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 |
|---|---|---|---|---|---|---|---|---|---|
1997 | T0 | 19 | null | IMO | 19. (IRE) Let $a_{1} \geq \cdots \geq a_{n} \geq a_{n+1}=0$ be a sequence of real numbers. Prove that $$ \sqrt{\sum_{k=1}^{n} a_{k}} \leq \sum_{k=1}^{n} \sqrt{k}\left(\sqrt{a_{k}}-\sqrt{a_{k+1}}\right) $$ | 19. Using that $a_{n+1}=0$ we can transform the desired inequality into $$ \begin{aligned} & \sqrt{a_{1}+}+a_{2}+\cdots+a_{n+1} \\ & \quad \leq \sqrt{1} \sqrt{a_{1}}+(\sqrt{2}-\sqrt{1}) \sqrt{a_{2}}+\cdots+(\sqrt{n+1}-\sqrt{n}) \sqrt{a_{n+1}} \end{aligned} $$ We shall prove by induction on $n$ that (1) holds for any $a... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 102 | 701 |
1997 | T0 | 2 | null | IMO | 2. (CAN) Let $R_{1}, R_{2}, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $R_{1}=(1)$, and if $R_{n-1}=\left(x_{1}, \ldots, x_{s}\right)$, then $$ R_{n}=\left(1,2, \ldots, x_{1}, 1,2, \ldots, x_{2}, \ldots, 1,2, \ldots, x_{s}, n\right) $$ For example, $R_{2}=(1,2), R_{... | 2. For any sequence $X=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ let us define $$ \bar{X}=\left(1,2, \ldots, x_{1}, 1,2, \ldots, x_{2}, \ldots, 1,2, \ldots, x_{n}\right) $$ Also, for any two sequences $A, B$ we denote their concatenation by $A B$. It clearly holds that $\overline{A B}=\bar{A} \bar{B}$. The sequences $R... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 230 | 924 |
1997 | T0 | 22 | null | IMO | 22. (UKR) (a) Do there exist functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ such that $$ f(g(x))=x^{2} \quad \text { and } \quad g(f(x))=x^{3} \quad \text { for all } x \in \mathbb{R} ? $$ (b) Do there exist functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{... | 22. (a) Suppose that $f$ and $g$ are such functions. From $g(f(x))=x^{3}$ we have $f\left(x_{1}\right) \neq f\left(x_{2}\right)$ whenever $x_{1} \neq x_{2}$. In particular, $f(-1), f(0)$, and $f(1)$ are three distinct numbers. However, since $f(x)^{2}=f(g(f(x)))=$ $f\left(x^{3}\right)$, each of the numbers $f(-1), f(0)... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 192 | 744 |
1997 | T0 | 23 | null | IMO | 23. (GBR) Let $A B C D$ be a convex quadrilateral and $O$ the intersection of its diagonals $A C$ and $B D$. If $$ O A \sin \angle A+O C \sin \angle C=O B \sin \angle B+O D \sin \angle D, $$ prove that $A B C D$ is cyclic. | 23. Let $K, L, M$, and $N$ be the projections of $O$ onto the lines $A B, B C, C D$, and $D A$, and let $\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}, \beta_{1}, \beta_{2}, \beta_{3}, \beta_{4}$ denote the angles $O A B$, $O B C, O C D, O D A, O A D, O B A, O C B, O D C$, respectively. We start with the following obs... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 86 | 604 |
1997 | T0 | 24 | null | IMO | 24. (LIT) ${ }^{\mathrm{IMO} 6}$ For a positive integer $n$, let $f(n)$ denote the number of ways to represent $n$ as the sum of powers of 2 with nonnegative integer exponents. Representations that differ only in the ordering in their summands are not considered to be distinct. (For instance, $f(4)=4$ because the numbe... | 24. There is a bijective correspondence between representations in the given form of $2 k$ and $2 k+1$ for $k=0,1, \ldots$, since adding 1 to every representation of $2 k$, we obtain a representation of $2 k+1$, and conversely, every representation of $2 k+1$ contains at least one 1 , which can be removed. Hence, $f(2 ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 177 | 863 |
1997 | T0 | 26 | null | IMO | 26. (ITA) For every integer $n \geq 2$ determine the minimum value that the sum $a_{0}+a_{1}+\cdots+a_{n}$ can take for nonnegative numbers $a_{0}, a_{1}, \ldots, a_{n}$ satisfying the condition $$ a_{0}=1, \quad a_{i} \leq a_{i+1}+a_{i+2} \quad \text { for } i=0, \ldots, n-2 $$ ### 3.39 The Thirty-Ninth IMO | 26. Let us first examine the case that all the inequalities in the problem are actually equalities. Then $a_{n-2}=a_{n-1}+a_{n}, a_{n-3}=2 a_{n-1}+a_{n}, \ldots, a_{0}=$ $F_{n} a_{n-1}+F_{n-1} a_{n}=1$, where $F_{n}$ is the $n$th Fibonacci number. Then it is easy to see (from $F_{1}+F_{2}+\cdots+F_{k}=F_{k+2}$ ) that $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 128 | 948 |
1997 | T0 | 4 | null | IMO | 4. (IRN) ${ }^{\mathrm{IMO} 4} \mathrm{An} n \times n$ matrix with entries from $\{1,2, \ldots, 2 n-1\}$ is called a coveralls matrix if for each $i$ the union of the $i$ th row and the $i$ th column contains $2 n-1$ distinct entries. Show that: (a) There exist no coveralls matrices for $n=1997$. (b) Coveralls matrices... | 4. (a) Suppose that an $n \times n$ coveralls matrix $A$ exists for some $n>1$. Let $x \in\{1,2, \ldots, 2 n-1\}$ be a fixed number that does not appear on the fixed diagonal of $A$. Such an element must exist, since the diagonal can contain at most $n$ different numbers. Let us call the union of the $i$ th row and the... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 122 | 1,105 |
1997 | T0 | 9 | null | IMO | 9. (USA) Let $A_{1} A_{2} A_{3}$ be a nonisosceles triangle with incenter $I$. Let $C_{i}$, $i=1,2,3$, be the smaller circle through $I$ tangent to $A_{i} A_{i+1}$ and $A_{i} A_{i+2}$ (the addition of indices being mod 3 ). Let $B_{i}, i=1,2,3$, be the second point of intersection of $C_{i+1}$ and $C_{i+2}$. Prove that... | 9. For $i=1,2,3$ (all indices in this problem will be modulo 3 ) we denote by $O_{i}$ the center of $C_{i}$ and by $M_{i}$ the midpoint of the $\operatorname{arc} A_{i+1} A_{i+2}$ that does not contain $A_{i}$. First we have that $O_{i+1} O_{i+2}$ is the perpendicular bisector of $I B_{i}$, and thus it contains the cir... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 172 | 805 |
1998 | T0 | 1 | null | IMO | 1. (LUX) ${ }^{\mathrm{IMO}} \mathrm{A}$ convex quadrilateral $A B C D$ has perpendicular diagonals. The perpendicular bisectors of $A B$ and $C D$ meet at a unique point $P$ inside $A B C D$. Prove that $A B C D$ is cyclic if and only if triangles $A B P$ and $C D P$ have equal areas. | 1. We begin with the following observation: Suppose that $P$ lies in $\triangle A E B$, where $E$ is the intersection of $A C$ and $B D$ (the other cases are similar). Let $M, N$ be the feet of the perpendiculars from $P$ to $A C$ and $B D$ respectively. We have $S_{A B P}=S_{A B E}-S_{A E P}-S_{B E P}=\frac{1}{2}(A E ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 93 | 717 |
1998 | T0 | 10 | null | IMO | 10. (AUS) Let $r_{1}, r_{2}, \ldots, r_{n}$ be real numbers greater than or equal to 1 . Prove that $$ \frac{1}{r_{1}+1}+\frac{1}{r_{2}+1}+\cdots+\frac{1}{r_{n}+1} \geq \frac{n}{\sqrt[n]{r_{1} r_{2} \cdots r_{n}}+1} $$ | 10. We shall first prove the inequality for $n$ of the form $2^{k}, k=0,1,2, \ldots$ The case $k=0$ is clear. For $k=1$, we have $$ \frac{1}{r_{1}+1}+\frac{1}{r_{2}+1}-\frac{2}{\sqrt{r_{1} r_{2}}+1}=\frac{\left(\sqrt{r_{1} r_{2}}-1\right)\left(\sqrt{r_{1}}-\sqrt{r_{2}}\right)^{2}}{\left(r_{1}+1\right)\left(r_{2}+1\righ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 110 | 737 |
1998 | T0 | 11 | null | IMO | 11. (RUS) Let $x, y$, and $z$ be positive real numbers such that $x y z=1$. Prove that $$ \frac{x^{3}}{(1+y)(1+z)}+\frac{y^{3}}{(1+z)(1+x)}+\frac{z^{3}}{(1+x)(1+y)} \geq \frac{3}{4} $$ | 11. The given inequality is equivalent to $x^{3}(x+1)+y^{3}(y+1)+z^{3}(z+1) \geq$ $\frac{3}{4}(x+1)(y+1)(z+1)$. By the A-G mean inequality, it will be enough to prove a stronger inequality: $$ x^{4}+x^{3}+y^{4}+y^{3}+z^{4}+z^{3} \geq \frac{1}{4}\left[(x+1)^{3}+(y+1)^{3}+(z+1)^{3}\right] . $$ If we set $S_{k}=x^{k}+y^{k... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 87 | 805 |
1998 | T0 | 12 | null | IMO | 12. (POL) Let $n \geq k \geq 0$ be integers. The numbers $c(n, k)$ are defined as follows: $$ \begin{aligned} c(n, 0) & =c(n, n)=1 & & \text { for all } n \geq 0 \\ c(n+1, k) & =2^{k} c(n, k)+c(n, k-1) & & \text { for } n \geq k \geq 1 \end{aligned} $$ Prove that $c(n, k)=c(n, n-k)$ for all $n \geq k \geq 0$. | 12. The assertion is clear for $n=0$. We shall prove the general case by induction on $n$. Suppose that $c(m, i)=c(m, m-i)$ for all $i$ and $m \leq n$. Then by the induction hypothesis and the recurrence formula we have $c(n+1, k)=2^{k} c(n, k)+c(n, k-1)$ and $c(n+1, n+1-k)=$ $2^{n+1-k} c(n, n+1-k)+c(n, n-k)=2^{n+1-k} ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 149 | 774 |
