problem stringlengths 182 1.09k | answer stringlengths 1 22 | source stringlengths 21 27 |
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Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_0 = 0$, $x_1 = 1$, and for each integer $k \geq 2$, there exists an integer $1 \leq t \leq k$ such that
\[ x_k = \frac{x_{k-1} + \dots + x_{k-t}}{t}. \]
Find the minimum possible value of $x_{2024} - x_{2025}$.
| $-\frac{2023}{2024^2}$ | imo-bench-algebra-008 |
Suppose that $g:\mathbb{Z}\to O$, where $O$ is the set of odd integers, satisfies
$$g(a + g(a) + b)-g(a+b) = g(a-b)-g(a-g(a)-b)$$
for all integers $a,b$. Furthermore, we have $g(0)=9, g(1)=27, g(2)=3$, and $g(10)=63$. Find all possible values of $g(2025)$. | 8109 | imo-bench-algebra-015 |
Let $a_1, a_2, \ldots, a_{2025}$ be positive integers such that for each positive integer $m$,
$$\left(\left (\sum^{2025}_{j=1} j a^m_j \right)-1\right)^{\frac{1}{m+1}}$$ is an integer. Find all possible value of $a_1+a_2+ \cdots +a_{2025}$.
| 4151879777 | imo-bench-algebra-019 |
We call $g: \mathbb{R} \rightarrow \mathbb{R}$ a good function if $g$ satisfies all the following conditions:
(1) For any two distinct real numbers $a, b$, if $g(ab) = 0$, then $g(a) = 0$ or $g(b) = 0$.
(2) For any two distinct real numbers $a, b$, if $g(ab) \neq 0$, then $$\frac{g(a)-g(b)}{a-b}=\frac{g(a)g(b)}{g(ab)}... | 16 | imo-bench-algebra-032 |
A sequence $f_1, f_2, \ldots, f_{1028}$ of non-negative real numbers is said to be concave if for each $1 < i < 1028$, $f_{i+1} \le 2f_i - f_{i-1}$. Find the minimum value of the constant $L$ such that the inequality
$$L \sum_{j=1}^{1028} i f_{i}^{2} \geq \sum_{i=1}^{1028} f_{i}^{2}$$
holds for all concave sequences... | $\frac{685}{176302}$ | imo-bench-algebra-042 |
Let $r$ be the unique positive root of $x^3+x^2+x-10=0$. Suppose that for non-negative integers $k_0, k_1, k_2, \ldots , k_{100}$, we have $$\sum_{i=0}^{101}k_i r^i=2025.$$ Find the minimum possible value of $\sum_{i=0}^{101}k_i$.
| 44 | imo-bench-algebra-050 |
Let $a_0, a_1, a_2, \ldots$ be an infinite sequence of integers that is not eventually periodic, which means there does not exist integers $N, d > 0$ such that $a_{n + d} = a_n$ for all integers $n \ge N$.
Let $S$ be the set of $2025$-tuples containing $(a_{n + 1}, \ldots, a_{n + 2025})$ for $n = 0, 1, \ldots$. Find t... | 2026 | imo-bench-algebra-054 |
For all positive integers $n$ and all real numbers $a_1, \ldots, a_n$ less than or equal to $2 / \sqrt{n}$ such that $\sum_{i=1}^n a_i^3 = 0$, find the maximum value of $\sum_{k=1}^n a_i^2$. | $\frac{16}{3}$ | imo-bench-algebra-058 |
Find the largest positive integer $n$ that satisfies the following condition:
There exist integers $t_1, \ldots, t_n, s_1, \ldots, s_n$ between 1 and 1000 (inclusive) such that for any non-negative real numbers $x_1, \ldots, x_{1000}$ satisfying $x_1 + \cdots + x_{1000} = 2014$, the following inequality holds:
\[
... | 496503 | imo-bench-algebra-085 |
Let $\{a_n\}_{n \ge 1}$ be the sequence of integers satisfying $a_1 = 0$ and
\[
a_n = \max_{1 \le i \le n - 1} \left\{a_i + a_{n - i} + \min(i, n - i) \right\}
\]
for all $n \ge 2$. Determine $a_{2025}$.
