id stringlengths 16 22 | problem stringlengths 57 1.26k | answer int64 0 991 | answer_str stringlengths 3 3 | tier stringclasses 4
values |
|---|---|---|---|---|
aimepp-aime-0001 | Let $X,Y,Z>1$ and $W>1$ satisfy $\log_X W=54$, $\log_Y W=24$, and $\log_{X^{3}Y^{2}Z^{2}}W=6$. Determine $\log_Z W$. | 72 | 072 | AIME |
aimepp-aime-0002 | For real $x$ with $-8\le x\le 25$, define $F(x)=|x+4|+2|x-10|+|x-12|+2|x-21|$. Find the minimum value of $F(x)$. | 38 | 038 | AIME |
aimepp-aime-0003 | Find the product of all real roots of \[x^2-5x+15=8\sqrt{x^2-5x}\]. | 225 | 225 | AIME |
aimepp-aime-0004 | A circle has radius $25$. Two parallel chords of lengths $48$ and $30$ lie on opposite sides of the center. Find the distance between the chords. | 27 | 027 | AIME |
aimepp-aime-0005 | Complex numbers $x$ and $y$ satisfy $x^2+y^2=91$ and $x^3+y^3=836$. Find the largest real value that $x+y$ can have. | 11 | 011 | AIME |
aimepp-aime-0006 | A sequence is defined by $a_1=8$, $a_2=27$, and $a_n=3a_{n-1}+6a_{n-2}$ for $n\ge3$. Find the remainder when $a_{829}$ is divided by $1000$. | 293 | 293 | AIME |
aimepp-aime-0007 | Of $31$ people seated around a circular table, three are chosen uniformly at random. The probability that at least two chosen people were adjacent is $\frac{p}{q}$ in lowest terms. Find $p+q$. | 173 | 173 | AIME |
aimepp-aime-0008 | Find the largest two-digit prime divisor of $\binom{164}{82}$. | 97 | 097 | AIME |
aimepp-aime-0009 | Find the minimum value of $\dfrac{36x^2\sin^2x+25}{x\sin x}$ for $0<x<\pi$. | 60 | 060 | AIME |
aimepp-aime-0010 | Consider $5$-digit decimal integers beginning with $8$ that have exactly one pair of equal digits, with every other digit occurring once. Find the remainder when the number of such integers is divided by $1000$. | 40 | 040 | AIME |
aimepp-aime-0011 | A rectangular box has side lengths $23$, $25$, and $h$. The centers of the three faces meeting at one vertex form a triangle of area $\frac{579}{8}$. Find $h$. | 2 | 002 | AIME |
aimepp-aime-0012 | In a circle, a diameter has the two-digit base-$10$ length $\overline{ab}_{10}$, and a perpendicular chord has length $\overline{ba}_{10}$. The diameter is longer, the two digits sum to $11$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$. | 65 | 065 | AIME |
aimepp-aime-0013 | For every nonempty subset of $\{1,2,\ldots,14\}$, list its elements in decreasing order and alternately add and subtract, beginning with addition. Find the remainder when the sum of these alternating sums is divided by $1000$. | 688 | 688 | AIME |
aimepp-aime-0014 | In a circle, a diameter has the two-digit base-$9$ length $\overline{ab}_{9}$, and a perpendicular chord has length $\overline{ba}_{9}$. The diameter is longer, the two digits sum to $9$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$. | 65 | 065 | AIME |
aimepp-aime-0015 | In a circle, a diameter has the two-digit base-$8$ length $\overline{ab}_{8}$, and a perpendicular chord has length $\overline{ba}_{8}$. The diameter is longer, the two digits sum to $7$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$. | 35 | 035 | AIME |
aimepp-aime-0016 | The $8$ positive integer terms of an arithmetic sequence have total $420$, and the greatest term is $2$ times the least term. Find the greatest term. | 70 | 070 | AIME |
aimepp-aime-0017 | How many odd integers from $250$ through $999$ have no repeated decimal digit and have digit sum divisible by $5$? | 54 | 054 | AIME |
aimepp-aime-0018 | Find the area of a triangle with side lengths $10,13,13$. | 60 | 060 | AIME |
aimepp-aime-0019 | Positive integers $a\le b$ satisfy $\gcd(a,b)=3$ and $\operatorname{lcm}(a,b)=228$. Find the sum of all possible values of $a+b$. | 300 | 300 | AIME |
aimepp-aime-0020 | Let $X,Y,Z>1$ and $W>1$ satisfy $\log_X W=792$, $\log_Y W=352$, and $\log_{X^{3}YZ^{2}}W=96$. Determine $\log_Z W$. | 528 | 528 | AIME |
aimepp-aime-0021 | For real $x$ with $-21\le x\le 26$, define $F(x)=2|x+17|+3|x-8|+|x-18|+3|x-22|$. Find the minimum value of $F(x)$. | 102 | 102 | AIME |