1998 | T0 | 13 | null | IMO | 13. (BUL) ${ }^{\mathrm{IMO} 6}$ Determine the least possible value of $f(1998)$, where $f$ is a function from the set $\mathbb{N}$ of positive integers into itself such that for all $m, n \in \mathbb{N}$, $$ f\left(n^{2} f(m)\right)=m[f(n)]^{2} $$ | 13. Denote by $\mathcal{F}$ the set of functions considered. Let $f \in \mathcal{F}$, and let $f(1)=a$. Putting $n=1$ and $m=1$ we obtain $f(f(z))=a^{2} z$ and $f\left(a z^{2}\right)=f(z)^{2}$ for all $z \in \mathbb{N}$. These equations, together with the original one, imply $f(x)^{2} f(y)^{2}=f(x)^{2} f\left(a y^{2}\r... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 93 | 807 |
1998 | T0 | 17 | null | IMO | 17. (GBR) A sequence of integers $a_{1}, a_{2}, a_{3}, \ldots$ is defined as follows: $a_{1}=1$, and for $n \geq 1, a_{n+1}$ is the smallest integer greater than $a_{n}$ such that $a_{i}+a_{j} \neq 3 a_{k}$ for any $i, j, k$ in $\{1,2, \ldots, n+1\}$, not necessarily distinct. Determine $a_{1998}$. | 17. Initially, we determine that the first few values for $a_{n}$ are $1,3,4,7,10$, $12,13,16,19,21,22,25$. Since these are exactly the numbers of the forms $3 k+1$ and $9 k+3$, we conjecture that this is the general pattern. In fact, it is easy to see that the equation $x+y=3 z$ has no solution in the set $K=\{3 k+1,9... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 128 | 549 |
1998 | T0 | 19 | null | IMO | 19. (BLR) ${ }^{\mathrm{IMO} 3}$ For any positive integer $n$, let $\tau(n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau\left(n^{2}\right)}{\tau(n)}=m$. | 19. For $n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{r}^{\alpha_{r}}$, where $p_{i}$ are distinct primes and $\alpha_{i}$ natural numbers, we have $\tau(n)=\left(\alpha_{1}+1\right) \cdots\left(\alpha_{r}+1\right)$ and $\tau\left(n^{2}\right)=\left(2 \alpha_{1}+1\right) \ldots\left(2 \alpha_{r}+1\right)$. Putting... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 83 | 605 |
1998 | T0 | 2 | null | IMO | 2. (POL) Let $A B C D$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $A B$ and $C D$, respectively, such that $A E: E B=C F$ : $F D$. Let $P$ be the point on the segment $E F$ such that $P E: P F=A B$ : $C D$. Prove that the ratio between the areas of triangles $A P D$ and $B P C$ does not ... | 2. If $A D$ and $B C$ are parallel, then $A B C D$ is an isosceles trapezoid with $A B=C D$, so $P$ is the midpoint of $E F$. Let $M$ and $N$ be the midpoints of $A B$ and $C D$. Then $M N \| B C$, and the distance $d(E, M N)$ equals the distance $d(F, M N)$ because $B$ and $D$ are the same distance from $M N$ and $E M... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 124 | 676 |
1998 | T0 | 20 | null | IMO | 20. (ARG) Prove that for each positive integer $n$, there exists a positive integer with the following properties: (i) It has exactly $n$ digits. (ii) None of the digits is 0 . (iii) It is divisible by the sum of its digits. | 20. We first consider the special case $n=3^{r}$. Then the simplest choice $\frac{10^{n}-1}{9}=$ $11 \ldots 1$ ( $n$ digits) works. This can be shown by induction: it is true for $r=$ 1, while the inductive step follows from $10^{3^{r}}-1=\left(10^{3^{r-1}}-1\right)\left(10^{2 \cdot 3^{r-1}}+\right.$ $10^{3^{r-1}}+1$ )... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 61 | 762 |
1998 | T0 | 21 | null | IMO | 21. (CAN) Let $a_{0}, a_{1}, a_{2}, \ldots$ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2 a_{j}+4 a_{k}$, where $i, j, k$ are not necessarily distinct. Determine $a_{1998}$. | 21. Such a sequence is obviously strictly increasing. We note that it must be unique. Indeed, given $a_{0}, a_{1}, \ldots, a_{n-1}$, then $a_{n}$ is the least positive integer not of the form $a_{i}+2 a_{j}+4 a_{k}, i, j, k<n$. We easily get that the first few $a_{n}$ 's are $0,1,8,9,64,65,72,73, \ldots$ Let $\left\{c_... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 86 | 807 |
1998 | T0 | 22 | null | IMO | 22. (UKR) A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil$ or $\lfloor x\rfloor$ so that the row sums and column sums remain unchanged. (Note that $\lceil x\rceil$ i... | 22. We can obviously change each $x$ into $\lfloor x\rfloor$ or $\lceil x\rceil$ so that the column sums remain unchanged. However, this does not necessarily match the row sums as well, so let us consider the sum $S$ of the absolute values of the changes in the row sums. It is easily seen that $S$ is even, and we want ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 120 | 600 |
1998 | T0 | 23 | null | IMO | 23. (BLR) Let $n$ be an integer greater than 2. A positive integer is said to be attainable if it is 1 or can be obtained from 1 by a sequence of operations with the following properties: (i) The first operation is either addition or multiplication. (ii) Thereafter, additions and multiplications are used alternately. (... | 23. (a) If $n$ is even, then every odd integer is unattainable. Assume that $n \geq 9$ is odd. Let $a$ be obtained by addition from some $b$, and $b$ from $c$ by multiplication. Then $a$ is $2 c+2,2 c+n, n c+2$, or $n c+n$, and is in every case congruent to $2 c+2$ modulo $n-2$. In particular, if $a \equiv-2$ $(\bmod n... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 183 | 718 |
1998 | T0 | 24 | null | IMO | 24. (SWE) Cards numbered 1 to 9 are arranged at random in a row. In a move, one may choose any block of consecutive cards whose numbers are in ascending or descending order, and switch the block around. For example, $91 \underline{6532748}$ may be changed to $91 \underline{3562748}$. Prove that in at most 12 moves, one... | 24. Let $f(n)$ be the minimum number of moves needed to monotonize any permutation of $n$ distinct numbers. Let us be given a permutation $\pi$ of $\{1,2, \ldots, n\}$, and let $k$ be the first element of $\pi$. In $f(n-1)$ moves, we can transform $\pi$ to either $(k, 1,2, \ldots, k-1, k+1, \ldots, n)$ or $(k, n, n-1, ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 114 | 514 |
1998 | T0 | 27 | null | IMO | 27. (BLR) Ten points such that no three of them lie on a line are marked in the plane. Each pair of points is connected with a segment. Each of these segments is painted with one of $k$ colors in such a way that for any $k$ of the ten points, there are $k$ segments each joining two of them with no two being painted the... | 27. Since this is essentially a graph problem, we call the points and segments vertices and edges of the graph. We first prove that the task is impossible if $k \leq 4$. Cases $k \leq 2$ are trivial. If $k=3$, then among the edges from a vertex $A$ there are two of the same color, say $A B$ and $A C$, so we don't have ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 108 | 714 |
1998 | T0 | 3 | null | IMO | 3. (UKR) ${ }^{\mathrm{IMO}}$ Let $I$ be the incenter of triangle $A B C$. Let $K, L$, and $M$ be the points of tangency of the incircle of $A B C$ with $A B, B C$, and $C A$, respectively. The line $t$ passes through $B$ and is parallel to $K L$. The lines $M K$ and $M L$ intersect $t$ at the points $R$ and $S$. Prove... | 3. Lemma. If $U, W, V$ are three points on a line $l$ in this order, and $X$ a point in the plane with $X W \perp U V$, then $\angle U X V<90^{\circ}$ if and only if $X W^{2}>U W \cdot V W$. Proof. Let $X W^{2}>U W \cdot V W$, and let $X_{0}$ be a point on the segment $X W$ such that $X_{0} W^{2} \geq U W \cdot V W$. T... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 126 | 684 |
1998 | T0 | 6 | null | IMO | 6. (POL) Let $A B C D E F$ be a convex hexagon such that $\angle B+\angle D+\angle F=$ $360^{\circ}$ and $$ \frac{A B}{B C} \cdot \frac{C D}{D E} \cdot \frac{E F}{F A}=1 $$ Prove that $$ \frac{B C}{C A} \cdot \frac{A E}{E F} \cdot \frac{F D}{D B}=1 $$ | 6. Let $P$ be the point such that $\triangle C D P$ and $\triangle C B A$ are similar and equally oriented. Since then $\angle D C P=\angle B C A$ and $\frac{B C}{C A}=\frac{D C}{C P}$, it follows that $\angle A C P=\angle B C D$ and $\frac{A C}{C P}=\frac{B C}{C D}$, so $\triangle A C P \sim \triangle B C D$. In parti... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 114 | 554 |
1998 | T0 | 7 | null | IMO | 7. (GBR) Let $A B C$ be a triangle such that $\angle A C B=2 \angle A B C$. Let $D$ be the point on the side $B C$ such that $C D=2 B D$. The segment $A D$ is extended to $E$ so that $A D=D E$. Prove that $$ \angle E C B+180^{\circ}=2 \angle E B C . $$ | 7. We shall use the following result. Lemma. In a triangle $A B C$ with $B C=a, C A=b$, and $A B=c$, i. $\angle C=2 \angle B$ if and only if $c^{2}=b^{2}+a b$; ii. $\angle C+180^{\circ}=2 \angle B$ if and only if $c^{2}=b^{2}-a b$. Proof. i. Take a point $D$ on the extension of $B C$ over $C$ such that $C D=b$. The con... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 100 | 627 |