| 11059 | imo-bench-algebra-095 |
There are 345 students participating in the National Olympiad. The organizer needs to divide the students into $n$ classes so that each student is in one class. Each student knows exactly 9 other students. (If student A knows student B, then student B does not necessarily know student A.) Find the smallest $n$ such tha... | 19 | imo-bench-combinatorics-007 |
Two players, Boris and Natasha, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with Boris. On Boris's turn, Boris selects one white unit square and colors it blue. On Natasha's turn, Natasha selects two white unit squares and colors them red. Th... | 4 | imo-bench-combinatorics-010 |
Suppose $X$ is a set with $|X| = 56$. In a Chinese mathematics competition, students are given 15 subsets of $X$. Find the minimum value of $n$, so that if the cardinality of the union of any 7 of these subsets is greater or equal to $n$, then there exists 3 of them whose intersection is nonempty. | 41 | imo-bench-combinatorics-016 |
Two rational numbers \(\tfrac{m}{n}\) and \(\tfrac{n}{m}\) are written on a blackboard, where \(m\) and \(n\) are relatively prime positive integers. At any point, Lin may pick two of the numbers \(x\) and \(y\) written on the board and write either their arithmetic mean \(\tfrac{x+y}{2}\) or their harmonic mean \(\tf... | 2048 | imo-bench-combinatorics-027 |
Several positive integers are written on a chalkboard in a row from left to right. A transformation is performed as follows: For every pair of consecutive integers on the chalkboard, the sum of those two numbers is written between them. After 2013 such transformations, how many times does the number 2013 appear on the... | 1198 | imo-bench-combinatorics-028 |
A conference hall is used for hosting events, and each event is assigned a specific time slot (represented as an interval that is a subset of $[0,1]$) within the full schedule from $0$ to $1$ (representing a day on a normalized scale). The hall manager designs a schedule (which is a set of intervals representing time s... | 27648 | imo-bench-combinatorics-051 |
In the mystical land of Azeroth, an ancient ritual requires the Sky Guardians to form sacred constellations using $n$ magical stars scattered randomly in the sky. These stars are carefully positioned such that no three of them lie in a straight line.
Each star is imbued with either red or blue energy, assigned randoml... | 8 | imo-bench-combinatorics-053 |
Consider a checkerboard consisting of $38$ by $38$ unit squares. At the midpoints of some of these unit squares, there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions meet, they both turn $90^{\circ}$ clockwise and continue... | 56 | imo-bench-combinatorics-054 |
Let $t_0,t_1,\ldots ,t_{86}$ be integers satisfying $0=t_0<t_1<\ldots <t_{86}=171$. Find the smallest possible number of elements in the set $\{ t_i+t_j \mid 0\le i \le j \le 86 \}$. | 258 | imo-bench-combinatorics-060 |
A sequence of positive integers $a_1, a_2, \ldots, a_n$ is called a Korean sequence if it satisfies the following conditions:
\begin{itemize}
\item $a_1 < a_2 < \ldots < a_n$
\item For each $1 \leq k < n$, we define two sets $A_k = \{a_1, a_2, \ldots, a_k\}$ and $B_k = \{a_{k+1}, a_{k+2}, \ldots, a_n\}$. A part... | 3024 | imo-bench-combinatorics-061 |
Let $h$ be the smallest height of a triangular pyramid, and let $d$ be the shortest distance between its opposite edges. Let $S$ be the set of values of $t$ such that the inequality $d > th$ is possible. Find the supremum of $S$.
| $\frac{3}{2}$ | imo-bench-geometry-001 |
Let XYZ be a fixed triangle in the plane. Let P be an arbitrary point in the plane. The circle with center P, passing through X, meets XY and XZ again at points X_y and X_z respectively. Points Y_x, Y_z, Z_x and Z_y are defined similarly. A point P is called special if the points X_y, X_z, Y_x, Y_z, Z_x, and Z_y are co... | 4 | imo-bench-geometry-012 |
Let $l$ and $m$ be parallel lines with $100$ distinct points marked on $l$ and $100$ distinct points marked on $m$. Find the greatest possible number of acute-angled triangles all of whose vertices are marked.
| 333300 | imo-bench-geometry-020 |
A triangle $OPQ$ with $\angle P=90^{\circ}$ lies inside another triangle with vertex $O$. The altitude of $OPQ$ from $P$ until it meets the side of angle $O$ at $N$. The distances from $N$ and $Q$ to the second side of angle $O$ are $2$ and $1$ respectively. Find the length of $OP$.
| $\sqrt{2}$ | imo-bench-geometry-022 |
Let $PQRS$ be a convex quadrilateral. The circumcenter and the incenter of triangle $PQR$ coincide with the incenter and the circumcenter of triangle $PRS$ respectively. It is known that $PQ = 1$. What is the value of angle P in degrees?
| 72 | imo-bench-geometry-024 |
In $\triangle XYZ$ with $XY=XZ$, point $P$ lies strictly between $X$ and $Z$ on side $\overline{XZ}$, and point $Q$ lies strictly between $X$ and $Y$ on side $\overline{XY}$ such that $XQ=QP=PY=YZ$. The degree measure of $\angle XYZ$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$... | 547 | imo-bench-geometry-053 |
Let $EFGH$ be a parallelogram with $\angle FEH < 90^{\circ}$. A circle tangent to sides $\overline{HE}$, $\overline{EF}$, and $\overline{FG}$ intersects diagonal $\overline{EG}$ at points $X$ and $Y$ with $EX < EY$. Suppose that $EX = 3$, $XY = 9$, and $YG = 16$. Then the area of $EFGH$ can be expressed in the form $a\... | 150 | imo-bench-geometry-060 |
Let $WXYZ$ be a tetrahedron such that $WX = YZ = \sqrt{41}$, $WY = XZ = \sqrt{80}$, and $WZ = XY = \sqrt{89}$. There exists a point $P$ inside the tetrahedron such that the distances from $P$ to each of the faces of the tetrahedron are all equal. This distance can be written in the form $\frac{a \sqrt{b}}{c}$, when $a$... | 104 | imo-bench-geometry-070 |
In the triangle $DEF$, $\angle{F}=90^{\circ},\angle {D}=30^{\circ}$ and $EF=1$. Find the minimum value of the longest side of all inscribed triangles (i.e. triangles with vertices on each of three sides) of the triangle $DEF$.
