id stringlengths 16 16 | problem stringlengths 0 13.6k | answer stringlengths 0 1.87k | solution stringlengths 0 14.4k | source stringlengths 3 73 | origin stringclasses 2
values | domain stringclasses 683
values | category stringclasses 9
values | difficulty float64 -1 10 ⌀ | extra stringlengths 124 397 |
|---|---|---|---|---|---|---|---|---|---|
8b389ea01798202c | Let \( x_{1}, x_{2}, \ldots, x_{n} \) be positive real numbers, and let
\[
S = x_{1} + x_{2} + \cdots + x_{n}
\]
Prove that
\[
\left(1 + x_{1}\right)\left(1 + x_{2}\right) \cdots \left(1 + x_{n}\right) \leq 1 + S + \frac{S^{2}}{2!} + \frac{S^{3}}{3!} + \cdots + \frac{S^{n}}{n!}
\] | proof | Let $\sigma_{k}$ be the $k$ th symmetric polynomial, namely
$$
\sigma_{k}=\sum_{\substack{|S|=k \\ S \subseteq\{1,2, \ldots, n\}}} \prod_{i \in S} x_{i},
$$
and more explicitly
$$
\sigma_{1}=S, \quad \sigma_{2}=x_{1} x_{2}+x_{1} x_{3}+\cdots+x_{n-1} x_{n}, \quad \text { and so on. }
$$
Then
$$
\left(1+x_{1}\right)... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
a52ed83b49137467 | Prove that the equation
$$
6\left(6 a^{2}+3 b^{2}+c^{2}\right)=5 n^{2}
$$
has no solutions in integers except \(a=b=c=n=0\). | proof | We can suppose without loss of generality that \(a, b, c, n \geq 0\). Let \((a, b, c, n)\) be a solution with minimum sum \(a+b+c+n\). Suppose, for the sake of contradiction, that \(a+b+c+n > 0\). Since 6 divides \(5 n^{2}\), \(n\) is a multiple of 6. Let \(n = 6 n_{0}\). Then the equation reduces to
\[
6 a^{2} + 3 b^... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
3de39bfcc01f271b | Let $S$ be a set consisting of $m$ pairs $(a, b)$ of positive integers with the property that $1 \leq a < b \leq n$. Show that there are at least
$$
4 m \frac{\left(m-\frac{n^{2}}{4}\right)}{3 n}
$$
triples $(a, b, c)$ such that $(a, b),(a, c)$, and $(b, c)$ belong to $S$. | 4 m \frac{\left(m-\frac{n^{2}}{4}\right)}{3 n} | Call a triple $(a, b, c)$ good if and only if $(a, b),(a, c)$, and $(b, c)$ all belong to $S$. For $i$ in $\{1,2, \ldots, n\}$, let $d_{i}$ be the number of pairs in $S$ that contain $i$, and let $D_{i}$ be the set of numbers paired with $i$ in $S$ (so $\left|D_{i}\right|=d_{i}$ ). Consider a pair $(i, j) \in S$. Our g... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
f1e0e05630a99c61 | In $\triangle A B C$, let $D, E, F$ be the midpoints of $B C, A C, A B$ respectively and let $G$ be the centroid of the triangle.
For each value of $\angle B A C$, how many non-similar triangles are there in which $A E G F$ is a cyclic quadrilateral? | not found | Let $I$ be the intersection of $A G$ and $E F$.
Let $\delta=A I \cdot I G - F I \cdot I E$. Then
$$
A I = A D / 2, \quad I G = A D / 6, \quad F I = B C / 4 = I E
$$
Further, applying the cosine rule to triangles $A B D, A C D$ we get
$$
\begin{aligned}
A B^{2} & = B C^{2} / 4 + A D^{2} - A D \cdot B C \cdot \cos \an... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
8de0941639dba9b6 | Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers, and let $S_{k}$ be the sum of products of $a_{1}, a_{2}, \ldots, a_{n}$ taken $k$ at a time.
Show that
$$
S_{k} S_{n-k} \geq\binom{ n}{k}^{2} a_{1} a_{2} \ldots a_{n}, \quad \text { for } \quad k=1,2, \ldots, n-1
$$ | proof | $$
\binom{n}{k} a_{1} a_{2} \ldots a_{n}
$$
$$2=\sum_{1 \leq i_{1}<i_{2}<\ldots<i_{k} \leq n} a_{i_{1}} a_{i_{2}} \ldots a_{i_{k}} \cdot \frac{a_{1} a_{2} \ldots a_{n}}{a_{i_{1}} a_{i_{2}} \ldots a_{i_{k}}}$$
(and using the Cauchy-Schwarz inequality)
$$
\begin{aligned}
& \leq\left(\sum_{1 \leq i_{1}<i_{2}<\ldots<i_{k... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
a454ee370ce278a9 | Consider all the triangles $A B C$ which have a fixed base $A B$ and whose altitude from $C$ is a constant $h$. For which of these triangles is the product of its altitudes a maximum? | not found | Let $h_{a}$ and $h_{b}$ be the altitudes from $A$ and $B$, respectively. Then
$$
\begin{aligned}
A B \cdot h \cdot A C \cdot h_{b} \cdot B C \cdot h_{a} & =8 \cdot (\text{area of } \triangle A B C)^{3} \\
& =(A B \cdot h)^{3},
\end{aligned}
$$
which is a constant. So the product $h \cdot h_{a} \cdot h_{b}$ attains it... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
35b7403e1c1d708f | A set of 1990 persons is divided into non-intersecting subsets in such a way that
(a) no one in a subset knows all the others in the subset;
(b) among any three persons in a subset, there are always at least two who do not know each other; and
(c) for any two persons in a subset who do not know each other, there is exa... | 398 | (i) Let \( S \) be a subset of persons satisfying conditions (a), (b), and (c). Let \( x \in S \) be one who knows the maximum number of persons in \( S \).
Assume that \( x \) knows \( x_{1}, x_{2}, \ldots, x_{n} \). By (b), \( x_{i} \) and \( x_{j} \) are strangers if \( i \neq j \). For each \( x_{i} \), let \( N_{i... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
22fe199da10da564 | Show that for every integer $n \geq 6$, there exists a convex hexagon which can be dissected into exactly $n$ congruent triangles. | proof | (provided by the Canadian Problems Committee).
The basic building blocks will be right-angled triangles with sides \( p, q \) (which are positive integers) adjacent to the right angle.
In the... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
e6c4307bef309436 | Let $G$ be the centroid of triangle $A B C$ and $M$ be the midpoint of $B C$. Let $X$ be on $A B$ and $Y$ on $A C$ such that the points $X, Y$, and $G$ are collinear and $X Y$ and $B C$ are parallel. Suppose that $X C$ and $G B$ intersect at $Q$ and $Y B$ and $G C$ intersect at $P$. Show that triangle $M P Q$ is simila... | proof | Let $R$ be the midpoint of $A C$; so $B R$ is a median and contains the centroid $G$.
It is well known that $\frac{A G}{A M}=\frac{2}{3}$; thus the ratio of the similarity between $A X Y$ and ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
3c25bf9bbedeed10 | Suppose there are 997 points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least 1991 red points in the plane. Can you find a special case with exactly 1991 red points? | 1991 | Embed the points in the Cartesian plane such that no two points have the same $y$-coordinate. Let $P_{1}, P_{2}, \ldots, P_{997}$ be the points and $y_{1}<y_{2}<\ldots<y_{997}$ be their respective $y$-coordinates. Then the $y$-coordinate of the midpoint of $P_{i} P_{i+1}, i=1,2, \ldots, 996$ is $\frac{y_{i}+y_{i+1}}{2}... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
6b01cc22fc488884 | Let $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots+a_{n}=b_{1}+b_{2}+$ $\cdots+b_{n}$. Show that
$$
\frac{a_{1}^{2}}{a_{1}+b_{1}}+\frac{a_{2}^{2}}{a_{2}+b_{2}}+\cdots+\frac{a_{n}^{2}}{a_{n}+b_{n}} \geq \frac{a_{1}+a_{2}+\cdots+a_{n}}{2}
$$ | proof | By the Cauchy-Schwartz inequality,
$$
\left(\frac{a_{1}^{2}}{a_{1}+b_{1}}+\frac{a_{2}^{2}}{a_{2}+b_{2}}+\cdots+\frac{a_{n}^{2}}{a_{n}+b_{n}}\right)\left(\left(a_{1}+b_{1}\right)+\left(a_{2}+b_{2}\right)+\cdots+\left(a_{n}+b_{n}\right)\right) \geq\left(a_{1}+a_{2}+\cdots+a_{n}\right)^{2} .
$$
Since $\left(\left(a_{1}+... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
53a69f6ca6fcb5c3 | A triangle with sides $a, b$, and $c$ is given. Denote by $s$ the semiperimeter, that is $s=(a+b+c) / 2$. Construct a triangle with sides $s-a, s-b$, and $s-c$. This process is repeated until a triangle can no longer be constructed with the sidelengths given.
For which original triangles can this process be repeated in... | a = b = c | The perimeter of each new triangle constructed by the process is $(s-a)+(s-b)+(s-c) = 3s - (a+b+c) = 3s - 2s = s$, that is, it is halved. Consider a new equivalent process in which a similar triangle with side lengths $2(s-a), 2(s-b), 2(s-c)$ is constructed, so the perimeter is kept invariant.