1999 | T0 | 10 | Geometry | IMO | 10. G4 (GBR) For a triangle $T=A B C$ we take the point $X$ on the side $(A B)$ such that $A X / X B=4 / 5$, the point $Y$ on the segment $(C X)$ such that $C Y=2 Y X$, and, if possible, the point $Z$ on the ray ( $C A$ such that $\measuredangle C X Z=180^{\circ}-\measuredangle A B C$. We denote by $\Sigma$ the set of ... | 10. We use the following lemma. Lemma. Let $A B C$ be a triangle and $X \in A B$ such that $\overrightarrow{A X}: \overrightarrow{X B}=m: n$. Then $(m+n) \cot \angle C X B=n \cot A-m \cot B$ and $m \cot \angle A C X=$ $(n+m) \cot C+n \cot A$. Proof. Let $C D$ be the altitude from $C$ and $h$ its length. Then using orie... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 163 | 550 |
1999 | T0 | 12 | Geometry | IMO | 12. G6 (RUS) ${ }^{\mathrm{IMO} 5}$ Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in $M$ and $N$, and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B . M A$ and $M B$ intersect $\Omega_{1}$ in... | 12. We first introduce the following lemmas. Lemma 1. Let $A B C$ be a triangle, $I$ its inenter and $I_{a}$ the center of the excircle touching $B C$. Let $A^{\prime}$ be the center of the arc $\widehat{B C}$ of the circumcircle not containing $A$. Then $A^{\prime} B=A^{\prime} C=A^{\prime} I=A^{\prime} I_{a}$. Proof.... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 131 | 1,202 |
1999 | T0 | 14 | Geometry | IMO | 14. G8 (RUS) Points $A, B, C$ divide the circumcircle $\Omega$ of the triangle $A B C$ into three arcs. Let $X$ be a variable point on the $\operatorname{arc} A B$, and let $O_{1}, O_{2}$ be the incenters of the triangles $C A X$ and $C B X$. Prove that the circumcircle of the triangle $X O_{1} O_{2}$ intersects $\Omeg... | 14. We first introduce the same lemma as in problem 12 and state it here without proof. Lemma. Let $A B C$ be a triangle and $I$ the center of its incircle. Let $M$ be the center of the $\operatorname{arc} \widehat{B C}$ of the circumcircle not containing $A$. Then $M B=M C=M I$. Let the circle $X O_{1} O_{2}$ intersec... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 112 | 510 |
1999 | T0 | 15 | Algebra | IMO | 15. A1 (POL) ${ }^{\mathrm{IMO} 2}$ Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such that the inequality $$ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} $$ holds for every $x_{1}, \ldots, x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ s... | 15. For all $x_{i}=0$ any $C$ will do, so we may assume the contrary. Since the equation is symmetric and homogeneous, we may assume $\sum_{i} x_{i}=1$. The equation now becomes $F\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right)=$ $\sum_{i} x_{i}^{2} \sum_{j \neq i} x_{j}=... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 153 | 764 |
1999 | T0 | 17 | Algebra | IMO | 17. A3 (FIN) A game is played by $n$ girls $(n \geq 2)$, everybody having a ball. Each of the $\binom{n}{2}$ pairs of players, in an arbitrary order, exchange the balls they have at that moment. The game is called nice if at the end nobody has her own ball, and it is called tiresome if at the end everybody has her init... | 17. A game is determined by the ordering $t_{1}, \ldots, t_{N}$ of the $N=\binom{n}{2}$ transpositions $(i, j)$ of the set $\{1,2, \ldots, n\}$. The game is nice if the permutation $P=t_{N} t_{N-1} \ldots t_{1}$ has no fixed point, and tiresome if $P$ is the identity (denoted by $I$ ). Recall that every permutation can... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 115 | 873 |
1999 | T0 | 18 | Algebra | IMO | 18. A4 (BLR) Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that for any two integers $x, y$ taken from two different subsets, the number $x^{2}-x y+y^{2}$ belongs to the third subset. | 18. Define $f(x, y)=x^{2}-x y+y^{2}$. Let us assume that three such sets $A, B$, and $C$ do exist and that w.l.o.g. $1, b$, and $c(c>b)$ are respectively their smallest elements. Lemma 1. Numbers $x, y$, and $x+y$ cannot belong to three different sets. Proof. The number $f(x, x+y)=f(y, x+y)$ must belong to both the set... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 62 | 595 |
1999 | T0 | 20 | Algebra | IMO | 20. A6 (SWE) For $n \geq 3$ and $a_{1} \leq a_{2} \leq \cdots \leq a_{n}$ given real numbers we have the following instructions: (1) place the numbers in some order in a circle; (2) delete one of the numbers from the circle; (3) if just two numbers are remaining in the circle, let $S$ be the sum of these two numbers. O... | 20. We first introduce some useful notation. An arrangement around the circle will be denoted by $x=\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}$, where the elements are arranged clockwise and $x_{1}$ is fixed to be the smallest number. We will call an arrangement balanced if $x_{1} \leq x_{n} \leq x_{2} \leq x_{n-1} \le... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 264 | 1,375 |
1999 | T0 | 21 | Combinatorics | IMO | 21. C1 (IND) Let $n \geq 1$ be an integer. A path from $(0,0)$ to $(n, n)$ in the $x y$ plane is a chain of consecutive unit moves either to the right (move denoted by $E$ ) or upwards (move denoted by $N$ ), all the moves being made inside the half-plane $x \geq y$. A step in a path is the occurrence of two consecutiv... | 21. Let us call $f(n, s)$ the number of paths from $(0,0)$ to $(n, n)$ that contain exactly $s$ steps. Evidently, for all $n$ we have $f(n, 1)=f(2,2)=1$, in accordance with the formula. Let us thus assume inductively for a given $n>2$ that for all $s$ we have $f(n, s)=\frac{1}{s}\binom{n-1}{s-1}\binom{n}{s-1}$. We shal... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 174 | 527 |
1999 | T0 | 23 | Combinatorics | IMO | 23. C3 (GBR) A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that: (i) the first fly is caught after a resting period of one minute; (ii) the resting period before catching the $2 m$ th fly is the same as the resting period before catching the $m$ th fly an... | 23. Let $r(m)$ denote the rest period before the $m$ th catch, $t(m)$ the number of minutes before the $m$ th catch, and $f(n)$ as the number of flies caught in $n$ minutes. We have $r(1)=1, r(2 m)=r(m)$, and $r(2 m+1)=f(m)+1$. We then have by induction that $r(m)$ is the number of ones in the binary representation of ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 176 | 604 |
1999 | T0 | 24 | Combinatorics | IMO | 24. C4 (GBR) Let $A$ be a set of $N$ residues $\left(\bmod N^{2}\right)$. Prove that there exists a set $B$ of $N$ residues $\left(\bmod N^{2}\right)$ such that the set $A+B=\{a+b \mid$ $a \in A, b \in B\}$ contains at least half of all residues $\left(\bmod N^{2}\right)$. | 24. Let $S=\left\{0,1, \ldots, N^{2}-1\right\}$ be the group of residues (with respect to addition modulo $N^{2}$ ) and $A$ an $n$-element subset. We will use $|X|$ to denote the number of elements of a subset $X$ of $S$, and $\bar{X}$ to refer to the complement of $X$ in $S$. For $i \in S$ we also define $A_{i}=\{a+i ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 104 | 762 |
1999 | T0 | 27 | Combinatorics | IMO | 27. C7 (IRE) Let $p>3$ be a prime number. For each nonempty subset $T$ of $\{0,1,2,3, \ldots, p-1\}$ let $E(T)$ be the set of all $(p-1)$-tuples $\left(x_{1}, \ldots, x_{p-1}\right)$, where each $x_{i} \in T$ and $x_{1}+2 x_{2}+\cdots+(p-1) x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in ... | 27. Denote $A=\{0,1,2\}$ and $B=\{0,1,3\}$. Let $f_{T}(x)=\sum_{a \in T} x^{a}$. Then define $F_{T}(x)=f_{T}(x) f_{T}\left(x^{2}\right) \cdots f_{T}\left(x^{p-1}\right)$. We can write $F_{T}(x)=\sum_{i=0}^{p(p-1)} a_{i} x^{i}$, where $a_{i}$ is the number of ways to select an array $\left\{x_{1}, \ldots, x_{p-1}\right\... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 192 | 1,841 |
1999 | T0 | 4 | Number Theory | IMO | 4. N4 (FRA) Denote by $S$ the set of all primes $p$ such that the decimal representation of $1 / p$ has its fundamental period divisible by 3 . For every $p \in S$ such that $1 / p$ has its fundamental period $3 r$ one may write $1 / p=$ $0 . a_{1} a_{2} \ldots a_{3 r} a_{1} a_{2} \ldots a_{3 r} \ldots$, where $r=r(p)$... | 4. (a) The fundamental period of $p$ is the smallest integer $d(p)$ such that $p \mid 10^{d(p)}-1$. Let $s$ be an arbitrary prime and set $N_{s}=10^{2 s}+10^{s}+1$. In that case $N_{s} \equiv 3(\bmod 9)$. Let $p_{s} \neq 37$ be a prime dividing $N_{s} / 3$. Clearly $p_{s} \neq 3$. We claim that such a prime exists and ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 217 | 835 |