| $\sqrt{\frac{3}{7}}$ | imo-bench-geometry-075 |
Consider a regular $2024$-sided polygon. We define a line as suitable if it intersects the interior of this polygon. Draw $n$ distinct suitable lines, which divide the polygon into several smaller polygons. Determine the minimum value of $n$ such that at least one of the smaller polygons has either $3$ or $4$ sides.
| 2020 | imo-bench-geometry-085 |
Find the smallest positive integer $n$ for which there exist integer-coefficient polynomials $P_1, \dots, P_n$ such that $\frac{{P_1(x)}^2 + \cdots + {P_n(x)}^2}{x^2 + 7}$ is a perfect square of an positive integer.
| 5 | imo-bench-number_theory-025 |
We call a triple $(x, y, z)$ of (not necessarily positive) integers a \textit{good triple} if $y^x z^y x^z$ is a positive prime number. We enumerate good triples as $(x_1, y_1, z_1), (x_2, y_2, z_2), \dots$ in a way that $x_k^2 + y_k^2 + z_k^2 \le x_{k+1}^2 + y_{k+1}^2 + z_{k+1}^2$ holds for all positive integer $k$. E... | 18771 | imo-bench-number_theory-070 |
Given a positive integer n, perform the following operation:
(i) Remove the last digit of n.
(ii) Add 3 times the removed digit to the remaining number.
For example, if $n = 1013$, the operation yields $101 + 9 = 110$. If $n = 2$, the operation yields $0 + 6 = 6$.
Starting with $260^{135}$, repeatedly apply this op... | 8 | imo-bench-number_theory-077 |
Find the number of integer-coefficient polynomials $P$ that satisfy all of the following conditions:
(1) $\deg P \le 3$, and all coefficients of $P$ are between $0$ and $2012$, inclusive.
(2) $P(x) + P(-x)$ is a multiple of $2013$ for all positive integers $x$ less than or equal to $100$.
(3) The remainders when $P(0)... | 7200 | imo-bench-number_theory-082 |
For a positive integer $n$, find the sum of all $n$ such that the number of ordered pairs of positive integers $(a,b)$ where the greatest common divisor of $a$ and $b$ is $2023$ and the least common multiple of $a$ and $b$ is $n!$ does not exceed 4000.
| 820 | imo-bench-number_theory-088 |
Find all positive integers $m > 1$ such that there exists a permutation $(a_1, a_2, \dots, a_n)$ of the all divisors of $m$ for which $a_1 + \dots + a_i$ is a perfect square for all $1 \leq i \leq n$.
| 3 | imo-bench-number_theory-001 |
Find the number of all positive integers $2\leq m \leq 2000$ that satisfy the following condition: For any $m$ distinct positive integers $(n_1, \ldots, n_m)$, at least one of the following two conditions holds: $n_1 + \ldots + n_m$ is a multiple of $m$, or there exists a permutation $(k_1, \ldots, k_m)$ such that $k_1... | 1009 | imo-bench-number_theory-002 |
Does there exist a positive integer $n$ satisfying the following condition? If so, find the smallest such $n$.
(Condition) There exist infinitely many ordered $n$-tuples of positive rational numbers $(x_1, \dots, x_n)$ such that both $\sum_{i=1}^n i \cdot x_i$ and $\frac{1}{x_1 + \dots + x_n} + \frac{1}{x_2 + \dots + ... | 3 | imo-bench-number_theory-008 |
$\alpha$ is a real number that is not rational, and $k$ is a positive integer. A pair of positive integers $(m, n)$ is called "pretty" if $m\lceil n\alpha \rceil - n \lfloor m\alpha \rfloor =k$. A pretty pair $(m, n)$ is called "cool" if neither $(m-n, n)$ nor $(m, n-m)$ are pretty pairs. Find the number of cool pairs ... | 6912 | imo-bench-number_theory-017 |
Find the smallest positive integer $m$ that satisfies the following condition:
(Condition) There exist $m$ distinct positive integers $a_1, \dots, a_m$ such that $(1-\frac{1}{a_1})\cdots (1-\frac{1}{a_m}) = \frac{42}{2010}$.
| 48 | imo-bench-number_theory-023 |
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