Suppose without loss of g... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
590c3cbdb22b71a7 | In a circle \( C \) with centre \( O \) and radius \( r \), let \( C_{1}, C_{2} \) be two circles with centres \( O_{1}, O_{2} \) and radii \( r_{1}, r_{2} \) respectively, so that each circle \( C_{i} \) is internally tangent to \( C \) at \( A_{i} \) and so that \( C_{1}, C_{2} \) are externally tangent to each other... | proof | Because of the tangencies, the following triples of points (two centers and a tangency point) are collinear:
$$
O_{1} ; O_{2} ; A, \quad O ; O_{1} ; A_{1}, \quad O ; O_{2} ; A_{2}
$$
Because of that we can ignore the circles and only draw their centers and tangency points.
![](https://cdn.mathpix.com/cropped/2024_11_... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
132f6aad383d6209 | Let $n$ be an integer such that $n>3$. Suppose that we choose three numbers from the set $\{1,2, \ldots, n\}$. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all possible combinations.
(a) Show that if we choose all three numbers greater than $n / 2$, then t... | proof | In both items, the smallest chosen number is at least 2: in part (a), \( n / 2 > 1 \) and in part (b), \( p \) is a prime. So let \( 11, (m-1)(n-1) \geq 1 \cdot 2 \Longrightarrow m n > m + n, t n + m - (t m + n) = (t-1)(n-m) > 0 \Longrightarrow t n + m > t m + n \), and \((t + m) n - (t + n) m = t(n - m) > 0\),
\[
x +... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
b4efad976b3d6d7e | Let $A B C D$ be a quadrilateral such that all sides have equal length and angle $\angle A B C$ is 60 degrees. Let $\ell$ be a line passing through $D$ and not intersecting the quadrilateral (except at $D)$. Let $E$ and $F$ be the points of intersection of $\ell$ with $A B$ and $B C$ respectively. Let $M$ be the point ... | proof | 
Triangles $A E D$ and $C D F$ are similar, because $A D \| C F$ and $A E \| C D$. Thus, since $A B C$ and $A C D$ are equilateral triangles,
$$
\frac{A E}{C D}=\frac{A D}{C F} \Longleftrighta... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
1c3a2f7a1bc5e678 | Find the total number of different integer values the function
$$
f(x)=[x]+[2 x]+\left[\frac{5 x}{3}\right]+[3 x]+[4 x]
$$
takes for real numbers \( x \) with \( 0 \leq x \leq 100 \).
Note: \([t]\) is the largest integer that does not exceed \( t \).
Answer: 734. | 734 | Note that, since $[x+n]=[x]+n$ for any integer $n$,
$$
f(x+3)=[x+3]+[2(x+3)]+\left[\frac{5(x+3)}{3}\right]+[3(x+3)]+[4(x+3)]=f(x)+35,
$$
one only needs to investigate the interval $[0,3)$.
The numbers in this interval at which at least one of the real numbers $x, 2 x, \frac{5 x}{3}, 3 x, 4 x$ is an integer are
- $0,... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
180a0e043d27a998 | Let
$$
f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \quad \text { and } \quad g(x)=c_{n+1} x^{n+1}+c_{n} x^{n}+\cdots+c_{0}
$$
be non-zero polynomials with real coefficients such that $g(x)=(x+r) f(x)$ for some real number $r$. If $a=\max \left(\left|a_{n}\right|, \ldots,\left|a_{0}\right|\right)$ and $c=\max \left(... | proof | Expanding $(x+r) f(x)$, we find that $c_{n+1}=a_{n}, c_{k}=a_{k-1}+r a_{k}$ for $k=1,2, \ldots, n$, and $c_{0}=r a_{0}$. Consider three cases:
- $r=0$. Then $c_{0}=0$ and $c_{k}=a_{k-1}$ for $k=1,2, \ldots, n$, and $a=c \Longrightarrow \frac{a}{c}=1 \leq n+1$.
- $|r| \geq 1$. Then
$$
\begin{gathered}
\left|a_{0}\righ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Algebra", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
0585a97954f1859b | Determine all positive integers $n$ for which the equation
$$
x^{n}+(2+x)^{n}+(2-x)^{n}=0
$$
has an integer as a solution.
## Answer: $n=1$. | n=1 | If $n$ is even, $x^{n}+(2+x)^{n}+(2-x)^{n}>0$, so $n$ is odd.
For $n=1$, the equation reduces to $x+(2+x)+(2-x)=0$, which has the unique solution $x=-4$.
For $n>1$, notice that $x$ is even, because $x, 2-x$, and $2+x$ have all the same parity. Let $x=2 y$, so the equation reduces to
$$
y^{n}+(1+y)^{n}+(1-y)^{n}=0 .
$$... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Algebra", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
668f1af7d00e9461 | Let $P_{1}, P_{2}, \ldots, P_{1993}=P_{0}$ be distinct points in the $x y$-plane with the following properties:
(i) both coordinates of $P_{i}$ are integers, for $i=1,2, \ldots, 1993$;
(ii) there is no point other than $P_{i}$ and $P_{i+1}$ on the line segment joining $P_{i}$ with $P_{i+1}$ whose coordinates are both i... | proof | Call a point $(x, y) \in \mathbb{Z}^{2}$ even or odd according to the parity of $x+y$. Since there are an odd number of points, there are two points $P_{i}=(a, b)$ and $P_{i+1}=(c, d), 0 \leq i \leq 1992$ with the same parity. This implies that $a+b+c+d$ is even. We claim that the midpoint of $P_{i} P_{i+1}$ is the des... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
60e9936cc159c4b3 | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that
(i) For all $x, y \in \mathbb{R}$,
$$
f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y)
$$
(ii) For all $x \in[0,1), f(0) \geq f(x)$,
(iii) $-f(-1)=f(1)=1$.
Find all such functions $f$.
Answer: $f(x)=\lfloor x\rfloor$, the largest integer that does not exceed $... | f(x)=\lfloor x \rfloor | Plug $y \rightarrow 1$ in (i):
$$
f(x)+f(1)+1 \geq f(x+1) \geq f(x)+f(1) \Longleftrightarrow f(x)+1 \leq f(x+1) \leq f(x)+2
$$
Now plug $y \rightarrow -1$ and $x \rightarrow x+1$ in (i):
$$
f(x+1)+f(-1)+1 \geq f(x) \geq f(x+1)+f(-1) \Longleftrightarrow f(x) \leq f(x+1) \leq f(x)+1
$$
Hence $f(x+1)=f(x)+1$ and we on... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Algebra", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
a647b0ac52c04610 | Given a nondegenerate triangle $A B C$, with circumcentre $O$, orthocentre $H$, and circumradius $R$, prove that $|O H|<3 R$. | O H<3 R | Embed $A B C$ in the complex plane, with $A, B$ and $C$ on the circle $|z|=R$, so $O$ is the origin. Represent each point by its lowercase letter. It is well known that $h=a+b+c$, so
$$
O H=|a+b+c| \leq|a|+|b|+|c|=3 R .
$$
The equality cannot occur because $a, b$, and $c$ are not collinear, so $O H<3 R$. | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
7d31ed00934f5a11 | Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?
Answer: Yes. | proof | The answer is yes and we present the following construction: the idea is considering points in the unit circle of the form \( P_{n} = (\cos (2 n \theta), \sin (2 n \theta)) \) for an appropriate \(\theta\). Then the distance \( P_{m} P_{n} \) is the length of the chord with central angle \((2 m - 2 n) \theta \bmod \pi\... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
78b9bd7dfab7d6eb | You are given three lists $A, B$, and $C$. List $A$ contains the numbers of the form $10^{k}$ in base 10, with $k$ any integer greater than or equal to 1. Lists $B$ and $C$ contain the same numbers translated into base 2 and 5 respectively:
| $A$ | $B$ | $C$ |
| :--- | :--- | :--- |
| 10 | 1010 | 20 |
| 100 | 1100100 ... | proof | Let $b_{k}$ and $c_{k}$ be the number of digits in the $k$ th term in lists $B$ and $C$, respectively. Then
$$
2^{b_{k}-1} \leq 10^{k}<2^{b_{k}} \Longleftrightarrow \log _{2} 10^{k}<b_{k} \leq \log _{2} 10^{k}+1 \Longleftrightarrow b_{k}=\left\lfloor k \cdot \log _{2} 10\right\rfloor+1
$$
and, similarly
$$
c_{k}=\le... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
58f505c304305b93 | Find the smallest positive integer $n$ with the following property: There does not exist an arithmetic progression of 1993 terms of real numbers containing exactly $n$ integers. | 70 | and Marking Scheme:
We first note that the integer terms of any arithmetic progression are "equally spaced", because if the $i$ th term $a_{i}$ and the $(i+j)$ th term $a_{i+j}$ of an arithmetic progression are both integers, then so is the $(i+2 j)$ th term $a_{i+2 j}=a_{i+j}+\left(a_{i+j}-a_{i}\right)$.
1 POINT for ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
07321f93db0e2de7 | Let $a_{1}, a_{2}, \cdots$ be a sequence of real numbers satisfying $a_{i+j} \leq a_{i}+a_{j}$ for all $i, j=1,2, \cdots$. Prove that
$$
a_{1}+\frac{a_{2}}{2}+\frac{a_{3}}{3}+\cdots+\frac{a_{n}}{n} \geq a_{n}
$$
for each positive integer $n$. | proof | and Marking Scheme:
Letting $b_{i}=a_{i} / i,(i=1,2, \cdots)$, we prove that
$$
b_{1}+\cdots+b_{n} \geq a_{n} \quad(n=1,2, \cdots)
$$
by induction on $n$. For $n=1, b_{1}=a_{1} \geq a_{1}$, and the induction starts. Assume that
$$
b_{1}+\cdots+b_{k} \geq a_{k}
$$
for all $k=1,2, \cdots, n-1$. It suffices to prove ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
8219f78c15b94bda | Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles intersecting at $P$ and $Q$. The common tangent, closer to $P$, of $\Gamma_{1}$ and $\Gamma_{2}$ touches $\Gamma_{1}$ at $A$ and $\Gamma_{2}$ at $B$. The tangent of $\Gamma_{1}$ at $P$ meets $\Gamma_{2}$ at $C$, which is different from $P$ and the extension of $A P$ meet... | proof | and Marking Scheme:
Let $\alpha=\angle P A B, \beta=\angle A B P$ and $\gamma=\angle Q A P$. Then, since $P C$ is tangent to $\Gamma_{1}$, we have $\angle Q P C=$ $\angle Q B C=\gamma$. Thus $A, B, R, Q$ are concyclic.