1999 | T0 | 6 | Number Theory | IMO | 6. N6 (BLR) Prove that for every real number $M$ there exists an infinite arithmetic progression such that: (i) each term is a positive integer and the common difference is not divisible by 10 ; (ii) the sum of the digits of each term (in decimal representation) exceeds $M$. | 6. Let $N$ be the smallest integer greater than $M$. We take the difference of the numbers in the progression to be of the form $10^{m}+1, m \in \mathbb{N}$. Hence we can take $a_{n}=a_{0}+n\left(10^{m}+1\right)=\overline{b_{s} b_{s-1} \ldots b_{0}}$ where $a_{0}$ is the initial term in the progression and $\overline{b... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 68 | 901 |
1999 | T0 | 9 | Geometry | IMO | 9. G3 (EST) ${ }^{\mathrm{IMO} 1}$ A set $S$ of points in space will be called completely symmetric if it has at least three elements and satisfies the following condition: For every two distinct points $A, B$ from $S$ the perpendicular bisector of the segment $A B$ is an axis of symmetry for $S$. Prove that if a compl... | 9. Let $r_{P Q}$ denote a reflection about the planar bisector of $P Q$ with $P, Q \in S$. Let $G$ be the centroid of $S$. From $r_{P Q}(S)=S$ it follows that $r_{P Q}(G)=G$. Hence $G$ belongs to the perpendicular bisector of $P Q$ and thus $G P=G Q$. Consequently the whole of $S$ lies on a sphere $\Sigma$ centered at ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 117 | 857 |
2000 | T0 | 1 | null | IMO | 1. $\mathbf{C 1}$ (HUN) ${ }^{\mathrm{IMO}}$ A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one, and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those... | 1. In order for the trick to work, whenever $x+y=z+t$ and the cards $x, y$ are placed in different boxes, either $z, t$ are in these boxes as well or they are both in the remaining box. Case 1. The cards $i, i+1, i+2$ are in different boxes for some $i$. Since $i+(i+3)=(i+1)+(i+2)$, the cards $i$ and $i+3$ must be in t... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 127 | 553 |
2000 | T0 | 10 | Algebra | IMO | 10. A4 (GBR) The function $F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: For every $n \geq 0$, (i) $F(4 n)=F(2 n)+F(n)$; (ii) $F(4 n+2)=F(4 n)+1$; (iii) $F(2 n+1)=F(2 n)+1$. Prove that for each positive integer $m$, the number of integers $n$... | 10. Clearly $F(0)=0$ by (i). Moreover, it follows by induction from (i) that $F\left(2^{n}\right)=f_{n+1}$ where $f_{n}$ denotes the $n$th Fibonacci's number. In general, if $n=\epsilon_{k} 2^{k}+\epsilon_{k-1} 2^{k-1}+\cdots+\epsilon_{1} \cdot 2+\epsilon_{0}$ (where $\epsilon_{i} \in\{0,1\}$ ), it is straightforward t... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 156 | 1,111 |
2000 | T0 | 11 | Algebra | IMO | 11. A5 (BLR) ${ }^{\mathrm{IMO} 3}$ Let $n \geq 2$ be a positive integer and $\lambda$ a positive real number. Initially there are $n$ fleas on a horizontal line, not all at the same point. We define a move of choosing two fleas at some points $A$ and $B$, with $A$ to the left of $B$, and letting the flea from $A$ jump... | 11. We claim that for $\lambda \geq \frac{1}{n-1}$ we can take all fleas as far to the right as we want. In every turn we choose the leftmost flea and let it jump over the rightmost one. Let $d$ and $\delta$ denote the maximal and the minimal distances between two fleas at some moment. Clearly, $d \geq(n-1) \delta$. Af... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 175 | 575 |
2000 | T0 | 12 | Algebra | IMO | 12. A6 (IRE) A nonempty set $A$ of real numbers is called a $B_{3}$-set if the conditions $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6} \in A$ and $a_{1}+a_{2}+a_{3}=a_{4}+a_{5}+a_{6}$ imply that the sequences $\left(a_{1}, a_{2}, a_{3}\right)$ and $\left(a_{4}, a_{5}, a_{6}\right)$ are identical up to a permutation. Let $... | 12. Since $D(A)=D(B)$, we can define $f(i)>g(i) \geq 0$ that satisfy $b_{i}-b_{i-1}=$ $a_{f(i)}-a_{g(i)}$ for all $i$. The number $b_{i+1}-b_{i-1} \in D(B)=D(A)$ can be written in the form $a_{u}-a_{v}, u>v \geq 0$. Then $b_{i+1}-b_{i-1}=b_{i+1}-b_{i}+b_{i}-b_{i-1}$ implies $a_{f(i+1)}+a_{f(i)}+a_{v}=a_{g(i+1)}+a_{g(i)... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 259 | 858 |
2000 | T0 | 13 | Algebra | IMO | 13. A7 (RUS) For a polynomial $P$ of degree 2000 with distinct real coefficients let $M(P)$ be the set of all polynomials that can be produced from $P$ by permutation of its coefficients. A polynomial $P$ will be called $n$-independent if $P(n)=0$ and we can get from any $Q$ in $M(P)$ a polynomial $Q_{1}$ such that $Q_... | 13. One can easily find $n$-independent polynomials for $n=0,1$. For example, $P_{0}(x)=2000 x^{2000}+\cdots+2 x^{2}+x+0$ is 0 -independent (for $Q \in M\left(P_{0}\right)$ it suffices to exchange the coefficient 0 of $Q$ with the last term), and $P_{1}(x)=2000 x^{2000}+\cdots+2 x^{2}+x-(1+2+\cdots+2000)$ is 1-independ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 136 | 1,899 |
2000 | T0 | 15 | Number Theory | IMO | 15. N2 (FRA) For a positive integer $n$, let $d(n)$ be the number of all positive divisors of $n$. Find all positive integers $n$ such that $d(n)^{3}=4 n$. | 15. Let $n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{k}^{\alpha_{k}}$ be the factorization of $n$ onto primes $\left(p_{1}<p_{2}<\right.$ $\cdots<p_{k}$ ). Since $4 n$ is a perfect cube, we deduce that $p_{1}=2$ and $\alpha_{1}=$ $3 \beta_{1}+1, \alpha_{2}=3 \beta_{2}, \ldots, \alpha_{k}=3 \beta_{k}$ for some int... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 52 | 723 |
2000 | T0 | 16 | Number Theory | IMO | 16. N3 (RUS) ${ }^{\mathrm{IMO} 5}$ Does there exist a positive integer $n$ such that $n$ has exactly 2000 prime divisors and $2^{n}+1$ is divisible by $n$ ? | 16. More generally, we will prove by induction on $k$ that for each $k \in \mathbb{N}$ there exists $n_{k} \in \mathbb{N}$ that has exactly $k$ distinct prime divisors such that $n_{k} \mid 2^{n_{k}}+1$ and $3 \mid n_{k}$. For $k=1, n_{1}=3$ satisfies the given conditions. Now assume that $k \geq 1$ and $n_{k}=3^{\alph... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 60 | 507 |
2000 | T0 | 24 | Geometry | IMO | 24. G5 (GBR) The tangents at $B$ and $A$ to the circumcircle of an acuteangled triangle $A B C$ meet the tangent at $C$ at $T$ and $U$ respectively. $A T$ meets $B C$ at $P$, and $Q$ is the midpoint of $A P ; B U$ meets $C A$ at $R$, and $S$ is the midpoint of $B R$. Prove that $\angle A B Q=\angle B A S$. Determine, i... | 24. Since $\angle A B T=180^{\circ}-\gamma$ and $\angle A C T=180^{\circ}-\beta$, the law of sines gives $\frac{B P}{P C}=\frac{S_{A B T}}{S_{A C T}}=\frac{A B \cdot B T \cdot \sin \gamma}{A B \cdot B T \cdot \sin \beta}=\frac{A B \sin \gamma}{A C \sin \beta}=\frac{c^{2}}{b^{2}}$, which implies $B P=\frac{c^{2} a}{b^{2... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 134 | 594 |
2000 | T0 | 26 | Geometry | IMO | 26. G7 (IRN) Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o'clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed? | 26. The problem can be reformulated in the following way: Given a set $S$ of ten points in the plane such that the distances between them are all distinct, for each point $P \in S$ we mark the point $Q \in S \backslash\{P\}$ nearest to $P$. Find the least possible number of marked points. Observe that each point $A \in... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 69 | 816 |
2000 | T0 | 27 | Geometry | IMO | 27. G8 (RUS) ${ }^{\mathrm{IMO}} A_{1} A_{2} A_{3}$ is an acute-angled triangle. The foot of the altitude from $A_{i}$ is $K_{i}$, and the incircle touches the side opposite $A_{i}$ at $L_{i}$. The line $K_{1} K_{2}$ is reflected in the line $L_{1} L_{2}$. Similarly, the line $K_{2} K_{3}$ is reflected in $L_{2} L_{3}$... | 27. Denote by $\alpha_{1}, \alpha_{2}, \alpha_{3}$ the angles of $\triangle A_{1} A_{2} A_{3}$ at vertices $A_{1}, A_{2}, A_{3}$ respectively. Let $T_{1}, T_{2}, T_{3}$ be the points symmetric to $L_{1}, L_{2}, L_{3}$ with respect to $A_{1} I, A_{2} I$, and $A_{3} I$ respectively. We claim that $T_{1} T_{2} T_{3}$ is t... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 165 | 742 |