aimepp-aime-0022 | A polynomial $P$ has degree at most $4$ and satisfies $P(0)=1$, $P(1)=-1$, $P(2)=-47$, $P(3)=-257$, $P(4)=-823$. Find the remainder when $P(5)$ is divided by $1000$. | 991 | 991 | AIME |
aimepp-aime-0023 | The nonreal roots of $z^2-7z+24=0$ are $\alpha$ and $\beta$. Find the remainder when $\alpha^{36}+\beta^{36}$ is divided by $1000$. | 977 | 977 | AIME |
aimepp-aime-0024 | A rectangular box has side lengths $3$, $8$, and $h$. The centers of the three faces meeting at one vertex form a triangle of area $\frac{61}{4}$. Find $h$. | 14 | 014 | AIME |
aimepp-aime-0025 | How many pairs of positive integers $(x,y)$ with $x\le y$ satisfy $\frac1x+\frac1y=\frac1{147}$? | 8 | 008 | AIME |
aimepp-aime-0026 | Three distinct vertices of a regular $10$-gon are chosen uniformly at random. The probability that they form an obtuse triangle is $\frac{m}{q}$ in lowest terms. Find $m+q$. | 3 | 003 | AIME |
aimepp-aime-0027 | How many subsets $S$ of $\{1,2,\ldots,10\}$ satisfy $\sum_{s\in S}s\equiv 4\pmod{7}$? The empty set is allowed. | 146 | 146 | AIME |
aimepp-aime-0028 | Angles $u$ and $v$ satisfy $\tan u+\tan v=34$ and $\cot u+\cot v=68$. Find $\tan(u+v)$. | 68 | 068 | AIME |
aimepp-aime-0029 | Find the smallest positive integer whose cube has last three digits $237$. | 933 | 933 | AIME |
aimepp-aime-0030 | A faculty has three departments: department 1 has 2 men and 3 women; department 2 has 3 men and 2 women; department 3 has 3 men and 3 women. A six-person committee must contain exactly two people from each department and exactly three men. How many committees are possible? | 558 | 558 | AIME |
aimepp-aime-0031 | A $6\times 5\times 9$ rectangular block is painted on all six faces and cut into unit cubes. How many unit cubes have exactly two painted faces? | 56 | 056 | AIME |
aimepp-aime-0032 | A faculty has three departments: department 1 has 2 men and 3 women; department 2 has 3 men and 2 women; department 3 has 4 men and 2 women. A six-person committee must contain exactly two people from each department and exactly three men. How many committees are possible? | 536 | 536 | AIME |
aimepp-aime-0033 | How many rectangles have all four vertices among the vertices of a regular $32$-gon? | 120 | 120 | AIME |
aimepp-aime-0034 | A polynomial $P$ has degree at most $4$ and satisfies $P(0)=-2$, $P(1)=8$, $P(2)=30$, $P(3)=40$, $P(4)=-10$. Find the remainder when $P(5)$ is divided by $1000$. | 808 | 808 | AIME |
aimepp-hard-0001 | Define $K_N=\displaystyle\prod_{j=1}^N(j!)^j$. Write $K_{236}=19^E U$, where $19\nmid U$. Let $Z$ be the least nonnegative residue of $E+U$ modulo $997$. Define $P(x)=x^{5} - 9 x^{4} + 349 x^{3} - 3 x^{2} + 7 x$ and $Q(x)=x^{4} + 692 x^{2} + 8 x - 63$. Find the least nonnegative residue modulo $1000$ of the resultant $... | 929 | 929 | AIME Hard |
aimepp-hard-0002 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of reduced fractions $a/b$ that satisfy $\frac{1}{5}<a/b<\frac{3}{5}$, $1\le b\le 554$, $\omega(b)=2$, and $a+b\equiv 1\pmod{7}$. Here $\omega(b)$ is the number of distinct prime divisors of $b$. Let $p=1361$, let $A$ and $B$ be the least nonnegative ... | 835 | 835 | AIME Hard |
aimepp-hard-0003 | A partition of $\{1,2,\ldots,9\}$ has a crossing quadruple when $a<b<c<d$, the numbers $a,c$ lie in one block, and $b,d$ lie in a different block. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of set partitions that have exactly $4$ blocks, exactly $0$ singleton blocks, and a number of crossing q... | 284 | 284 | AIME Hard |
aimepp-hard-0004 | Let $C_{253}=\frac1{254}\binom{506}{253}$ be a Catalan number. Write $C_{253}=5^e u$ with $5\nmid u$. Let $Z$ be the least nonnegative residue of $u+84e$ modulo $1000$. Write $Z=100d_2+10d_1+d_0$ with $0\le d_0,d_1,d_2\le9$. Let $\lambda=(12+d_2+d_1+d_0,8+d_1+d_0,4+d_0,4,2)$. Find the least nonnegative residue modulo $... | 192 | 192 | AIME Hard |