3 POINTS for proving that $A, B, R, Q$ are concyclic.
Since $A B$ is a common tangent to $\Gamma_{1... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
3c0c03a76fbad2b0 | Let $S$ be a set of $2 n+1$ points in the plane such that no three are collinear and no four concyclic. A circle will be called good if it has 3 points of $S$ on its circumference, $n-1$ points in its interior and $n-1$ in its exterior. Prove that the number of good circles has the same parity as $n$. | proof | and Marking Scheme:
Lemma 1. Let $P$ and $Q$ be two points of $S$. The number of good circles that contain $P$ and $Q$ on their circumference is odd.
## Proof of Lemma 1.
Let $N$ be the nu... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
02875169dbe8f29f | Compute the sum $S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{1-3 x_{i}+3 x_{i}^{2}}$ for $x_{i}=\frac{i}{101}$.
Answer: $S=51$. | 51 | Since $x_{101-i}=\frac{101-i}{101}=1-\frac{i}{101}=1-x_{i}$ and
$$
1-3 x_{i}+3 x_{i}^{2}=\left(1-3 x_{i}+3 x_{i}^{2}-x_{i}^{3}\right)+x_{i}^{3}=\left(1-x_{i}\right)^{3}+x_{i}^{3}=x_{101-i}^{3}+x_{i}^{3},
$$
we have, by replacing $i$ by $101-i$ in the second sum,
$$
2 S=S+S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{x_{101-i}... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Algebra", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
8dd0e75df38340b7 | Given the following arrangement of circles:
Each of the numbers \(1, 2, \ldots, 9\) is to be written into one of these circles, so that each circle contains exactly one of these numbers and
(i... | 48 | Let $a, b$, and $c$ be the numbers in the vertices of the triangular arrangement. Let $s$ be the sum of the numbers on each side and $t$ be the sum of the squares of the numbers on each side. Summing the numbers (or their squares) on the three sides repeats each once the numbers on the vertices (or their squares):
$$
... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
13f0524faf9b8eea | Let \( n, k \) be given positive integers with \( n > k \). Prove that
\[
\frac{1}{n+1} \cdot \frac{n^{n}}{k^{k}(n-k)^{n-k}} < \frac{n!}{k!(n-k)!} < \frac{n^{n}}{k^{k}(n-k)^{n-k}} .
\] | proof | The inequality is equivalent to
$$
\frac{n^{n}}{n+1}1 \Longleftrightarrow(n-i+1) k>i(n-k) \Longleftrightarrow iT_{k+2}>\cdots>T_{n+1}
$$
that is, $T_{k+1}=\binom{n}{k} k^{k}(n-k)^{n-k}$ is the largest term in the expansion. The maximum term is greater than the average, which is the sum $n^{n}$ divided by the quantity... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
4e7edeaed357c97c | Given a permutation $\left(a_{0}, a_{1}, \ldots, a_{n}\right)$ of the sequence $0,1, \ldots, n$. A transposition of $a_{i}$ with $a_{j}$ is called legal if $a_{i}=0$ for $i>0$, and $a_{i-1}+1=a_{j}$. The permutation $\left(a_{0}, a_{1}, \ldots, a_{n}\right)$ is called regular if after a number of legal transpositions i... | n=2 \text{ and } n=2^{k}-1, k \text{ positive integer} | A legal transposition consists of looking at the number immediately before 0 and exchanging 0 and its successor; therefore, we can perform at most one legal transposition to any permutation, and a legal transposition is not possible only and if only 0 is preceded by \( n \).
If \( n=1 \) or \( n=2 \) there is nothing t... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
597c08b03a79a1cc | Determine all finite nonempty sets $S$ of positive integers satisfying
$$
\frac{i+j}{(i, j)} \text { is an element of } S \text { for all } i, j \text { in } S \text {, }
$$
where $(i, j)$ is the greatest common divisor of $i$ and $j$.
Answer: $S=\{2\}$. | S=\{2\} | Let $k \in S$. Then $\frac{k+k}{(k, k)}=2$ is in $S$ as well.
Suppose for the sake of contradiction that there is an odd number in $S$, and let $k$ be the largest such odd number. Since $(k, 2)=1, \frac{k+2}{(k, 2)}=k+2>k$ is in $S$ as well, a contradiction. Hence $S$ has no odd numbers.
Now suppose that $\ell>2$ is th... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
cb2933a3ef1d5432 | Let $O$ be the circumcentre and $H$ the orthocentre of an acute triangle $A B C$. Prove that the area of one of the triangles $\mathrm{AOH}, \mathrm{BOH}$ and COH is equal to the sum of the areas of the other two. | proof | Suppose, without loss of generality, that $B$ and $C$ lie on the same side of line $O H$. This line is the Euler line of $\triangle ABC$, so the centroid $G$ lies on this line.
Let $M$ be the ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
f6f8b4e140982f4a | Let a set $S$ of 2004 points in the plane be given, no three of which are collinear. Let $\mathcal{L}$ denote the set of all lines (extended indefinitely in both directions) determined by pairs of points from the set. Show that it is possible to colour the points of $S$ with at most two colours, such that for any point... | proof | Choose any point $p$ from $S$ and color it, say, blue. Let $n(q, r)$ be the number of lines from $\mathcal{L}$ that separates $q$ and $r$. Then color any other point $q$ blue if $n(p, q)$ is odd and red if $n(p, q)$ is even.
Now it remains to show that $q$ and $r$ have the same color if and only if $n(q, r)$ is odd for... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
f7143e0fb1a29e57 | Prove that
$$
\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9(a b+b c+c a)
$$
for all real numbers \(a, b, c > 0\). | proof | Let $p=a+b+c, q=ab+bc+ca$, and $r=abc$. The inequality simplifies to
$$
a^{2} b^{2} c^{2}+2\left(a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}\right)+4\left(a^{2}+b^{2}+c^{2}\right)+8-9(ab+bc+ca) \geq 0
$$
Since $a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}=q^{2}-2pr$ and $a^{2}+b^{2}+c^{2}=p^{2}-2q$,
$$
r^{2}+2q^{2}-4pr+4p^{2}-8q+8-9... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
4790637be9735959 | Let \(a, b\) and \(c\) be positive real numbers such that \(a b c=8\). Prove that
\[
\frac{a^{2}}{\sqrt{\left(1+a^{3}\right)\left(1+b^{3}\right)}}+\frac{b^{2}}{\sqrt{\left(1+b^{3}\right)\left(1+c^{3}\right)}}+\frac{c^{2}}{\sqrt{\left(1+c^{3}\right)\left(1+a^{3}\right)}} \geq \frac{4}{3} .
\] | \frac{4}{3} | Observe that
$$
\frac{1}{\sqrt{1+x^{3}}} \geq \frac{2}{2+x^{2}}
$$
In fact, this is equivalent to \(\left(2+x^{2}\right)^{2} \geq 4\left(1+x^{3}\right)\), or \(x^{2}(x-2)^{2} \geq 0\). Notice that equality holds in (1) if and only if \(x=2\).
We substitute \(x\) by \(a, b, c\) in (1), respectively, to find
$$
\begi... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
e827b39d7f9ac1d5 | Let $n$ be a positive integer. Find the largest nonnegative real number $f(n)$ (depending on $n$) with the following property: whenever $a_{1}, a_{2}, \ldots, a_{n}$ are real numbers such that $a_{1}+a_{2}+\cdots+a_{n}$ is an integer, there exists some $i$ such that $\left|a_{i}-\frac{1}{2}\right| \geq f(n)$. | f(n)=\left\{\begin{array}{cl}
0 & \text { if } n \text { is even, } \\
\frac{1}{2 n} & \text { if } n \text { is odd. }
\end{array}\right.} | The answer is
$$
f(n)=\left\{\begin{array}{cl}
0 & \text { if } n \text { is even, } \\
\frac{1}{2 n} & \text { if } n \text { is odd. }
\end{array}\right.
$$
First, assume that $n$ is even. If $a_{i}=\frac{1}{2}$ for all $i$, then the sum $a_{1}+a_{2}+\cdots+a_{n}$ is an integer. Since $\left|a_{i}-\frac{1}{2}\right... | olympiads_ref | yoonholee/math-corpus-combined | 8 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
7923960c1f940282 | Let $S$ be a set of 9 distinct integers all of whose prime factors are at most 3. Prove that $S$ contains 3 distinct integers such that their product is a perfect cube. | proof | Without loss of generality, we may assume that $S$ contains only positive integers. Let
$$
S=\left\{2^{a_{i}} 3^{b_{i}} \mid a_{i}, b_{i} \in \mathbb{Z}, a_{i}, b_{i} \geq 0,1 \leq i \leq 9\right\}
$$
It suffices to show that there are $1 \leq i_{1}, i_{2}, i_{3} \leq 9$ such that
$$
a_{i_{1}}+a_{i_{2}}+a_{i_{3}} \e... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
e1afc0f2f478b716 | Let $A B C$ be an acute angled triangle with $\angle B A C=60^{\circ}$ and $A B>A C$. Let $I$ be the incenter, and $H$ the orthocenter of the triangle $A B C$. Prove that
$$
2 \angle A H I=3 \angle A B C .