2000 | T0 | 5 | Combinatorics | IMO | 5. C5 (RUS) In the plane we have $n$ rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is nice if it has at least one of the vertices of the $n$ rectangles on its boundary. Prove that the sum of the num... | 5. We say that a vertex of a nice region is convex if the angle of the region at that vertex equals $90^{\circ}$; otherwise (if the angle is $270^{\circ}$ ), we say that a vertex is concave. For a simple broken line $C$ contained in the boundary of a nice region $R$ we call the pair $(R, C)$ a boundary pair. Such a pai... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 108 | 645 |
2000 | T0 | 6 | Combinatorics | IMO | 6. C6 (FRA) Let $p$ and $q$ be relatively prime positive integers. A subset $S$ of $\{0,1,2, \ldots\}$ is called ideal if $0 \in S$ and for each element $n \in S$, the integers $n+p$ and $n+q$ belong to $S$. Determine the number of ideal subsets of $\{0,1,2 \ldots\}$. | 6. Every integer $z$ has a unique representation $z=p x+q y$, where $x, y \in \mathbb{Z}$, $0 \leq x \leq q-1$. Consider the region $T$ in the $x y$-plane defined by the last inequality and $p x+q y \geq 0$. There is a bijective correspondence between lattice points of this region and nonnegative integers given by $(x,... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 98 | 773 |
2000 | T0 | 8 | Algebra | IMO | 8. A2 (GBR) Let $a, b, c$ be positive integers satisfying the conditions $b>2 a$ and $c>2 b$. Show that there exists a real number $t$ with the property that all the three numbers $t a, t b, t c$ have their fractional parts lying in the interval (1/3,2/3]. | 8. We note that $\{t a\}$ lies in $\left(\frac{1}{3}, \frac{2}{3}\right]$ if and only if there is an integer $k$ such that $k+\frac{1}{3}<t a \leq k+\frac{2}{3}$, i.e., if and only if $t \in I_{k}=\left(\frac{k+1 / 3}{a}, \frac{k+2 / 3}{a}\right]$ for some $k$. Similarly, $t$ should belong to the sets $J_{m}=\left(\fra... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 80 | 733 |
2000 | T0 | 9 | Algebra | IMO | 9. A3 (BLR) Find all pairs of functions $f: \mathbb{R} \rightarrow \mathbb{R}, g: \mathbb{R} \rightarrow \mathbb{R}$ such that $$ f(x+g(y))=x f(y)-y f(x)+g(x) \quad \text { for all } x, y \in R $$ | 9. Let us first solve the problem under the assumption that $g(\alpha)=0$ for some $\alpha$. Setting $y=\alpha$ in the given equation yields $g(x)=(\alpha+1) f(x)-x f(\alpha)$. Then the given equation becomes $f(x+g(y))=(\alpha+1-y) f(x)+(f(y)-f(\alpha)) x$, so setting $y=\alpha+1$ we get $f(x+n)=m x$, where $n=g(\alph... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 83 | 651 |
2001 | T0 | 10 | Combinatorics | IMO | 10. C4 (NZL) A set of three nonnegative integers $\{x, y, z\}$ with $x<y<z$ is called historic if $\{z-y, y-x\}=\{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of disjoint historic sets. | 10. For convenience let us write $a=1776, b=2001,0<a<b$. There are two types of historic sets: $$ \text { (1) }\{x, x+a, x+a+b\} \quad \text { and } \quad \text { (2) }\{x, x+b, x+a+b\} \text {. } $$ We construct a sequence of historic sets $H_{1}, H_{2}, H_{3}, \ldots$ inductively as follows: (i) $H_{1}=\{0, a, a+b\}$... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 79 | 507 |
2001 | T0 | 15 | Geometry | IMO | 15. G1 (UKR) Let $A_{1}$ be the center of the square inscribed in acute triangle $A B C$ with two vertices of the square on side $B C$. Thus one of the two remaining vertices of the square is on side $A B$ and the other is on $A C$. Points $B_{1}, C_{1}$ are defined in a similar way for inscribed squares with two verti... | 15. Let $M N P Q$ be the square inscribed in $\triangle A B C$ with $M \in A B, N \in A C$, $P, Q \in B C$, and let $A A_{1}$ meet $M N, P Q$ at $K, X$ respectively. Put $M K=$ $P X=m, N K=Q X=n$, and $M N=d$. Then $$ \frac{B X}{X C}=\frac{m}{n}=\frac{B X+m}{X C+n}=\frac{B P}{C Q}=\frac{d \cot \beta+d}{d \cot \gamma+d}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 128 | 655 |
2001 | T0 | 17 | Geometry | IMO | 17. G3 (GBR) Let $A B C$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $A B C$ such that $$ A P \cdot A G+B P \cdot B G+C P \cdot C G $$ is a minimum, and express this minimum value in terms of the side lengths of $A B C$. | 17. Let us investigate a more general problem, in which $G$ is any point of the plane such that $A G, B G, C G$ are sides of a triangle. Let $F$ be the point in the plane such that $B C: C F: F B=A G: B G: C G$ and $F, A$ lie on different sides of $B C$. Then by Ptolemy's inequality, on $B P C F$ we have $A G \cdot A P... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 90 | 736 |
2001 | T0 | 18 | Geometry | IMO | 18. G4 (FRA) Let $M$ be a point in the interior of triangle $A B C$. Let $A^{\prime}$ lie on $B C$ with $M A^{\prime}$ perpendicular to $B C$. Define $B^{\prime}$ on $C A$ and $C^{\prime}$ on $A B$ similarly. Define $$ p(M)=\frac{M A^{\prime} \cdot M B^{\prime} \cdot M C^{\prime}}{M A \cdot M B \cdot M C} $$ Determin... | 18. Let $\alpha_{1}, \beta_{1}, \gamma_{1}, \alpha_{2}, \beta_{2}, \gamma_{2}$ denote the angles $\angle M A B, \angle M B C, \angle M C A$, $\angle M A C, \angle M B A, \angle M C B$ respectively. Then $\frac{M B^{\prime} \cdot M C^{\prime}}{M A^{2}}=\sin \alpha_{1} \sin \alpha_{2}$, $\frac{M C^{\prime} \cdot M A^{\pr... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 176 | 507 |
2001 | T0 | 2 | Algebra | IMO | 2. A2 (POL) Let $a_{0}, a_{1}, a_{2}, \ldots$ be an arbitrary infinite sequence of positive numbers. Show that the inequality $1+a_{n}>a_{n-1} \sqrt[n]{2}$ holds for infinitely many positive integers $n$. | 2. It follows from Bernoulli's inequality that for each $n \in \mathbb{N},\left(1+\frac{1}{n}\right)^{n} \geq 2$, or $\sqrt[n]{2} \leq 1+\frac{1}{n}$. Consequently, it will be enough to show that $1+a_{n}>$ $\left(1+\frac{1}{n}\right) a_{n-1}$. Assume the opposite. Then there exists $N$ such that for each $n \geq N$, $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 65 | 615 |
2001 | T0 | 20 | Geometry | IMO | 20. G6 (IND) Let $A B C$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose $A P, B P, C P$ meet the sides $B C, C A, A B$ (or extensions thereof) in $D, E, F$, respectively. Suppose further that the areas of triangles $P B D, P C E, P A F$ are all equal. Prove that each of these areas is equ... | 20. By Ceva's theorem, we can choose real numbers $x, y, z$ such that $$ \frac{\overrightarrow{B D}}{\overrightarrow{D C}}=\frac{z}{y}, \frac{\overrightarrow{C E}}{\overrightarrow{E A}}=\frac{x}{z}, \text { and } \frac{\overrightarrow{A F}}{\overrightarrow{F B}}=\frac{y}{x} $$ The point $P$ lies outside the triangle $A... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 115 | 1,045 |
2001 | T0 | 21 | Geometry | IMO | 21. G7 (BUL) Let $O$ be an interior point of acute triangle $A B C$. Let $A_{1}$ lie on $B C$ with $O A_{1}$ perpendicular to $B C$. Define $B_{1}$ on $C A$ and $C_{1}$ on $A B$ similarly. Prove that $O$ is the circumcenter of $A B C$ if and only if the perimeter of $A_{1} B_{1} C_{1}$ is not less than any one of the p... | 21. We denote by $p(X Y Z)$ the perimeter of a triangle $X Y Z$. If $O$ is the circumcenter of $\triangle A B C$, then $A_{1}, B_{1}, C_{1}$ are the midpoints of the corresponding sides of the triangle, and hence $p\left(A_{1} B_{1} C_{1}\right)=$ $p\left(A B_{1} C_{1}\right)=p\left(A_{1} B C_{1}\right)=p\left(A_{1} B_... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 153 | 940 |
2001 | T0 | 25 | Number Theory | IMO | 25. N3 (GBR) Let $a_{1}=11^{11}, a_{2}=12^{12}, a_{3}=13^{13}$, and $$ a_{n}=\left|a_{n-1}-a_{n-2}\right|+\left|a_{n-2}-a_{n-3}\right|, \quad n \geq 4 $$ Determine $a_{14^{14}}$. | 25. Define $b_{n}=\left|a_{n+1}-a_{n}\right|$ for $n \geq 1$. From the equalities $a_{n+1}=b_{n-1}+b_{n-2}$, from $a_{n}=b_{n-2}+b_{n-3}$ we obtain $b_{n}=\left|b_{n-1}-b_{n-3}\right|$. From this relation we deduce that $b_{m} \leq \max \left(b_{n}, b_{n+1}, b_{n+2}\right)$ for all $m \geq n$, and consequently $b_{n}$ ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 105 | 825 |
2001 | T0 | 26 | Number Theory | IMO | 26. N4 (VIE) Let $p \geq 5$ be a prime number. Prove that there exists an integer $a$ with $1 \leq a \leq p-2$ such that neither $a^{p-1}-1$ nor $(a+1)^{p-1}-1$ is divisible by $p^{2}$. | 26. Let $C$ be the set of those $a \in\{1,2, \ldots, p-1\}$ for which $a^{p-1} \equiv 1\left(\bmod p^{2}\right)$. At first, we observe that $a, p-a$ do not both belong to $C$, regardless of the value of $a$. Indeed, by the binomial formula, $$ (p-a)^{p-1}-a^{p-1} \equiv-(p-1) p a^{p-2} \not \equiv 0 \quad\left(\bmod p^... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 81 | 513 |
