All Questions ⁺⁰235535 Questions

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Geometry

In right triangle $ABC,$ $\angle C = 90^\circ$. Median $\overline{AM}$ has a length of 1, and median $\overline{BN}$ has a length of 1. What is the length of the hypotenuse of the triangle?

Rangcr897 Oct 29, 2024

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+306

Algebra

Let a and b be the solutions to 7x^2 + x - 5 = -3x^2 - 9x - 4. Compute (a - 4)/(b - 4) + (b - 4)/(a - 4).

nathanl6656 Oct 29, 2024

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+306

Algebra

Seven years ago, Grogg's dad was $9$ times as old as Grogg. Four years ago, Grogg's dad was $6$ times as old as Grogg. How old is Grogg's dad currently?

nathanl6656 Oct 29, 2024

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2

+306

Algebra

Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to 28/9 times the largest of the seven integers. What is the smallest integer that Pearl wrote down?

nathanl6656 Oct 29, 2024

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+306

Algebra

Find the number of ordered pairs $(a,b)$ of integers such that
\frac{a + 2}{a + 1} = \frac{b}{8}.

nathanl6656 Oct 29, 2024

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+219

Number Theory

Find the 4000th digit following the decimal point in the expansion of \frac{1}{17}.

bIueb3rry Oct 29, 2024

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+219

Number Theory

How many bases b \ge 2 are there such that 100_b + 1_b is prime?

bIueb3rry Oct 29, 2024

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+219

Number Theory

Let $p$ be a prime. What are the possible remainders when $p$ is divided by $17?$ Select all that apply.

bIueb3rry Oct 29, 2024

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+527

Number Theory

Find the number of ordered triples (a,b,c) of positive integers that satisfy
a = \gcd(b,c) + 33
b = \gcd(a,c) + 25
c = \gcd(a,b) + 35

Pythagorearn Oct 29, 2024

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+527

Number Theory

Let a, b, c, d be distinct positive integers such that {lcm}(a, b, c, d) < 1000 and a + b + c + d = 1000. Find the largest possible value of a + b.

Pythagorearn Oct 29, 2024

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+595

Number Theory

Find the largest positive integer n for which there exist positive integers a, b, and c such that
\gcd(a + 3b, b + 3c, c + 3a) = nabc

siIviajendeukie Oct 29, 2024

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+595

Number Theory

Let N be the number of ordered 6-tuples (a_1, a_2, a_3, a_4, a_5, a_6) of positive integers that satisfy
\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \frac{1}{a_4} + \frac{1}{a_5} + \frac{1}{a_6} = 6.
Find the remainder when N is divided read more ..

siIviajendeukie Oct 29, 2024

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+595

Number Theory

Let a, b, and c be integers such that 7a + 4b = 3c. Find the largest integer that always divides $abc$.

siIviajendeukie Oct 29, 2024

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+826

Counting

Xavier and Yvonne meet one day at a cafe. For any day that Xavier goes to the cafe, the probability that Xavier goes the next day is $\frac{1}{2},$ and the probability that Xavier returns in two days is $\frac{1}{2}.$ For any day that Yvonne read more ..

RedDragonl Oct 29, 2024

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+826

Counting

Find the number of subsets S of \{0, 1, 2, 3, \dots, 15\} that have the following property: If n is in S, and m \equiv n + 1 \pmod{16}$ with $0 \le m \le 15,$ then m is also in S.

RedDragonl Oct 29, 2024

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+430

Counting

Let A and B be two fixed points in the plane such that the distance between them is $1.$ Point $P$ is chosen at random on the circle centered at $A$ with radius $1,$ and point $Q$ is chosen at random on the circle centered at $B$ with radius $1.$ read more ..

onyuIee Oct 29, 2024

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+430

Counting

In the $4 \times 4$ grid of points below, each point is one unit away from its closest neighbor. In how many ways are there to choose four of these points, such that the distance between any two chosen points is at most \sqrt{2}?

onyuIee Oct 29, 2024

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+430

Counting

Five workers $A,$ $B,$ $C,$ $D,$ and $E$ at a coffee stand are working out a schedule for the next $10$ days. The schedule must satisfy the following conditions:
* There must be exactly two workers every day, and every pair of workers must be read more ..

onyuIee Oct 29, 2024

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+140

Geometry

The sides of convex quadrilateral ABCD are AB = 8, BC = 5, $CD = 5,$ and $DA = 8.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$ so that $BE:ED = 1:1.$ Find the area of ABCD.

AUnVerifedTaxPayer Oct 29, 2024

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+140

Geometry

Let H be the orthocenter of acute triangle ABC, and let M be the midpoint of \overline{AC}. Ray $\overrightarrow{MH}$ intersects the circumcircle of triangle $ABC$ at $P.$ If $BC < AB,$ $\angle ABP = 60^\circ,$ $MH = 14,$ and $HP = 3,$ then read more ..

AUnVerifedTaxPayer Oct 29, 2024

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+479

Geometry

Triangle ABC has a right angle at B. Point D lies on side \overline{AC} such that CD = 6. The circle with diameter $\overline{CD}$ intersects $\overline{AB}$ at two distinct points, $E$ and $F,$ with $AE < AF.$ If AE = 6 and DE = 4, read more ..

ChiIIBill Oct 29, 2024

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+479

Geometry

In right triangle ABC, \angle ACB = 90^\circ, and the legs are a = BC and b = AC. Let $S$ and $T$ be points on $\overline{AC}$ and $\overline{BC},$ respectively. Let $R$ be the foot of the altitude from $S$ to $\overline{AB},$ and let $U$ be read more ..

ChiIIBill Oct 29, 2024

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+479

Geometry

In tetrahedron ABCD, AB = AC = AD = 12 and BC = BD = CD = 12. There is a sphere that is tangent to all six edges of the tetrahedron. Find the radius of this sphere.

ChiIIBill Oct 29, 2024

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+518

Algebra

Let the roots of z^{99} = 1 be z_k = x_k + iy_k for 1 \le k \le 99, where $x_k$ and $y_k$ are real. Let $P(z)$ be the monic polynomial whose roots are x_k - iy_k,$ for $1 \le k \le 99.$ Compute P(-2).

cooIcooIcooI17 Oct 29, 2024

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+666

Algebra

Find the number of positive integers n \le 10000 that satisfy
\lfloor \log_4 n \rfloor + \lfloor \log_8 n \rfloor + \lfloor \log_{64} n \rfloor = 5

booboo44 Oct 29, 2024

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