$$ | proof | Let $D$ be the intersection point of the lines $A H$ and $B C$. Let $K$ be the intersection point of the circumcircle $O$ of the triangle $A B C$ and the line $A H$. Let the line through $I$ perpendicular to $B C$ meet $B C$ and the minor arc $B C$ of the circumcircle $O$ at $E$ and $N$, respectively. We have
$\angle B... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
e2275d4aada61be8 | Let \( x, y \) and \( z \) be positive real numbers such that \( \sqrt{x} + \sqrt{y} + \sqrt{z} = 1 \). Prove that
\[
\frac{x^{2} + y z}{\sqrt{2 x^{2} (y + z)}} + \frac{y^{2} + z x}{\sqrt{2 y^{2} (z + x)}} + \frac{z^{2} + x y}{\sqrt{2 z^{2} (x + y)}} \geq 1
\] | proof | We first note that
$$
\begin{aligned}
\frac{x^{2}+y z}{\sqrt{2 x^{2}(y+z)}} & =\frac{x^{2}-x(y+z)+y z}{\sqrt{2 x^{2}(y+z)}}+\frac{x(y+z)}{\sqrt{2 x^{2}(y+z)}} \\
& =\frac{(x-y)(x-z)}{\sqrt{2 x^{2}(y+z)}}+\sqrt{\frac{y+z}{2}} \\
& \geq \frac{(x-y)(x-z)}{\sqrt{2 x^{2}(y+z)}}+\frac{\sqrt{y}+\sqrt{z}}{2} .
\end{aligned}
$... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
65bf4c71f61cfcef | Let $A B C$ be a triangle with $\angle A<60^{\circ}$. Let $X$ and $Y$ be the points on the sides $A B$ and $A C$, respectively, such that $C A + A X = C B + B X$ and $B A + A Y = B C + C Y$. Let $P$ be the point in the plane such that the lines $P X$ and $P Y$ are perpendicular to $A B$ and $A C$, respectively. Prove t... | proof | Let $I$ be the incenter of $\triangle A B C$, and let the feet of the perpendiculars from $I$ to $A B$ and to $A C$ be $D$ and $E$, respectively. (Without loss of generality, we may assume that $A C$ is the longest side. Then $X$ lies on the line segment $A D$. Although $P$ may or may not lie inside $\triangle A B C$, ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
0ecec022603d8df8 | Students in a class form groups each of which contains exactly three members such that any two distinct groups have at most one member in common. Prove that, when the class size is 46, there is a set of 10 students in which no group is properly contained. | proof | We let $C$ be the set of all 46 students in the class and let
$$
s:=\max \{|S|: S \subseteq C \text { such that } S \text { contains no group properly }\}
$$
Then it suffices to prove that $s \geq 10$. (If $|S|=s>10$, we may choose a subset of $S$ consisting of 10 students.)
Suppose that $s \leq 9$ and let $S$ be a ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
215b5f7d253042bd | Consider the function $f: \mathbb{N}_{0} \rightarrow \mathbb{N}_{0}$, where $\mathbb{N}_{0}$ is the set of all non-negative integers, defined by the following conditions:
(i) $f(0)=0$,
(ii) $f(2 n)=2 f(n)$ and
(iii) $f(2 n+1)=n+2 f(n)$ for all $n \geq 0$.
(a) Determine the three sets $L:=\{n \mid f(n)f(n+1)\}$.
(b) For... | a_{k} = k 2^{k-1} - 2^{k} + 1 | (a) Let
$$
L_{1}:=\{2 k: k>0\}, \quad E_{1}:=\{0\} \cup\{4 k+1: k \geq 0\}, \quad \text { and } G_{1}:=\{4 k+3: k \geq 0\} .
$$
We will show that $L_{1}=L, E_{1}=E$, and $G_{1}=G$. It suffices to verify that $L_{1} \subseteq E, E_{1} \subseteq E$, and $G_{1} \subseteq G$ because $L_{1}, E_{1}$, and $G_{1}$ are mutual... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
1c3fa78fd23d3072 | Let \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) be real numbers satisfying the following equations:
$$
\frac{a_{1}}{k^{2}+1}+\frac{a_{2}}{k^{2}+2}+\frac{a_{3}}{k^{2}+3}+\frac{a_{4}}{k^{2}+4}+\frac{a_{5}}{k^{2}+5}=\frac{1}{k^{2}} \text { for } k=1,2,3,4,5
$$
Find the value of \(\frac{a_{1}}{37}+\frac{a_{2}}{38}+\frac{a_{3}}... | \frac{187465}{6744582} | Let $R(x):=\frac{a_{1}}{x^{2}+1}+\frac{a_{2}}{x^{2}+2}+\frac{a_{3}}{x^{2}+3}+\frac{a_{4}}{x^{2}+4}+\frac{a_{5}}{x^{2}+5}$. Then $R( \pm 1)=1$, $R( \pm 2)=\frac{1}{4}, R( \pm 3)=\frac{1}{9}, R( \pm 4)=\frac{1}{16}, R( \pm 5)=\frac{1}{25}$ and $R(6)$ is the value to be found. Let's put $P(x):=\left(x^{2}+1\right)\left(x^... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Algebra", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
50c58f52588298b2 | Prove that for any positive integer $k$, there exists an arithmetic sequence
$$
\frac{a_{1}}{b_{1}}, \quad \frac{a_{2}}{b_{2}}, \ldots, \quad \frac{a_{k}}{b_{k}}
$$
of rational numbers, where $a_{i}, b_{i}$ are relatively prime positive integers for each $i=1,2, \ldots, k$, such that the positive integers $a_{1}, b_{... | proof | For $k=1$, there is nothing to prove. Henceforth assume $k \geq 2$.
Let $p_{1}, p_{2}, \ldots, p_{k}$ be $k$ distinct primes such that
$$
kN^{2}. Consider the following sequence:
$$
\frac{x+1}{N}, \frac{x+2}{N}, \quad, \ldots, \frac{x+k}{N}
$$
This sequence is obviously an arithmetic sequence of positive rational nu... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
7e3b1c71c53ef9cf | For a positive integer $k$, call an integer a pure $k$-th power if it can be represented as $m^{k}$ for some integer $m$. Show that for every positive integer $n$ there exist $n$ distinct positive integers such that their sum is a pure 2009-th power, and their product is a pure 2010-th power. | proof | For the sake of simplicity, let us set $k=2009$.
First of all, choose $n$ distinct positive integers $b_{1}, \cdots, b_{n}$ suitably so that their product is a pure $k+1$-th power (for example, let $b_{i}=i^{k+1}$ for $i=1, \cdots, n$). Then we have $b_{1} \cdots b_{n}=t^{k+1}$ for some positive integer $t$. Set $b_{1}... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
94fd6e82d3ae0552 | Let $A B C$ be an acute triangle satisfying the condition $A B > B C$ and $A C > B C$. Denote by $O$ and $H$ the circumcenter and the orthocenter, respectively, of the triangle $A B C$. Suppose that the circumcircle of the triangle $A H C$ intersects the line $A B$ at $M$ different from $A$, and that the circumcircle o... | proof | In the sequel, we denote $\angle B A C=\alpha, \angle C B A=\beta, \angle A C B=\gamma$. Let $O^{\prime}$ be the circumcenter of the triangle $M N H$. The lengths of line segments starting from the point $H$ will be treated as signed quantities.
Let us denote by $M^{\prime}, N^{\prime}$ the point of intersection of $C... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
2935754b06295f1c | For $2 k$ real numbers $a_{1}, a_{2}, \ldots, a_{k}, b_{1}, b_{2}, \ldots, b_{k}$ define the sequence of numbers $X_{n}$ by
$$
X_{n}=\sum_{i=1}^{k}\left[a_{i} n+b_{i}\right] \quad(n=1,2, \ldots)
$$
If the sequence $X_{n}$ forms an arithmetic progression, show that $\sum_{i=1}^{k} a_{i}$ must be an integer. Here $[r]$... | A \text{ is an integer} | Let us write $A=\sum_{i=1}^{k} a_{i}$ and $B=\sum_{i=1}^{k} b_{i}$. Summing the corresponding terms of the following inequalities over $i$,
$$
a_{i} n+b_{i}-1<\left[a_{i} n+b_{i}\right] \leq a_{i} n+b_{i}
$$
we obtain $A n+B-k<X_{n}<A n+B$. Now suppose that $\left\{X_{n}\right\}$ is an arithmetic progression with the... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
9a1ee96cf015e907 | Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying:
(i) $A$ and $B$ are disjoint;
(ii) if an integer $i$ belongs either to $A$ or to $B$, then $i+a$ belongs to $A$ or $i-b$ belongs to $B$.
Prove that $a|A|=b|B|$. (Here $|X|$ denotes the number of elements in the set $X$.) | a|A|=b|B| | Let $A^{*}=\{n-a: n \in A\}$ and $B^{*}=\{n+b: n \in B\}$. Then, by (ii), $A \cup B \subseteq A^{*} \cup B^{*}$ and by (i),
$$
|A \cup B| \leq\left|A^{*} \cup B^{*}\right| \leq\left|A^{*}\right|+\left|B^{*}\right|=|A|+|B|=|A \cup B|
$$
Thus, $A \cup B=A^{*} \cup B^{*}$ and $A^{*}$ and $B^{*}$ have no element in commo... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
7ef20ccb002fe99b | For a positive integer $m$ denote by $S(m)$ and $P(m)$ the sum and product, respectively, of the digits of $m$. Show that for each positive integer $n$, there exist positive integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying the following conditions:
$$
S\left(a_{1}\right)<S\left(a_{2}\right)<\cdots<S\left(a_{n}\right)... | proof | Let $k$ be a sufficiently large positive integer. Choose for each $i=2,3, \ldots, n$, $a_{i}$ to be a positive integer among whose digits the number 2 appears exactly $k+i-2$ times and the number 1 appears exactly $2^{k+i-1}-2(k+i-2)$ times, and nothing else. Then, we have $S\left(a_{i}\right)=2^{k+i-1}$ and $P\left(a_... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
55414be37e6dad79 | Let $n$ and $b$ be positive integers. We say $n$ is $b$-discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$.