2001 | T0 | 3 | Algebra | IMO | 3. A3 (ROM) Let $x_{1}, x_{2}, \ldots, x_{n}$ be arbitrary real numbers. Prove the inequality $$ \frac{x_{1}}{1+x_{1}^{2}}+\frac{x_{2}}{1+x_{1}^{2}+x_{2}^{2}}+\cdots+\frac{x_{n}}{1+x_{1}^{2}+\cdots+x_{n}^{2}}<\sqrt{n} $$ | 3. By the arithmetic-quadratic mean inequality, it suffices to prove that $$ \frac{x_{1}^{2}}{\left(1+x_{1}^{2}\right)^{2}}+\frac{x_{2}^{2}}{\left(1+x_{1}^{2}+x_{2}^{2}\right)^{2}}+\cdots+\frac{x_{n}^{2}}{\left(1+x_{1}^{2}+\cdots+x_{n}^{2}\right)^{2}}<1 . $$ Observe that for $k \geq 2$ the following holds: $$ \begin{al... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 108 | 835 |
2001 | T0 | 6 | Algebra | IMO | 6. A6 (KOR) ${ }^{\mathrm{IMO} 2}$ Prove that for all positive real numbers $a, b, c$, $$ \frac{a}{\sqrt{a^{2}+8 b c}}+\frac{a}{\sqrt{b^{2}+8 c a}}+\frac{c}{\sqrt{c^{2}+8 a b}} \geq 1 $$ | 6. We shall determine a constant $k>0$ such that $$ \frac{a}{\sqrt{a^{2}+8 b c}} \geq \frac{a^{k}}{a^{k}+b^{k}+c^{k}} \quad \text { for all } a, b, c>0 $$ This inequality is equivalent to $\left(a^{k}+b^{k}+c^{k}\right)^{2} \geq a^{2 k-2}\left(a^{2}+8 b c\right)$, which further reduces to $$ \left(a^{k}+b^{k}+c^{k}\rig... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 93 | 1,878 |
2002 | T0 | 11 | Geometry | IMO | 11. G5 (AUS) For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$. What is the minimum possible value of $M(S) / m(S)$ ? | 11. When $S$ is the set of vertices of a regular pentagon, then it is easily verified that $\frac{M(S)}{m(S)}=\frac{1+\sqrt{5}}{2}=\alpha$. We claim that this is the best possible. Let $A, B, C, D, E$ be five arbitrary points, and assume that $\triangle A B C$ has the area $M(S)$. We claim that some triangle has area l... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 75 | 581 |
2002 | T0 | 12 | Geometry | IMO | 12. G6 (UKR) ${ }^{\mathrm{IMO} 6}$ Let $n \geq 3$ be a positive integer. Let $C_{1}, C_{2}, C_{3}, \ldots, C_{n}$ be unit circles in the plane, with centers $O_{1}, O_{2}, O_{3}, \ldots, O_{n}$ respectively. If no line meets more than two of the circles, prove that $$ \sum_{1 \leq i<j \leq n} \frac{1}{O_{i} O_{j}} \l... | 12. Let $l(M N)$ denote the length of the shorter $\operatorname{arc} M N$ of a given circle. Lemma. Let $P R, Q S$ be two chords of a circle $k$ of radius $r$ that meet each other at a point $X$, and let $\angle P X Q=\angle R X S=2 \alpha$. Then $l(P Q)+$ $l(R S)=4 \alpha r$. Proof. Let $O$ be the center of the circl... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 146 | 650 |
2002 | T0 | 13 | Geometry | IMO | 13. G7 (BUL) The incircle $\Omega$ of the acute-angled triangle $A B C$ is tangent to $B C$ at $K$. Let $A D$ be an altitude of triangle $A B C$ and let $M$ be the midpoint of $A D$. If $N$ is the other common point of $\Omega$ and $K M$, prove that $\Omega$ and the circumcircle of triangle $B C N$ are tangent at $N$. | 13. Let $k$ be the circle through $B, C$ that is tangent to the circle $\Omega$ at point $N^{\prime}$. We must prove that $K, M, N^{\prime}$ are collinear. Since the statement is trivial for $A B=A C$, we may assume that $A C>A B$. As usual, $R, r, \alpha, \beta, \gamma$ denote the circumradius and the inradius and the... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 105 | 819 |
2002 | T0 | 16 | Algebra | IMO | 16. A2 (YUG) Let $a_{1}, a_{2}, \ldots$ be an infinite sequence of real numbers for which there exists a real number $c$ with $0 \leq a_{i} \leq c$ for all $i$ such that $$ \left|a_{i}-a_{j}\right| \geq \frac{1}{i+j} \quad \text { for all } i, j \text { with } i \neq j $$ Prove that $c \geq 1$. | 16. For $n \geq 2$, let $\left(k_{1}, k_{2}, \ldots, k_{n}\right)$ be the permutation of $\{1,2, \ldots, n\}$ with $a_{k_{1}} \leq a_{k_{2}} \leq \cdots \leq a_{k_{n}}$. Then from the condition of the problem, using the Cauchy-Schwarz inequality, we obtain $$ \begin{aligned} c & \geq a_{k_{n}}-a_{k_{1}}=\left|a_{k_{n}}... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 122 | 659 |
2002 | T0 | 18 | Algebra | IMO | 18. A4 (IND) ${ }^{\text {IMO5 }}$ Find all functions $f$ from the reals to the reals such that $$ (f(x)+f(z))(f(y)+f(t))=f(x y-z t)+f(x t+y z) $$ for all real $x, y, z, t$. | 18. Putting $x=z=0$ and $t=y$ into the given equation gives $4 f(0) f(y)=$ $2 f(0)$ for all $y$. If $f(0) \neq 0$, then we deduce $f(y)=\frac{1}{2}$, i.e., $f$ is identically equal to $\frac{1}{2}$. Now we suppose that $f(0)=0$. Setting $z=t=0$ we obtain $$ f(x y)=f(x) f(y) \quad \text { for all } x, y \in \mathbb{R} $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 76 | 516 |
2002 | T0 | 19 | Algebra | IMO | 19. A5 (IND) Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a, b, c$ by $$ a=\sqrt[3]{n}, \quad b=\frac{1}{a-[a]}, \quad c=\frac{1}{b-[b]}, $$ where $[x]$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers... | 19. Write $m=[\sqrt[3]{n}]$. To simplify the calculation, we shall assume that $[b]=1$. Then $a=\sqrt[3]{n}, b=\frac{1}{\sqrt[3]{n}-m}=\frac{1}{n-m^{3}}\left(m^{2}+m \sqrt[3]{n}+\sqrt[3]{n^{2}}\right), c=\frac{1}{b-1}=$ $u+v \sqrt[3]{n}+w \sqrt[3]{n^{2}}$ for certain rational numbers $u, v, w$. Obviously, integers $r, ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 125 | 568 |
2002 | T0 | 20 | Algebra | IMO | 20. A6 (IRN) Let $A$ be a nonempty set of positive integers. Suppose that there are positive integers $b_{1}, \ldots, b_{n}$ and $c_{1}, \ldots, c_{n}$ such that (i) for each $i$ the set $b_{i} A+c_{i}=\left\{b_{i} a+c_{i} \mid a \in A\right\}$ is a subset of $A$, and (ii) the sets $b_{i} A+c_{i}$ and $b_{j} A+c_{j}$ a... | 20. Assume to the contrary that $\frac{1}{b_{1}}+\cdots+\frac{1}{b_{n}}>1$. Certainly $n \geq 2$ and $A$ is infinite. Define $f_{i}: A \rightarrow A$ as $f_{i}(x)=b_{i} x+c_{i}$ for each $i$. By condition (ii), $f_{i}(x)=f_{j}(y)$ implies $i=j$ and $x=y$; iterating this argument, we deduce that $f_{i_{1}}\left(\ldots f... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 177 | 1,086 |
2002 | T0 | 24 | Combinatorics | IMO | 24. C4 (BUL) Let $T$ be the set of ordered triples $(x, y, z)$, where $x, y, z$ are integers with $0 \leq x, y, z \leq 9$. Players $A$ and $B$ play the following guessing game: Player $A$ chooses a triple $(x, y, z)$ in $T$, and Player $B$ has to discover A's triple in as few moves as possible. A move consists of the f... | 24. Two moves are not sufficient. Indeed, the answer to each move is an even number between 0 and 54 , so the answer takes at most 28 distinct values. Consequently, two moves give at most $28^{2}=784$ distinct outcomes, which is less than $10^{3}=1000$. We now show that three moves are sufficient. With the first move $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 188 | 552 |
2002 | T0 | 27 | Combinatorics | IMO | 27. C7 (NZL) Among a group of 120 people, some pairs are friends. A weak quartet is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets? | 27. For a graph $G$ on 120 vertices (i.e., people at the party), write $q(G)$ for the number of weak quartets in $G$. Our solution will consist of three parts. First, we prove that some graph $G$ with maximal $q(G)$ breaks up as a disjoint union of complete graphs. This will follow if we show that any two adjacent vert... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 52 | 907 |
2002 | T0 | 3 | Number Theory | IMO | 3. N3 (MON) Let $p_{1}, p_{2}, \ldots, p_{n}$ be distinct primes greater than 3 . Show that $2^{p_{1} p_{2} \cdots p_{n}}+1$ has at least $4^{n}$ divisors. | 3. We observe that if $a, b$ are coprime odd numbers, then $\operatorname{gcd}\left(2^{a}+1,2^{b}+1\right)=$ 3. In fact, this g.c.d. divides $\operatorname{gcd}\left(2^{2 a}-1,2^{2 b}-1\right)=2^{\operatorname{gcd}(2 a, 2 b)}-1=2^{2}-1=$ 3 , while 3 obviously divides both $2^{a}+1$ and $2^{b}+1$. In particular, if $3 \... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 67 | 641 |
2002 | T0 | 4 | Number Theory | IMO | 4. N4 (GER) Is there a positive integer $m$ such that the equation $$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a b c}=\frac{m}{a+b+c} $$ has infinitely many solutions in positive integers $a, b, c$ ? | 4. For $a=b=c=1$ we obtain $m=12$. We claim that the given equation has infinitely many solutions in positive integers $a, b, c$ for this value of $m$. After multiplication by $a b c(a+b+c)$ the equation $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a b c}-\frac{12}{a+b+c}=0$ becomes $$ a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b)... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 77 | 697 |
