id int64 2 3.28M | problem stringlengths 27 6.88k | solution stringlengths 2 38.6k | problem_vector listlengths 1.02k 1.02k | solution_vector listlengths 1.02k 1.02k | last_modified stringdate 2025-08-13 00:00:00 2025-08-13 00:00:00 |
|---|---|---|---|---|---|
2,759,383 | Given an integer $n\geq2$, let $x_1<x_2<\cdots<x_n$ and $y_1<y_2<\cdots<y_n$ be positive reals. Prove that for every value $C\in (-2,2)$ (by taking $y_{n+1}=y_1$) it holds that
$\hspace{122px}\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_i+y_i^2}<\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_{i+1}+y_{i+1}^2}$.
[i]Proposed by Mirko Petrusevski[/... | We can use a similar argument as in the proof of rearrangement inequality. Letting $f(x,y)=\sqrt{x^2+Cxy+y^2}$, it suffices to show the case $n=2$, which corresponds to a single transposition in the general case.
Al we have to show that if $a<b$ and $c<d$, then
$$\sqrt{a^2+Cac+c^2}+\sqrt{b^2+Cbd+d^2}<\sqrt{a^2+Cad+d^2... | [
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-0.049... | [
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0.12027298659086227,
0.08431415259838104,
0.03226931393146515,
-0.0005405861884355545,
... | 2025-08-13 |
2,759,385 | Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$.
[i]Proposed by Viktor Simjanoski[/i] | [hide="Solution (using Vieta's Jumping Root Method)"]
$\wedge$ means 'and'.
$\mathbb{N*}$ means $\{n | n\in \mathbb{Z} \wedge n>0\}$.
$(*)$ stands for the equation $x^2+y^2+x+y+z=xyz+1$.
Define $g(x,y):=\frac{x^2+y^2+x+y-1}{xy-1}$.
WLOG assume $x\geq y$.
$\textbf{Case 1.}$ $y=1$.
$x^2+x+z+2=xz+1 \Rightarrow (x-1)z=x^2+... | [
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-... | [
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-0.... | 2025-08-13 |
2,759,387 | Find all positive integers $n$ such that the set $S=\{1,2,3, \dots 2n\}$ can be divided into $2$ disjoint subsets $S_1$ and $S_2$, i.e. $S_1 \cap S_2 = \emptyset$ and $S_1 \cup S_2 = S$, such that each one of them has $n$ elements, and the sum of the elements of $S_1$ is divisible by the sum of the elements in $S_2$.
... | We claim the answer is all $n \not\equiv 5 \pmod 6$. Let $\sum_{i \in S_1} i=A$ and $\sum_{i \in S_2} i=B$. Then, $A+B=n(2n+1)$ and $A \mid B$. Note that
$A \geq 1+2+\ldots+n=\dfrac{n(n+1)}{2}$
and
$B \leq 2n+(2n-1)+\ldots+(n+1)=\dfrac{n(3n+1)}{2}.$
Therefore,
$B \leq \dfrac{n(3n+1)}{2} <\dfrac{3n(n+1)}{2} \leq 3... | [
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-0.... | [
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0.03005078062415123,
0.015332605689764023,
-0... | 2025-08-13 |
2,759,392 | We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with strictly more $1$'s than $0$'s (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$'s than $0$'s). Are there infinitely many memorable perfect squares?
[i]Proposed by Nikola Velov[/i] | $n^2=2^k \cdot a_k + ... + 2^1 \cdot a_1 + 2^0 a_0$
Next number$$\boxed {(2^{k+2} + 1)n}$$ | [
0.05805108696222305,
0.005184285808354616,
0.009502636268734932,
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0.0672762468457222,
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-0.00668... | [
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-0.02901... | 2025-08-13 |
2,777,211 | "Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients of $2$ and $-2$, respectively. Th(...TRUNCATED) | "Mine.\n\n[hide=\"Solution\"]The polynomial $R(x) := P(x) + Q(x)$ is linear, and we may compute $R(1(...TRUNCATED) | [-0.002993835136294365,0.019381651654839516,0.021293753758072853,-0.022528525441884995,-0.0142195168(...TRUNCATED) | [-0.00018698335043154657,-0.02624564617872238,-0.001497828052379191,-0.0066256532445549965,-0.038050(...TRUNCATED) | 2025-08-13 |
2,777,219 | "Let $w = \\frac{\\sqrt{3}+i}{2}$ and $z=\\frac{-1+i\\sqrt{3}}{2}$, where $i=\\sqrt{-1}$. Find the n(...TRUNCATED) | "[hide=Sketch]Rewrite in exponential form using $e^{i\\theta}=\\cos{\\theta}+i\\sin{\\theta}$. Take (...TRUNCATED) | [-0.01794128492474556,0.004107029177248478,0.01799754798412323,-0.030381303280591965,-0.006243055220(...TRUNCATED) | [0.0031980194617062807,-0.0026694494299590588,0.011544609442353249,-0.06636080145835876,-0.041016101(...TRUNCATED) | 2025-08-13 |
2,777,232 | "Let $ABCD$ be a parallelogram with $\\angle BAD < 90^{\\circ}$. A circle tangent to sides $\\overli(...TRUNCATED) | "Mine.\n\n[hide=\"Solution\"]\nLet $X$, $Y$, and $Z$ denote the tangency points of $\\omega$ with $A(...TRUNCATED) | [-0.030748888850212097,-0.016265293583273888,0.03184623271226883,-0.009412916377186775,-0.0220094509(...TRUNCATED) | [0.012515556998550892,-0.008496584370732307,0.013024513609707355,-0.012677568010985851,-0.0312300808(...TRUNCATED) | 2025-08-13 |
2,782,948 | "Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A l(...TRUNCATED) | "Of the three labels $a<b<c$ at the vertices of such a triangle, let $p_1=b-a$ be the distance betwe(...TRUNCATED) | [-0.015248993411660194,-0.004060782957822084,-0.007858255878090858,-0.015608984977006912,-0.00629547(...TRUNCATED) | [0.01415644958615303,-0.021235832944512367,0.04215333238244057,-0.007219011429697275,-0.022813443094(...TRUNCATED) | 2025-08-13 |
2,793,705 | "Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along the wire. ((...TRUNCATED) | "The probability of placing the birds on a $10$-meter wire such that they are all more than a meter (...TRUNCATED) | [0.0270835068076849,0.03395358845591545,-0.004546529147773981,-0.02390521951019764,0.001757429679855(...TRUNCATED) | [0.019506897777318954,0.05267248675227165,-0.0015969660598784685,-0.0226998720318079,-0.021289106458(...TRUNCATED) | 2025-08-13 |
2,352,936 | "For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\\frac{(...TRUNCATED) | "[b](a)[/b] $a_{n+1} =\\frac{a_n +2}{a_n + 1}$\n$\\frac43 \\le \\frac{a_1 +2}{a_1 + 1} \\le \\frac32(...TRUNCATED) | [0.015385467559099197,0.03444742038846016,0.032713450491428375,-0.009648566134274006,0.0181335508823(...TRUNCATED) | [0.011885537765920162,0.0012671657605096698,-0.006396659649908543,0.013905386440455914,0.02335941977(...TRUNCATED) | 2025-08-13 |
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