(a) Prove that 8 is 100-discerning.
(b) Prove... | proof | (a) Take \( S = \{3, 6, 12, 24, 48, 95, 96, 97\} \), i.e.
\[
S = \left\{3 \cdot 2^{k}: 0 \leq k \leq 5\right\} \cup \left\{3 \cdot 2^{5} - 1, 3 \cdot 2^{5} + 1\right\}
\]
As \( k \) ranges between 0 to 5, the sums obtained from the numbers \( 3 \cdot 2^{k} \) are \( 3t \), where \( 1 \leq t \leq 63 \). These are 63 n... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
7d08b3bf49e6894d | Circles $\omega$ and $\Omega$ meet at points $A$ and $B$. Let $M$ be the midpoint of the $\operatorname{arc} A B$ of circle $\omega$ ( $M$ lies inside $\Omega$ ). A chord $M P$ of circle $\omega$ intersects $\Omega$ at $Q(Q$ lies inside $\omega)$. Let $\ell_{P}$ be the tangent line to $\omega$ at $P$, and let $\ell_{Q}... | proof | Denote $X=A B \cap \ell_{P}, Y=A B \cap \ell_{Q}$, and $Z=\ell_{P} \cap \ell_{Q}$. Without loss of generality we have $A X<B X$. Let $F=M P \cap A B$.
Denote by $R$ the second point of inter... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
f49fe55ffc76e3ff | A sequence of real numbers $a_{0}, a_{1}, \ldots$ is said to be good if the following three conditions hold.
(i) The value of $a_{0}$ is a positive integer.
(ii) For each non-negative integer $i$ we have $a_{i+1}=2 a_{i}+1$ or $a_{i+1}=\frac{a_{i}}{a_{i}+2}$.
(iii) There exists a positive integer $k$ such that $a_{k}=2... | 60 | Note that
$$
a_{i+1}+1=2\left(a_{i}+1\right) \text { or } a_{i+1}+1=\frac{a_{i}+a_{i}+2}{a_{i}+2}=\frac{2\left(a_{i}+1\right)}{a_{i}+2} .
$$
Hence
$$
\frac{1}{a_{i+1}+1}=\frac{1}{2} \cdot \frac{1}{a_{i}+1} \text { or } \frac{1}{a_{i+1}+1}=\frac{a_{i}+2}{2\left(a_{i}+1\right)}=\frac{1}{2} \cdot \frac{1}{a_{i}+1}+\fra... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
94466cf7ccbe673f | A positive integer is called fancy if it can be expressed in the form
$$
2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{100}}
$$
where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct.
Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.
Answer: The answe... | 2^{101}-1 | Let $k$ be any positive integer less than $2^{101}-1$. Then $k$ can be expressed in binary notation using at most 100 ones, and therefore there exists a positive integer $r$ and non-negative integers $a_{1}, a_{2}, \ldots, a_{r}$ such that $r \leq 100$ and $k=2^{a_{1}}+\cdots+2^{a_{r}}$. Notice that for a positive inte... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
267e8755c8022630 | Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$
(z+1) f(x+y)=f(x f(z)+y)+f(y f(z)+x)
$$
for all positive real numbers $x, y, z$.
Answer: The only solution is $f(x)=x$ for all positive real numbers $x$. | f(x)=x | The identity function $f(x)=x$ clearly satisfies the functional equation. Now, let $f$ be a function satisfying the functional equation. Plugging $x=y=1$ into (3) we get $2 f(f(z)+1)=(z+1)(f(2))$ for all $z \in \mathbb{R}^{+}$. Hence, $f$ is not bounded above.
Lemma. Let $a, b, c$ be positive real numbers. If $c$ is g... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Algebra", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
07b3684745b50d28 | We call a 5-tuple of integers arrangeable if its elements can be labeled $a$, $b, c, d, e$ in some order so that $a-b+c-d+e=29$. Determine all 2017-tuples of integers $n_{1}, n_{2}, \ldots, n_{2017}$ such that if we place them in a circle in clockwise order, then any 5 -tuple of numbers in consecutive positions on the ... | n_{1} = \cdots = n_{2017} = 29 | A valid 2017-tuple is \( n_{1} = \cdots = n_{2017} = 29 \). We will show that it is the only solution.
We first replace each number \( n_{i} \) in the circle by \( m_{i} := n_{i} - 29 \). Since the condition \( a - b + c - d + e = 29 \) can be rewritten as \( (a - 29) - (b - 29) + (c - 29) - (d - 29) + (e - 29) = 0 \)... | olympiads_ref | yoonholee/math-corpus-combined | 8 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
1f683fae0048b2b5 | Let $A B C$ be a triangle with $A B < A C$. Let $D$ be the intersection point of the internal bisector of angle $B A C$ and the circumcircle of $A B C$. Let $Z$ be the intersection point of the perpendicular bisector of $A C$ with the external bisector of angle $\angle B A C$. Prove that the midpoint of the segment $A ... | proof | . Let $N$ be the midpoint of $A C$. Let $M$ be the intersection point of the circumcircle of triangle $A D Z$ and the segment $A B$. We will show that $M$ is the midpoint of $A B$. To do this, let $D^{\prime}$ be the reflection of $D$ with respect to $M$. It suffices to show that $A D B D^{\prime}$ is a parallelogram.
... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
2fc7e560281e1139 | Let $H$ be the orthocenter of the triangle $A B C$. Let $M$ and $N$ be the midpoints of the sides $A B$ and $A C$, respectively. Assume that $H$ lies inside the quadrilateral $B M N C$ and that the circumcircles of triangles $B M H$ and $C N H$ are tangent to each other. The line through $H$ parallel to $B C$ intersect... | proof | Lemma 1. In a triangle $A B C$, let $D$ be the intersection of the interior angle bisector at $A$ with the circumcircle of $A B C$, and let $I$ be the incenter of $\triangle A B C$. Then
$$
D I=D B=D C
$$
![](https://cdn.mathpix.com/cropped/2024_11_22_c87ece96a341aa1506ccg-1.jpg?height=637&width=618&top_left_y=938&to... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
1cdb14ec9864d210 | Let $f(x)$ and $g(x)$ be given by
$$
f(x)=\frac{1}{x}+\frac{1}{x-2}+\frac{1}{x-4}+\cdots+\frac{1}{x-2018}
$$
and
$$
g(x)=\frac{1}{x-1}+\frac{1}{x-3}+\frac{1}{x-5}+\cdots+\frac{1}{x-2017}.
$$
Prove that
$$
|f(x)-g(x)|>2
$$
for any non-integer real number $x$ satisfying $0<x<2018$. | proof | There are two cases: $2 n-12$ whenever $2 n-10+0=0$
Hence it suffices to prove $d(x)>2$ for $1$ $\frac{1}{x-2 i}$ for $i=2,3, \ldots, 1008$. We also have $\frac{1}{x-2018}2 \\
\Leftrightarrow & \left(\frac{1}{x-1}+\frac{1}{2-x}\right)+\left(\frac{1}{x-3}-\frac{1}{x}\right)>2 \\
\Leftrightarrow & \frac{1}{(x-1)(2-x)}+\... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Algebra", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
622f9d3202bf1b6d | Let $\mathbb{Z}^{+}$ be the set of positive integers. Determine all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ such that $a^{2} + f(a) f(b)$ is divisible by $f(a) + b$ for all positive integers $a$ and $b$.
Answer: The answer is $f(n) = n$ for all positive integers $n$.
Clearly, $f(n) = n$ for all $n \in... | f(n) = n | First we perform the following substitutions on the original relation:
1. With \(a=b=1\), we find that \(f(1)+1 \mid f(1)^{2}+1\), which implies \(f(1)=1\).
2. With \(a=1\), we find that \(b+1 \mid f(b)+1\). In particular, \(b \leq f(b)\) for all \(b \in \mathbb{Z}^{+}\).
3. With \(b=1\), we find that \(f(a)+1 \mid a^... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
f24cf3514d577a46 | Let $A B C$
be a scalene triangle with circumcircle $\Gamma$. Let $M$ be the midpoint of $B C$. A variable point $P$ is selected in the line segment $A M$. The circumcircles of triangles $B P M$ and $C P M$ intersect $\Gamma$ again at points $D$ and $E$, respectively. The lines $D P$ and $E P$ intersect (a second time... | proof | Let $N$ be the radical center of the circumcircles of triangles $A B C, B M P$ and $C M P$. The pairwise radical axes of these circles are $B D, C E$ and $P M$, and hence they concur at $N$. Now, note that in directed angles:
$$
\angle M C E=\angle M P E=\angle M P Y=\angle M B Y .
$$
![](https://cdn.mathpix.com/crop... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
62976081c1ab60c7 | Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{2}+f(y)\right)=f(f(x))+f\left(y^{2}\right)+2 f(x y)
$$
for all real number $x$ and $y$.
Answer: The possible functions are $f(x)=0$ for all $x$ and $f(x)=x^{2}$ for all $x$. | f(x)=0 \text{ or } f(x)=x^2 | By substituting \(x=y=0\) in the given equation of the problem, we obtain that \(f(0)=0\). Also, by substituting \(y=0\), we get \(f\left(x^{2}\right)=f(f(x))\) for any \(x\).