2002 | T0 | 5 | Number Theory | IMO | 5. N5 (IRN) Let $m, n \geq 2$ be positive integers, and let $a_{1}, a_{2}, \ldots, a_{n}$ be integers, none of which is a multiple of $m^{n-1}$. Show that there exist integers $e_{1}, e_{2}, \ldots, e_{n}$, not all zero, with $\left|e_{i}\right|<m$ for all $i$, such that $e_{1} a_{1}+e_{2} a_{2}+\cdots+e_{n} a_{n}$ is ... | 5. Consider all possible sums $c_{1} a_{1}+c_{2} a_{2}+\cdots+c_{n} a_{n}$, where each $c_{i}$ is an integer with $0 \leq c_{i}<m$. There are $m^{n}$ such sums, and if any two of them give the same remainder modulo $m^{n}$, say $\sum c_{i} a_{i} \equiv \sum d_{i} a_{i}\left(\bmod m^{n}\right)$, then $\sum\left(c_{i}-d_... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 147 | 511 |
2002 | T0 | 6 | Number Theory | IMO | 6. N6 (ROM) ${ }^{\mathrm{IMO} 3}$ Find all pairs of positive integers $m, n \geq 3$ for which there exist infinitely many positive integers $a$ such that $$ \frac{a^{m}+a-1}{a^{n}+a^{2}-1} $$ is itself an integer. | 6. Suppose that $(m, n)$ is such a pair. Assume that division of the polynomial $F(x)=x^{m}+x-1$ by $G(x)=x^{n}+x^{2}-1$ gives the quotient $Q(x)$ and remainder $R(x)$. Since $\operatorname{deg} R(x)<\operatorname{deg} G(x)$, for $x$ large enough $|R(x)|<$ $|G(x)|$; however, $R(x)$ is divisible by $G(x)$ for infinitely... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 79 | 661 |
2003 | T0 | 1 | Algebra | IMO | 1. A1 (USA) Let $a_{i j}, i=1,2,3, j=1,2,3$, be real numbers such that $a_{i j}$ is positive for $i=j$ and negative for $i \neq j$. Prove that there exist positive real numbers $c_{1}, c_{2}, c_{3}$ such that the numbers $$ a_{11} c_{1}+a_{12} c_{2}+a_{13} c_{3}, \quad a_{21} c_{1}+a_{22} c_{2}+a_{23} c_{3}, \quad a_{... | 1. Consider the points $O(0,0,0), P\left(a_{11}, a_{21}, a_{31}\right), Q\left(a_{12}, a_{22}, a_{32}\right), R\left(a_{13}, a_{23}\right.$, $\left.a_{33}\right)$ in three-dimensional Euclidean space. It is enough to find a point $U\left(u_{1}, u_{2}, u_{3}\right)$ in the interior of the triangle $P Q R$ whose coordina... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 189 | 666 |
2003 | T0 | 10 | Combinatorics | IMO | 10. C4 (IRN) Let $x_{1}, \ldots, x_{n}$ and $y_{1}, \ldots, y_{n}$ be real numbers. Let $A=$ $\left(a_{i j}\right)_{1 \leq i, j \leq n}$ be the matrix with entries $$ a_{i j}= \begin{cases}1, & \text { if } x_{i}+y_{j} \geq 0 \\ 0, & \text { if } x_{i}+y_{j}<0\end{cases} $$ Suppose that $B$ is an $n \times n$ matrix ... | 10. Denote by $b_{i j}$ the entries of the matrix $B$. Suppose the contrary, i.e., that there is a pair $\left(i_{0}, j_{0}\right)$ such that $a_{i_{0}, j_{0}} \neq b_{i_{0}, j_{0}}$. We may assume without loss of generality that $a_{i_{0}, j_{0}}=0$ and $b_{i_{0}, j_{0}}=1$. Since the sums of elements in the $i_{0}$ t... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 190 | 511 |
2003 | T0 | 11 | Combinatorics | IMO | 11. C5 (ROM) Every point with integer coordinates in the plane is the center of a disk with radius $1 / 1000$. (a) Prove that there exists an equilateral triangle whose vertices lie in different disks. (b) Prove that every equilateral triangle with vertices in different disks has side length greater than 96. | 11. (a) By the pigeonhole principle there are two different integers $x_{1}, x_{2}$, $x_{1}>x_{2}$, such that $\left|\left\{x_{1} \sqrt{3}\right\}-\left\{x_{2} \sqrt{3}\right\}\right|<0.001$. Set $a=x_{1}-x_{2}$. Consider the equilateral triangle with vertices $(0,0),(2 a, 0),(a, a \sqrt{3})$. The points $(0,0)$ and $(... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 76 | 640 |
2003 | T0 | 12 | Combinatorics | IMO | 12. C6 (SAF) Let $f(k)$ be the number of integers $n$ that satisfy the following conditions: (i) $0 \leq n<10^{k}$, so $n$ has exactly $k$ digits (in decimal notation), with leading zeros allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by 11 . Prove that $f(2 m)=10 f(2... | 12. Denote by $\overline{a_{k-1} a_{k-2} \ldots a_{0}}$ the decimal representation of a number whose digits are $a_{k-1}, \ldots, a_{0}$. We will use the following well-known fact: $$ \overline{a_{k-1} a_{k-2} \ldots a_{0}} \equiv i(\bmod 11) \Longleftrightarrow \sum_{l=0}^{k-1}(-1)^{l} a_{l} \equiv i(\bmod 11) . $$ Le... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 117 | 835 |
2003 | T0 | 16 | Geometry | IMO | 16. G4 (ARM) Let $\Gamma_{1}, \Gamma_{2}, \Gamma_{3}, \Gamma_{4}$ be distinct circles such that $\Gamma_{1}, \Gamma_{3}$ are externally tangent at $P$, and $\Gamma_{2}, \Gamma_{4}$ are externally tangent at the same point $P$. Suppose that $\Gamma_{1}$ and $\Gamma_{2} ; \Gamma_{2}$ and $\Gamma_{3} ; \Gamma_{3}$ and $\G... | 16. Apply an inversion with center at $P$ and radius $r$; let $\widehat{X}$ denote the image of $X$. The circles $\Gamma_{1}, \Gamma_{2}, \Gamma_{3}, \Gamma_{4}$ are transformed into lines $\widehat{\Gamma_{1}}, \widehat{\Gamma_{2}}, \widehat{\Gamma_{3}}, \widehat{\Gamma}_{4}$, where $\widehat{\Gamma_{1}} \| \widehat{\... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 182 | 516 |
2003 | T0 | 19 | Geometry | IMO | 19. G7 (SAF) Let $A B C$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $B C, C A, A B$ are drawn outside of the triangle $A B C$. The circle tangent to all three semicircles has radius $t$. Prove that $$ \frac{s}{2}<t \leq \frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right) r $$ | 19. Let $D, E, F$ be the midpoints of $B C, C A, A B$, respectively. We construct smaller semicircles $\Gamma_{d}, \Gamma_{e}, \Gamma_{f}$ inside $\triangle A B C$ with centers $D, E, F$ and radii $d=\frac{s-a}{2}, e=\frac{s-b}{2}, f=\frac{s-c}{2}$ respectively. Since $D E=d+e, D F=d+f$, and $E F=e+f$, we deduce that $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 112 | 1,261 |
2003 | T0 | 2 | Algebra | IMO | 2. A2 (AUS) Find all nondecreasing functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that (i) $f(0)=0, f(1)=1$; (ii) $f(a)+f(b)=f(a) f(b)+f(a+b-a b)$ for all real numbers $a, b$ such that $a<1<b$. | 2. We can rewrite (ii) as $-(f(a)-1)(f(b)-1)=f(-(a-1)(b-1)+1)-1$. So putting $g(x)=f(x+1)-1$, this equation becomes $-g(a-1) g(b-1)=$ $g(-(a-1)(b-1))$ for $a<1<b$. Hence $$ -g(x) g(y)=g(-x y) \text { for } x<0<y $$ and $g$ is nondecreasing with $g(-1)=-1, g(0)=0$. Conversely, if $g$ satisfies (1), than $f$ is a solutio... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 89 | 699 |
2003 | T0 | 21 | Number Theory | IMO | 21. N2 (USA) Each positive integer $a$ undergoes the following procedure in order to obtain the number $d=d(a)$ : (1) move the last digit of $a$ to the first position to obtain the number $b$; (2) square $b$ to obtain the number $c$; (3) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers ... | 21. Let $a$ be a positive integer for which $d(a)=a^{2}$. Suppose that $a$ has $n+1$ digits, $n \geq 0$. Denote by $s$ the last digit of $a$ and by $f$ the first digit of $c$. Then $a=\overline{* \ldots * s}$, where $*$ stands for a digit that is not important to us at the moment. We have $\overline{\ldots * s^{2}}=a^{... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 186 | 752 |
2003 | T0 | 22 | Number Theory | IMO | 22. N3 (BUL) ${ }^{\mathrm{IMO} 2}$ Determine all pairs $(a, b)$ of positive integers such that $$ \frac{a^{2}}{2 a b^{2}-b^{3}+1} $$ is a positive integer. | 22. Let $a$ and $b$ be positive integers for which $\frac{a^{2}}{2 a b^{2}-b^{3}+1}=k$ is a positive integer. Since $k>0$, it follows that $2 a b^{2} \geq b^{3}$, so $2 a \geq b$. If $2 a>b$, then from $2 a b^{2}-b^{3}+1>0$ we see that $a^{2}>b^{2}(2 a-b)+1>b^{2}$, i.e. $a>b$. Therefore, if $a \leq b$, then $a=b / 2$. ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 63 | 547 |
2003 | T0 | 23 | Number Theory | IMO | 23. N4 (ROM) Let $b$ be an integer greater than 5 . For each positive integer $n$, consider the number $$ x_{n}=\underbrace{11 \ldots 1}_{n-1} \underbrace{22 \ldots 2}_{n} 5, $$ written in base $b$. Prove that the following condition holds if and only if $b=10$ : There exists a positive integer $M$ such that for ever... | 23. Assume that $b \geq 6$ has the required property. Consider the sequence $y_{n}=(b-1) x_{n}$. From the definition of $x_{n}$ we easily find that $y_{n}=b^{2 n}+$ $b^{n+1}+3 b-5$. Then $y_{n} y_{n+1}=(b-1)^{2} x_{n} x_{n+1}$ is a perfect square for all $n>M$. Also, straightforward calculation implies $$ \left(b^{2 n+... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 128 | 926 |
