neq n$. Prove that if $b^{m}-1$ and $b^{n}-1$ have the same prime divisors, then $b+1$ is a power of 2 . | 14. We use the following nonstandard notation: ( $1^{\circ}$ ) for $x, y \in \mathbb{N}, x \sim y$ means that $x$ and $y$ have the same prime divisors; $\left(2^{\circ}\right)$ for a prime $p$ and integers $r \geq 0$ and $x>0, p^{r} \| x$ means that $x$ is divisible by $p^{r}$, but not by $p^{r+1}$. First, $b^{m}-1 \si... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 71 | 1,219 |
1997 | T0 | 16 | null | IMO | 16. (BLR) In an acute-angled triangle $A B C$, let $A D, B E$ be altitudes and $A P, B Q$ internal bisectors. Denote by $I$ and $O$ the incenter and the circumcenter of the triangle, respectively. Prove that the points $D, E$, and $I$ are collinear if and only if the points $P, Q$, and $O$ are collinear. | 16. Let $d_{a}(X), d_{b}(X), d_{c}(X)$ denote the distances of a point $X$ interior to $\triangle A B C$ from the lines $B C, C A, A B$ respectively. We claim that $X \in P Q$ if and only if $d_{a}(X)+d_{b}(X)=d_{c}(X)$. Indeed, if $X \in P Q$ and $P X=$ $k P Q$ then $d_{a}(X)=k d_{a}(Q), d_{b}(X)=(1-k) d_{b}(P)$, and ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 100 | 989 |
1997 | T0 | 17 | null | IMO | 17. $(\mathbf{C Z E})^{\mathrm{IMO} 5}$ Find all pairs of integers $x, y \geq 1$ satisfying the equation $x^{y^{2}}=y^{x}$. | 17. We note first that $x$ and $y$ must be powers of the same positive integer. Indeed, if $x=p_{1}^{\alpha_{1}} \cdots p_{k}^{\alpha_{k}}$ and $y=p_{1}^{\beta_{1}} \cdots p_{k}^{\beta_{k}}$ (some of $\alpha_{i}$ and $\beta_{i}$ may be 0 , but not both for the same index $i$ ), then $x^{y^{2}}=y^{x}$ implies $\frac{\al... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 52 | 626 |
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