Furthermore, by letting \(y=1\) and simplifying, we get
\[
2 f(x)=f\left(x^{2}+f(1)\right)-f\left(x^{2}\right)-f(1)
\]
from which it follows ... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Algebra", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
237ee485fee2a4ab | Show that $r=2$ is the largest real number $r$ which satisfies the following condition:
If a sequence $a_{1}, a_{2}, \ldots$ of positive integers fulfills the inequalities
$$
a_{n} \leq a_{n+2} \leq \sqrt{a_{n}^{2}+r a_{n+1}}
$$
for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n... | proof | . First, let us assume that $r>2$, and take a positive integer $a \geq 1 /(r-2)$.
Then, if we let $a_{n}=a+\lfloor n / 2\rfloor$ for $n=1,2, \ldots$, the sequence $a_{n}$ satisfies the inequalities
$$
\sqrt{a_{n}^{2}+r a_{n+1}} \geq \sqrt{a_{n}^{2}+r a_{n}} \geq \sqrt{a_{n}^{2}+\left(2+\frac{1}{a}\right) a_{n}} \geq a... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
68b098d54d6292e1 | Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n>m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways. | k=2^{a} | We claim that $k=2^{a}$ for all $a \geq 0$.
Let $A=\{1,2,4,8, \ldots\}$ and $B=\mathbb{N} \backslash A$. For any set $T$, let $s(T)$ denote the sum of the elements of $T$. (If $T$ is empty, we let $s(T)=0$.)
We first show that any positive integer $k=2^{a}$ satisfies the desired property. Let $B^{\prime}$ be a subset o... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
41fb3db4a0d79c23 | Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property:
For any infinite sequence $a_{1}, a_{2}, \ldots$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i<j$ and an integer $k$ such that $a_{... | proof | Part 1: All polynomials with $\operatorname{deg} P=1$ satisfy the given property.
Suppose $P(x)=c x+d$, and assume without loss of generality that $c>d \geq 0$. Denote $s_{i}=a_{1}+a_{2}+\cdots+a_{i}(\bmod c)$. It suffices to show that there exist indices $i$ and $j$ such that $j-i \geq 2$ and $s_{j}-s_{i} \equiv d$ $(... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
dc472e714becb296 | Let $n \geq 3$ be a fixed integer. The number 1 is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$, replacing them with the numbers 1 and $a+b$, then adding one stone to the first bucket and $\operatorname... | \frac{t}{s} \in [1, n-1) | The answer is the set of all rational numbers in the interval $[1, n-1)$. First, we show that no other numbers are possible. Clearly the ratio is at least 1, since for every move, at least one stone is added to the second bucket. Note that the number $s$ of stones in the first bucket is always equal to $p-n$, where $p$... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
13b65e8131a03afa | Let $A B C D$ be a cyclic convex quadrilateral and $\Gamma$ be its circumcircle. Let $E$ be the intersection of the diagonals $A C$ and $B D$, let $L$ be the center of the circle tangent to sides $A B, B C$, and $C D$, and let $M$ be the midpoint of the arc $B C$ of $\Gamma$ not containing $A$ and $D$. Prove that the e... | proof | Let $L$ be the intersection of the bisectors of $\angle A B C$ and $\angle B C D$. Let $N$ be the $E$-excenter of $\triangle B C E$. Let $\angle B A C=\angle B D C=\alpha, \angle D B C=\beta$ and $\angle A C B=\gamma$.
We have the following:
$$
\begin{array}{r}
\angle C B L=\frac{1}{2} \angle A B C=90^{\circ}-\frac{1}... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
9a1c178a90bfedb2 | Given a $32 \times 32$ table, we put a mouse (facing up) at the bottom left cell and a piece of cheese at several other cells. The mouse then starts moving. It moves forward except that when it reaches a piece of cheese, it eats a part of it, turns right, and continues moving forward. We say that a subset of cells cont... | proof | (a) For the sake of contradiction, assume a good subset consisting of 888 cells exists. We call those cheese-cells and the other ones gap-cells. Observe that since each cheese-cell is visited once, each gap-cell is visited at most twice (once vertically and once horizontally). Define a finite sequence $s$ whose $i$-th ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
4c2984f1ae6d108a | Find all pairs $(a, b)$ of positive integers such that $a^{3}$ is a multiple of $b^{2}$ and $b-1$ is a multiple of $a-1$. Note: An integer $n$ is said to be a multiple of an integer $m$ if there is an integer $k$ such that $n=k m$. | proof | .1
By inspection, we see that the pairs $(a, b)$ with $a=b$ are solutions, and so too are the pairs $(a, 1)$. We will see that these are the only solutions.
- Case 1. Consider the case $b<a$. Since $b-1$ is a multiple of $a-1$, it follows that $b=1$. This yields the second set of solutions described above.
- Case 2. ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
1f509031d90cdab9 | Let $A B C$ be a right triangle with $\angle B=90^{\circ}$. Point $D$ lies on the line $C B$ such that $B$ is between $D$ and $C$. Let $E$ be the midpoint of $A D$ and let $F$ be the second intersection point of the circumcircle of $\triangle A C D$ and the circumcircle of $\triangle B D E$. Prove that as $D$ varies, t... | proof | .1
Let the line $E F$ intersect the line $B C$ at $P$ and the circumcircle of $\triangle A C D$ at $G$ distinct from $F$. We will prove that $P$ is the fixed point.
First, notice that $\triangle B E D$ is isosceles with $E B=E D$. This implies $\angle E B C=\angle E D P$.
Then, $\angle D A G=\angle D F G=\angle E B C=... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
2498aa420596ff3d | Find all positive integers $k<202$ for which there exists a positive integer $n$ such that
$$
\left\{\frac{n}{202}\right\}+\left\{\frac{2 n}{202}\right\}+\cdots+\left\{\frac{k n}{202}\right\}=\frac{k}{2}
$$
where $\{x\}$ denote the fractional part of $x$.
Note: $\{x\}$ denotes the real number $k$ with $0 \leq k<1$ su... | 1, 100, 101, 201 | Denote the equation in the problem statement as $\left(^{*}\right)$, and note that it is equivalent to the condition that the average of the remainders when dividing $n, 2 n, \ldots, k n$ by 202 is 101. Since $\left\{\frac{i n}{202}\right\}$ is invariant in each residue class modulo 202 for each $1 \leq i \leq k$, it s... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
4735b7d235679560 | Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1,2, \ldots, n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices.
Show that it is possible to arrange these squares in a way s... | proof | Set aside the squares with sidelengths \( n-3, n-2, n-1 \), and \( n \) and suppose we can split the remaining squares into two sets \( A \) and \( B \) such that the sum of the sidelengths of the squares in \( A \) is 1 or 2 units larger than the sum of the sidelengths of the squares in \( B \).
String the squares of ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
dd8eccc09fc1320b | Let $A B C D$ be a parallelogram. Let $W, X, Y$, and $Z$ be points on sides $A B, B C, C D$, and $D A$, respectively, such that the incenters of triangles $A W Z, B X W, C Y X$ and $D Z Y$ form a parallelogram. Prove that $W X Y Z$ is a parallelogram. | proof | Let the four incenters be \(I_{1}, I_{2}, I_{3}\), and \(I_{4}\) with inradii \(r_{1}, r_{2}, r_{3}\), and \(r_{4}\) respectively (in the order given in the question). Without loss of generality, let \(I_{1}\) be closer to \(AB\) than \(I_{2}\). Let the acute angle between \(I_{1} I_{2}\) and \(AB\) (and hence also the... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
b8ec45b2a99fccff | Let $A B C$ be an acute triangle. Let $D$ be a point on side $A B$ and $E$ be a point on side $A C$ such that lines $B C$ and $D E$ are parallel. Let $X$ be an interior point of $B C E D$. Suppose rays $D X$ and $E X$ meet side $B C$ at points $P$ and $Q$, respectively such that both $P$ and $Q$ lie between $B$ and $C$... | proof | Let $\ell$ be the radical axis of circles $B Q X$ and $C P X$. Since $X$ and $Y$ are on $\ell$, it is sufficient to show that $A$ is on $\ell$. Let line $A X$ intersect segments $B C$ and $D E$ at $Z$ and $Z^{\prime}$, respectively. Then it is sufficient to show that $Z$ is on $\ell$. By $B C \| D E$, we obtain
$$
\fr... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
3e41ff1f8acc2d4c | Let \( n \) be a positive integer and \( a_{1}, a_{2}, \ldots, a_{n} \) be positive real numbers. Prove that
\[
\sum_{i=1}^{n} \frac{1}{2^{i}}\left(\frac{2}{1+a_{i}}\right)^{2^{i}} \geq \frac{2}{1+a_{1} a_{2} \ldots a_{n}}-\frac{1}{2^{n}}
\] | proof | We first prove the following lemma:
Lemma 1. For $k$ a positive integer and $x, y > 0$,
$$
\left(\frac{2}{1+x}\right)^{2^{k}} + \left(\frac{2}{1+y}\right)^{2^{k}} \geq 2\left(\frac{2}{1+xy}\right)^{2^{k-1}}
$$
The proof goes by induction. For $k=1$, we have
$$
\left(\frac{2}{1+x}\right)^{2} + \left(\frac{2}{1+y}\rig... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
09c3490a5a60e177 | Prove that for every positive integer $t$ there is a unique permutation $a_{0}, a_{1}, \ldots, a_{t-1}$ of $0,1, \ldots, t-1$ such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2 a_{i}}$ is odd and $2 a_{i} \neq t+i$. | proof | We constantly make use of Kummer's theorem which, in particular, implies that $\binom{n}{k}$ is odd if and only if $k$ and $n-k$ have ones in different positions in binary. In other words, if $S(x)$ is the set of positions of the digits 1 of $x$ in binary (in which the digit multiplied by $2^{i}$ is in position $i),\bi... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
23c675bfd9930ae1 | Line $\ell$ intersects sides $B C$ and $A D$ of cyclic quadrilateral $A B C D$ in its interior points $R$ and $S$ respectively, and intersects ray $D C$ beyond point $C$ at $Q$, and ray $B A$ beyond point $A$ at $P$. Circumcircles of the triangles $Q C R$ and $Q D S$ intersect at $N \neq Q$, while circumcircles of the ... | proof | We start with the following lemma.