2003 | T0 | 24 | Number Theory | IMO | 24. N5 (KOR) An integer $n$ is said to be good if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented in infinitely many ways as a sum of three distinct good integers whose product is the square of an odd integer. | 24. Suppose that $m=u+v+w$ where $u, v, w$ are good integers whose product is a perfect square of an odd integer. Since $u v w$ is an odd perfect square, we have that $u v w \equiv 1(\bmod 4)$. Thus either two or none of the numbers $u, v, w$ are congruent to 3 modulo 4 . In both cases $u+v+w \equiv 3(\bmod 4)$. Hence ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 71 | 512 |
2003 | T0 | 25 | Number Theory | IMO | 25. N6 (FRA) ${ }^{\text {IMO6 }}$ Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^{p}-p$ is not divisible by $q$. | 25. Suppose that for every prime $q$, there exists an $n$ for which $n^{p} \equiv p(\bmod$ $q$ ). Assume that $q=k p+1$. By Fermat's theorem we deduce that $p^{k} \equiv$ $n^{k p}=n^{q-1} \equiv 1(\bmod q)$, so $q \mid p^{k}-1$. It is known that any prime $q$ such that $q \left\lvert\, \frac{p^{p}-1}{p-1}\right.$ must ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 63 | 567 |
2003 | T0 | 26 | Number Theory | IMO | 26. N7 (BRA) The sequence $a_{0}, a_{1}, a_{2}, \ldots$ is defined as follows: $$ a_{0}=2, \quad a_{k+1}=2 a_{k}^{2}-1 \quad \text { for } k \geq 0 $$ Prove that if an odd prime $p$ divides $a_{n}$, then $2^{n+3}$ divides $p^{2}-1$. | 26. Define the sequence $x_{k}$ of positive reals by $a_{k}=\cosh x_{k}$ ( $\cosh$ is the hyperbolic cosine defined by $\left.\cosh t=\frac{e^{t}+e^{-t}}{2}\right)$. Since $\cosh \left(2 x_{k}\right)=2 a_{k}^{2}-1=$ $\cosh x_{k+1}$, it follows that $x_{k+1}=2 x_{k}$ and thus $x_{k}=\lambda \cdot 2^{k}$ for some $\lambd... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 106 | 777 |
2003 | T0 | 27 | Number Theory | IMO | 27. N8 (IRN) Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions: (i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements; (ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$ th power. What is the largest p... | 27. Let $p_{1}, p_{2}, \ldots, p_{r}$ be distinct primes, where $r=p-1$. Consider the sets $B_{i}=\left\{p_{i}, p_{i}^{p+1}, \ldots, p_{i}^{(r-1) p+1}\right\}$ and $B=\bigcup_{i=1}^{r} B_{i}$. Then $B$ has $(p-1)^{2}$ elements and satisfies (i) and (ii). Now suppose that $|A| \geq r^{2}+1$ and that $A$ satisfies (i) an... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 99 | 1,163 |
2003 | T0 | 3 | Algebra | IMO | 3. A3 (GEO) Consider pairs of sequences of positive real numbers $a_{1} \geq$ $a_{2} \geq a_{3} \geq \cdots, b_{1} \geq b_{2} \geq b_{3} \geq \cdots$ and the sums $A_{n}=a_{1}+\cdots+a_{n}$, $B_{n}=b_{1}+\cdots+b_{n}, n=1,2, \ldots$. For any pair define $c_{i}=\min \left\{a_{i}, b_{i}\right\}$ and $C_{n}=c_{1}+\cdots+c... | 3. (a) Given any sequence $c_{n}$ (in particular, such that $C_{n}$ converges), we shall construct $a_{n}$ and $b_{n}$ such that $A_{n}$ and $B_{n}$ diverge. First, choose $n_{1}$ such that $n_{1} c_{1}>1$ and set $a_{1}=a_{2}=\cdots=a_{n_{1}}=$ $c_{1}$ : this uniquely determines $b_{2}=c_{2}, \ldots, b_{n_{1}}=c_{n_{1... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 311 | 610 |
2003 | T0 | 6 | Algebra | IMO | 6. A6 (USA) Let $n$ be a positive integer and let $\left(x_{1}, \ldots, x_{n}\right),\left(y_{1}, \ldots, y_{n}\right)$ be two sequences of positive real numbers. Suppose $\left(z_{2}, z_{3}, \ldots, z_{2 n}\right)$ is a sequence of positive real numbers such that $$ z_{i+j}^{2} \geq x_{i} y_{j} \quad \text { for all ... | 6. Set $X=\max \left\{x_{1}, \ldots, x_{n}\right\}$ and $Y=\max \left\{y_{1}, \ldots, y_{n}\right\}$. By replacing $x_{i}$ by $x_{i}^{\prime}=\frac{x_{i}}{X}, y_{i}$ by $y_{i}^{\prime}=\frac{y_{i}}{Y}$ and $z_{i}$ by $z_{i}^{\prime}=\frac{z_{i}}{\sqrt{X Y}}$, we may assume that $X=Y=1$. It is sufficient to prove that $... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 234 | 543 |
2003 | T0 | 8 | Combinatorics | IMO | 8. C2 (GEO) Let $D_{1}, \ldots, D_{n}$ be closed disks in the plane. (A closed disk is a region bounded by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most 2003 disks $D_{i}$. Prove that there exists disk $D_{k}$ that intersects at most $7 \cdot 2003-1$ other disk... | 8. Let $S$ be the disk with the smallest radius, say $s$, and $O$ the center of that disk. Divide the plane into 7 regions: one bounded by disk $s$ and 6 regions $T_{1}, \ldots, T_{6}$ shown in the figure. Any of the disks different from $S$, say $D_{k}$, has its center in one of the seven regions. If its center is ins... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 106 | 681 |
2004 | T0 | 11 | Combinatorics | IMO | 11. C4 (POL) Consider a matrix of size $n \times n$ whose entries are real numbers of absolute value not exceeding 1 , and the sum of all entries is 0 . Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not excee... | 11. Consider the matrix $A=\left(a_{i j}\right)_{i, j=1}^{n}$ such that $a_{i j}$ is equal to 1 if $i, j \leq n / 2$, -1 if $i, j>n / 2$, and 0 otherwise. This matrix satisfies the conditions from the problem and all row sums and column sums are equal to $\pm n / 2$. Hence $C \geq n / 2$. Let us show that $C=n / 2$. As... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 86 | 514 |
2004 | T0 | 12 | Combinatorics | IMO | 12. C5 (NZL) Let $N$ be a positive integer. Two players $A$ and $B$, taking turns, write numbers from the set $\{1, \ldots, N\}$ on a blackboard. $A$ begins the game by writing 1 on his first move. Then, if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2 n$ (provided the number... | 12. We say that a number $n \in\{1,2, \ldots, N\}$ is winning if the player who is on turn has a winning strategy, and losing otherwise. The game is of type $A$ if and only if 1 is a losing number. Let us define $n_{0}=N, n_{i+1}=\left[n_{i} / 2\right]$ for $i=0,1, \ldots$ and let $k$ be such that $n_{k}=1$. Consider t... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 203 | 712 |
2004 | T0 | 13 | Combinatorics | IMO | 13. C6 (IRN) For an $n \times n$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j, 1 \leq i, j \leq n$. We say that $A$ is golden if $X_{1}, \ldots, X_{n}, Y_{1}, \ldots, Y_{n}$ are distinct sets. Find the least integer $n$ such that there exists a $2004 \times 2004$... | 13. Since $X_{i}, Y_{i}, i=1, \ldots, 2004$, are 4008 distinct subsets of the set $S_{n}=$ $\{1,2, \ldots, n\}$, it follows that $2^{n} \geq 4008$, i.e. $n \geq 12$. Suppose $n=12$. Let $\mathcal{X}=\left\{X_{1}, \ldots, X_{2004}\right\}, \mathcal{Y}=\left\{Y_{1}, \ldots, Y_{2004}\right\}, \mathcal{A}=$ $\mathcal{X} \c... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 149 | 943 |
2004 | T0 | 15 | Combinatorics | IMO | 15. C8 (POL) For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that $$ g(G)^{3} \leq c \cdot f(G)^{4} \text { for every graph } G $$ | 15. Denote by $V_{1}, \ldots, V_{n}$ the vertices of a graph $G$ and by $E$ the set of its edges. For each $i=1, \ldots, n$, let $A_{i}$ be the set of vertices connected to $V_{i}$ by an edge, $G_{i}$ the subgraph of $G$ whose set of vertices is $A_{i}$, and $E_{i}$ the set of edges of $G_{i}$. Also, let $v_{i}, e_{i}$... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 82 | 778 |
2004 | T0 | 20 | Geometry | IMO | 20. G5 (SMN) Let $A_{1} A_{2} \ldots A_{n}$ be a regular $n$-gon. The points $B_{1}, \ldots, B_{n-1}$ are defined as follows: (i) If $i=1$ or $i=n-1$, then $B_{i}$ is the midpoint of the side $A_{i} A_{i+1}$. (ii) If $i \neq 1, i \neq n-1$, and $S$ is the intersection point of $A_{1} A_{i+1}$ and $A_{n} A_{i}$, then $B... | 20. We first establish the following lemma. Lemma. Let $A B C D$ be an isosceles trapezoid with bases $A B$ and $C D$. The diagonals $A C$ and $B D$ intersect at $S$. Let $M$ be the midpoint of $B C$, and let the bisector of the angle $B S C$ intersect $B C$ at $N$. Then $\angle A M D=\angle A N D$. Proof. It suffices ... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 252 | 709 |
2004 | T0 | 21 | Geometry | IMO | 21. G6 (GBR) Let $\mathcal{P}$ be a convex polygon. Prove that there is a convex hexagon that is contained in $\mathcal{P}$ and that occupies at least 75 percent of the area of $\mathcal{P}$. | 21. Let $A B C$ be the triangle of maximum area $S$ contained in $\mathcal{P}$ (it exists because of compactness of $\mathcal{P}$ ). Draw parallels to $B C, C A, A B$ through $A, B, C$, respectively, and denote the triangle thus obtained by $A_{1} B_{1} C_{1}$ ( $A \in B_{1} C_{1}$, etc.). Since each triangle with vert... | {
"problem_match": null,
"resource_path": "IMO/segmented/en-compendium.jsonl",
"solution_match": null
} | 60 | 873 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.