Lemma 1. Points \( M, N, P, Q \) are concyclic.
Point \( M \) is the Miquel point of lines \( A P = A B, P S = \ell, A S = A D \), and \( B R = B C \), and point \( N \) is the Miquel point of lines \( C Q = C D, R C = B C, Q R = \ell \), and \( D S = A D \). Both points \( M \) and \... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
cfc3e82d881d9acb | An acute-angled scalene triangle \(ABC\) is given, with \(AC > BC\). Let \(O\) be its circumcentre, \(H\) its orthocentre, and \(F\) the foot of the altitude from \(C\). Let \(P\) be the point (other than \(A\)) on the line \(AB\) such that \(AF = PF\), and \(M\) be the midpoint of \(AC\). We denote the intersection of... | proof | It is enough to show that $\mathrm{O} F \perp F X$.
Let $\mathrm{OE} \perp \mathrm{AB}$, then it is trivial that :
$$
C \mathrm{H}=2 \mathrm{OE} .
$$
Since from the hypothesis we have $\mathrm{P} F=\mathrm{A} F$ then we take $\mathrm{PB}=\mathrm{P} F-\mathrm{B} F$ or
$$
\mathrm{PB}=\mathrm{A} F-\mathrm{B} F
$$
Also... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
f7e4a7005cd01b1d | Does there exist a sequence $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ of positive real numbers satisfying both of the following conditions:
(i) $\sum_{i=1}^{n} a_{i} \leq n^{2}$, for every positive integer $n$;
(ii) $\sum_{i=1}^{n} \frac{1}{a_{i}} \leq 2008$, for every positive integer $n$? | proof | The answer is no.
It is enough to show that
if $\sum_{i=1}^{n} a_{i} \leq n^{2}$ for any $n$, then $\sum_{i=2}^{2^{n}} \frac{1}{a_{i}}>\frac{n}{4}$. (or any other precise estimate)
For this, we use that $\sum_{i=2^{k}+1}^{2^{k+1}} a_{i} \sum_{i=2^{k}+1}^{2^{k+1}} \frac{1}{a_{i}} \geq 2^{2 k}$ for any $k \geq 0$ by the ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
87f99994c7a7f18d | Let \( n \) be a positive integer. The rectangle \( A B C D \) with side lengths \( A B = 90 n + 1 \) and \( B C = 90 n + 5 \) is partitioned into unit squares with sides parallel to the sides of \( A B C D \). Let \( S \) be the set of all points which are vertices of these unit squares. Prove that the number of lines... | proof | Denote $90 n+1=m$. We investigate the number of the lines modulo 4 consecutively reducing different types of lines.
The vertical and horizontal lines are
$(m+5)+(m+1)=2(m+3)$ which is divisible by 4.
Moreover, every line which makes an acute angle to the axis $O x$ (i.e., that line has a positive angular coefficient) c... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
4528982a91008571 | Let $c$ be a positive integer. The sequence $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ is defined by $a_{1}=c$, and $a_{n+1}=a_{n}^{2}+a_{n}+c^{3}$, for every positive integer $n$. Find all values of $c$ for which there exist some integers $k \geq 1$ and $m \geq 2$, such that $a_{k}^{2}+c^{3}$ is the $m^{\text {th }}$ power... | c=s^{2}-1, s \geq 2, s \in \mathbb{N} | First, notice:
$$
a_{n+1}^{2}+c^{3}=\left(a_{n}^{2}+a_{n}+c^{3}\right)^{2}+c^{3}=\left(a_{n}^{2}+c^{3}\right)\left(a_{n}^{2}+2 a_{n}+1+c^{3}\right)
$$
We first prove that $a_{n}^{2}+c^{3}$ and $a_{n}^{2}+2 a_{n}+1+c^{3}$ are coprime.
We prove by induction that $4 c^{3}+1$ is coprime with $2 a_{n}+1$, for every $n \ge... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
d24f7ba2397d34f1 | We start by observing that $z$ must be even, so $z^{2}=3^{x}-5^{y} \equiv(-1)^{x}-1(\bmod 4)$ is divisible by 4, which implies that $x$ is even, say $x=2 t$. Then our equation can be rewritten as $\left(3^{t}-z\right)\left(3^{t}+z\right)=5^{y}$, which means that both $3^{t}-z=5^{k}$ and $3^{t}+z=5^{y-k}$ for some nonne... | (x, y, z)=(2,1,2) | We start by observing that $f$ is injective. From the known identity
$$
\left(a^{2}+2 b^{2}\right)\left(c^{2}+2 d^{2}\right)=(a c \pm 2 b d)^{2}+2(a d \mp b c)^{2}
$$
we obtain $f(a c+2 b d)^{2}+2 f(a d-b c)^{2}=f(a c-2 b d)^{2}+2 f(a d+b c)^{2}$, assuming that the arguments are positive integers. Specially, for $b=c... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
38698517e92950c1 | In a triangle $ABC$, points $M$ and $N$ on the sides $AB$ and $AC$ respectively are such that $MN \parallel BC$. Let $BN$ and $CM$ intersect at point $P$. The circumcircles of triangles $BMP$ and $CNP$ intersect at two distinct points $P$ and $Q$. Prove that $\angle BAQ = \angle CAP$.
(Moldova) | proof | Since the quadrilaterals $B M P Q$ and $C N P Q$ are cyclic, we have $\angle B Q N = \angle B Q P + \angle P Q N = \angle A M C + \angle M C A = 180^{\circ} - \angle C A B$, so $A B Q N$ is cyclic as well. Hence $\frac{\sin \angle B A Q}{\sin \angle N A Q} = \frac{B Q}{N Q}$. Moreover, triangles $M B Q$ and $C N Q$ are... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
0219ac075fa5db8b | A $9 \times 12$ rectangle is divided into unit squares. The centers of all the unit squares, except the four corner squares and the eight squares adjacent (by side) to them, are colored red. Is it possible to numerate the red centers by $C_{1}, C_{2}, \ldots, C_{96}$ so that the following two conditions are fulfilled:
... | proof | Place the given rectangle into the coordinate plane so that the center of the square at the intersection of $i$-th column and $j$-th row has the coordinates $(i, j)$. Suppose that a desired numeration of the red points exists; it corresponds to a path, i.e. a closed poligonal line consisting of 96 segments of length $\... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
dfbdfbae2fdea4e5 | Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying
$$
f\left(f(m)^{2}+2 f(n)^{2}\right)=m^{2}+2 n^{2} \quad \text { for all } m, n \in \mathbb{N} . \quad \text { (Bulgaria) }
$$
Time allowed: 270 minutes.
Each problem is worth 10 points.
## SOLUTIONS | f(n)=n | We start by observing that $f$ is injective. From the known identity
$$
\left(a^{2}+2 b^{2}\right)\left(c^{2}+2 d^{2}\right)=(a c \pm 2 b d)^{2}+2(a d \mp b c)^{2}
$$
we obtain $f(a c+2 b d)^{2}+2 f(a d-b c)^{2}=f(a c-2 b d)^{2}+2 f(a d+b c)^{2}$, assuming that the arguments are positive integers. Specially, for $b=c... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Algebra", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
e2369138f4f0b1d1 | The left-hand side is equal to
$$
\frac{a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3}-a^{3} b^{2} c-b^{3} c^{2} a-c^{3} a^{2} b}{(a+b)(b+c)(c+a)}
$$
so it is enough to show that \(a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3} \geq a^{3} b^{2} c+b^{3} c^{2} a+c^{3} a^{2} b\). The AM-GM inequality gives us \(a^{3} b^{3}+a^{3} b^{3}+a^{3} ... | proof | There are $n+1-\varphi(n)$ nonnegative integers not coprime with $n$, and whenever $r$ is among them, so is $n-r$. This gives us the formula $f(n)=\frac{1}{2} n(n+1-\varphi(n))$. Suppose that $f(n)=f(n+p)$. We observe first that $n$ and $n+p$ divide $2 f(n)q^{i-1}(q-1)+1>\frac{2}{3}\left(2 q^{i}+2\right)$. The other th... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
4d42670196aa896e | Let $A B C$ be an acute-angled triangle with orthocenter $H$ and let $M$ be the midpoint of $A C$. The foot of the altitude from $C$ is $C_{1}$. Point $H_{1}$ is symmetric to $H$ in $A B$. The projections of $C_{1}$ on lines $A H_{1}, A C$ and $B C$ are $P, Q$ and $R$ respectively. If $M_{1}$ is the circumcenter of tri... | proof | We shall use the following simple statement.
Lemma. Let $A_{1} A_{2} A_{3} A_{4}$ be a convex cyclic quadrilateral whose diagonals are orthogonal and meet at $X$. If $B_{i}$ is the midpoint of side $A_{i} A_{i+1}$ and $X_{i}$ the projection of $X$ on this side $\left(A_{5}=A_{1}\right)$, then the eight points $B_{i}, ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
cea90cc355a8d262 | We define a $w$-strip as the set of all points in the plane that are between or on two parallel lines at a mutual distance $w$. Let $S$ be a set of $n$ points in the plane such that any three points from $S$ can be covered by a 1-strip. Show that the entire set $S$ can be covered by a 2-strip.
(Romania) | proof | Of all triangles with the vertices in $S$, consider one with a maximum area, say $\triangle A B C$. Let $A^{\prime}, B^{\prime}, C^{\prime}$ be the points symmetric to $A, B, C$ with respect to the midpoints of $B C, C A, A B$, respectively. We claim that all points from $S$ must lie inside or on the boundary of $\tria... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
283edfcacc95a2b8 | For every integer $n \geq 2$, denote by $f(n)$ the sum of positive integers not exceeding $n$ that are not coprime to $n$. Prove that $f(n+p) \neq f(n)$ for any such $n$ and any prime number $p$.
(Turkey)
Time allowed: 270 minutes.
Each problem is worth 10 points.
## SOLUTIONS | proof | There are $n+1-\varphi(n)$ nonnegative integers not coprime with $n$, and whenever $r$ is among them, so is $n-r$. This gives us the formula $f(n)=\frac{1}{2} n(n+1-\varphi(n))$. Suppose that $f(n)=f(n+p)$. We observe first that $n$ and $n+p$ divide $2 f(n)q^{i-1}(q-1)+1>\frac{2}{3}\left(2 q^{i}+2\right)$. The other th... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
c4d15ee90b2b3a94 | Let $A B C D$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $A B$ and $C D$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $A B$. The feet of the perpendiculars from $E$ onto the lines $\ell$ and $C D$ are $H$ and $K$, respectively. Pr... | proof | The points $E, K, H, G$ are on the circle of diameter $G E$, so
$$
\angle E H K=\angle E G K
$$
Also, from $\angle D C A=\angle D B A$ and $\frac{C E}{C D}=\frac{B E}{B A}$ it follows
$$
\frac{C E}{C G}=\frac{2 C E}{C D}=\frac{2 B E}{B A}=\frac{B E}{B F},
$$
therefore $\triangle C G E \sim \triangle B F E$. In part... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
925dbff5b7c3cbda | Given real numbers $x, y, z$ such that $x+y+z=0$, show that
$$
\frac{x(x+2)}{2 x^{2}+1}+\frac{y(y+2)}{2 y^{2}+1}+\frac{z(z+2)}{2 z^{2}+1} \geq 0 .
$$
When does equality hold? | proof | The inequality is clear if $x y z=0$, in which case equality holds if and only if $x=y=z=0$.
Henceforth assume $x y z \neq 0$ and rewrite the inequality as
$$
\frac{(2 x+1)^{2}}{2 x^{2}+1}+\frac{(2 y+1)^{2}}{2 y^{2}+1}+\frac{(2 z+1)^{2}}{2 z^{2}+1} \geq 3 .
$$
Notice that (exactly) one of the products $x y, y z, z x... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
db1346df73046f11 | Let $S$ be a finite set of positive integers which has the following property: if $x$ is a member of $S$, then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is good if whenever $x, y \in T$ and $x<y$, the ratio $y / x$ is a power of a prime number. A non-empty subset $T$ of $S$ is bad if whenever $... | proof | Notice first that a bad subset of $S$ contains at most one element from a good one, to deduce that a partition of $S$ into bad subsets has at least as many members as a maximal good subset.
Notice further that the elements of a good subset of $S$ must be among the terms of a geometric sequence whose ratio is a prime: ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Combinatorics", "problem_type": "Combinatorics", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
4c7b66ab478510a9 | Let $A B C D E F$ be a convex hexagon of area 1, whose opposite sides are parallel. The lines $A B, C D$ and $E F$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $B C, D E$ and $F A$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two... | proof | Unless otherwise stated, throughout the proof indices take on values from 0 to 5 and are reduced modulo 6. Label the vertices of the hexagon in circular order, \(A_{0}, A_{1}, \cdots, A_{5}\), and let the lines of support of the alternate sides \(A_{i} A_{i+1}\) and \(A_{i+2} A_{i+3}\) meet at \(B_{i}\). To show that t... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
2e431e604a5732de | \quad$ Let $A, B$ and $C$ be points lying on a circle $\Gamma$ with centre $O$. Assume that $\angle A B C>90$. Let $D$ be the point of intersection of the line $A B$ with the line perpendicular to $A C$ at $C$. Let $l$ be the line through $D$ which is perpendicular to $A O$. Let $E$ be the point of intersection of $l$ ... | proof | Let $\ell \cap A O=\{K\}$ and $G$ be the other end point of the diameter of $\Gamma$ through $A$. Then $D, C, G$ are collinear. Moreover, $E$ is the orthocenter of triangle $A D G$. Therefore $G E \perp A D$ and $G, E, B$ are collinear.
![](https://cdn.mathpix.com/cropped/2024_12_10_ead8d0bf173f1f5b4aafg-1.jpg?height=1... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
2ea6ee6a115c24ac | Prove that
$$
\sum_{c y c}(x+y) \sqrt{(z+x)(z+y)} \geq 4(x y+y z+z x),
$$
for all positive real numbers \( x, y \) and \( z \). | proof | We will obtain the inequality by adding the inequalities
\[
(x+y) \sqrt{(z+x)(z+y)} \geq 2 x y+y z+z x
\]
for cyclic permutation of \(x, y, z\).
Squaring both sides of this inequality we obtain
\[
(x+y)^{2}(z+x)(z+y) \geq 4 x^{2} y^{2}+y^{2} z^{2}+z^{2} x^{2}+4 x y^{2} z+4 x^{2} y z+2 x y z^{2}
\]
which is equivale... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Inequalities", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
4d01da2531a8906c | Let $n$ be a positive integer. Let $P_{n}=\left\{2^{n}, 2^{n-1} \cdot 3,2^{n-2} \cdot 3^{2}, \ldots, 3^{n}\right\}$. For each subset $X$ of $P_{n}$, we write $S_{X}$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with ... | proof | Let $\alpha=3 / 2$ so $1+\alpha>\alpha^{2}$.
Given $y$, we construct $Y$ algorithmically. Let $Y=\varnothing$ and of course $S_{\varnothing}=0$. For $i=0$ to $m$, perform the following operation:
$$
\text{If } S_{Y}+2^{i} 3^{m-i} \leq y \text{, then replace } Y \text{ by } Y \cup\left\{2^{i} 3^{m-i}\right\}
$$
When t... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
740c0fad4a91c3fd | \quad$ Let $\mathbb{Z}^{+}$ be the set of positive integers. Find all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers. | proof | There are three such functions: the constant functions 1, 2 and the identity function $\mathrm{id}_{\mathbf{Z}^{+}}$. These functions clearly satisfy the conditions in the hypothesis. Let us prove that there are only ones.
Consider such a function $f$ and suppose that it has a fixed point $a \geq 3$, that is $f(a)=a$.... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
7666273117ea668c | We denote the angles of the triangle by $\alpha, \beta$ and $\gamma$ as usual. Since $\angle K M P=90^{\circ}-\frac{\beta}{2}$, it suffices to prove that $\angle K L P=90^{\circ}+\frac{\beta}{2}$, which is equivalent to $\angle K L C=\frac{\beta}{2}$.
Let $I$ be the incenter of triangle $A B C$ and let $D$ be the tange... | proof | Consider the graph $\mathcal{G}$ whose vertices are the contestants, where there is an edge between two contestants if and only if they are not friends.
Lemma. There is a vertex in graph $\mathcal{G}$ with degree at most 2.
Proof. Suppose that each vertex has a degree at least three. Consider the longest induced path ... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Geometry", "problem_type": "Geometry", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
e7fff6c6fbe44a14 | Determine all positive integers $x, y$ and $z$ such that
$$
x^{5}+4^{y}=2013^{z}
$$ | not found | Reducing modulo 11 yields \( x^{5} + 4^{y} \equiv 0 \pmod{11} \), where \( x^{5} \equiv \pm 1 \pmod{11} \), so we also have \( 4^{y} \equiv \pm 1 \pmod{11} \). The congruence \( 4^{y} \equiv -1 \pmod{11} \) does not hold for any \( y \), whereas \( 4^{y} \equiv 1 \pmod{11} \) holds if and only if \( 5 \mid y \).
Setti... | olympiads_ref | yoonholee/math-corpus-combined | 6 | {"dataset": "NuminaMath", "topic": "Number Theory", "problem_type": "Number Theory", "competition": null, "grade": null, "yoonholee_split": "train"} | ||
52b78d0e479fb726 | Let $S$ be the set of positive real numbers. Find all functions $f: S^{3} \rightarrow S$ such that, for all positive real numbers $x, y, z$ and $k$, the following three conditions are satisfied:
(i) $x f(x, y, z)=z f(z, y, x)$;
(ii) $f\left(x, k y, k^{2} z\right)=k f(x, y, z)$;
(iii) $f(1, k, k+1)=k+1$.
(United Kingdom... | \frac{y+\sqrt{y^{2}+4 x z}}{2 x} | It follows from the properties of function $f$ that, for all $x, y, z, a, b>0$, $f\left(a^{2} x, a b y, b^{2} z\right)=b f\left(a^{2} x, a y, z\right)=b \cdot \frac{z}{a^{2} x} f\left(z, a y, a^{2} x\right)=\frac{b z}{a x} f(z, y, x)=\frac{b}{a} f(x, y, z)$.
We shall choose $a$ and $b$ in such a way that the triple $\... | olympiads_ref | yoonholee/math-corpus-combined | 6.8 | {"dataset": "NuminaMath", "topic": "Algebra", "problem_type": "Algebra", "competition": null, "grade": null, "yoonholee_split": "train"} |
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