aimepp-hard-0005 | Consider standard Young tableaux of skew shape $(4,2,2,2,2)/(0,0,0,0,0)$. A descent is an integer $i$ for which the entry $i+1$ lies in a lower row than the entry $i$. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of such tableaux that have exactly $5$ descents. Put $L=18+(Z\bmod 13)$ and, for $1... | 24 | 024 | AIME Hard |
aimepp-hard-0006 | A tournament is formed by orienting every edge of the complete graph on $7$ labeled vertices. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of tournaments that have sorted outdegree sequence $(1, 1, 1, 4, 4, 5, 5)$ and exactly $3$ directed $3$-cycles. A Markov chain has transition matrix $\frac1{... | 771 | 771 | AIME Hard |
aimepp-hard-0007 | Let $\theta_k=\frac{2\pi k}{19}$ for $1\le k\le 18$. If $\displaystyle\sum_{k=1}^{18}\left(\frac1{(8-2\cos\theta_k)^2}+\frac{2}{8-2\cos\theta_k}\right)=\frac pq$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $m=8+(Z\bmod 9)$. Form a wheel with hub $0$ and rim vertices $1,2,\ldots... | 525 | 525 | AIME Hard |
aimepp-hard-0008 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of $5\times5$ zero-one matrices that have every row sum equal to $2$ or $3$, exactly $1$ rows of sum $3$, no zero column, rank $4$ over $\mathbb F_2$, trace $3$, an anti-diagonal sum congruent to $0$ modulo $3$, and exactly $5$ columns of odd sum. Fin... | 702 | 702 | AIME Hard |
aimepp-hard-0009 | Define $K_N=\displaystyle\prod_{j=1}^N(j!)^j$. Write $K_{195}=11^E U$, where $11\nmid U$. Let $Z$ be the least nonnegative residue of $E+U$ modulo $997$. Put $L=Z+30$. Define $G(n,0)=1$, $G(0,k)=0$ for $k>0$, and $G(n,k)=G(n-1,k)+7^{n-k}G(n-1,k-1)$. Find the least nonnegative residue modulo $1000$ of $G(L,11)+3G(L,10)$... | 208 | 208 | AIME Hard |
aimepp-hard-0010 | On an $8\times 8$ board, row $i$ contains exactly the first $b_i$ allowed squares, where $(b_1,\ldots,b_8)=(1, 2, 3, 7, 8, 8, 8, 8)$. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of ways to place $8$ nonattacking rooks, one in each row and column, so that exactly $4$ rooks lie on the main diagon... | 447 | 447 | AIME Hard |
aimepp-hard-0011 | Let $Y(x)$ be the unique formal power series with constant term $0$ satisfying $Y=x(1+Y)^{2}(1+Y^2)^{3}(1-Y)^{-1}$. Let $Z$ be the least nonnegative residue modulo $997$ of the coefficient of $x^{36}$ in $Y(x)^{10}$. Let $R$ be the least nonnegative residue modulo $10^{10}-1$ of $19^{Z^2+3Z+86}+31^{Z+86}+4510209742$, a... | 135 | 135 | AIME Hard |
aimepp-hard-0012 | Let $N=6452641$. For all coprime positive integer pairs $(x,y)$ with $x>y$ and $x^2-xy+y^2=N$, consider those satisfying $2x-y\equiv 11\pmod{13}$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of $3x+y$ over all such pairs. Define $u_0=81$ and $u_1=458$, and for $k\ge 0$ define $u_{k+2}$ by $u_{k+2}... | 545 | 545 | AIME Hard |
aimepp-hard-0013 | Let $Y(x)$ be the unique formal power series with constant term $0$ satisfying $Y=x(1+Y)^{2}(1+Y^2)^{3}(1-Y)^{-1}$. Let $Z$ be the least nonnegative residue modulo $991$ of the coefficient of $x^{43}$ in $Y(x)^{10}$. Let $p=2129$, let $A$ and $B$ be the least nonnegative residues modulo $p$ of $611Z+422$ and $1387Z+870... | 723 | 723 | AIME Hard |
aimepp-hard-0014 | Over the ring $(\mathbb Z/1000\mathbb Z)[x]$, let $c_0+c_1x+\cdots+c_6x^6$ be the remainder when $(x^2+x^4+3)^{154655378506}$ is divided by the monic polynomial $P(x)=x^{7} - 5 x^{6} - 6 x^{5} - 5 x^{4} - 2 x^{3} - 5 x - 5$. Let $Z$ be the least nonnegative residue of $9c_0+1c_1+2c_2+7c_3+5c_4+2c_5+9c_6$ modulo $1000$.... | 611 | 611 | AIME Hard |
aimepp-hard-0015 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of reduced fractions $a/b$ that satisfy $\frac{1}{5}<a/b<\frac{3}{5}$, $1\le b\le 466$, $\omega(b)=1$, and $a+b\equiv 6\pmod{13}$. Here $\omega(b)$ is the number of distinct prime divisors of $b$. Let $n=7+(Z\bmod 4)$ and let $M=(m_{ij})_{0\le i,j<n}$... | 171 | 171 | AIME Hard |
aimepp-hard-0016 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of $5\times5$ zero-one matrices that have every row sum equal to $2$ or $3$, exactly $5$ rows of sum $3$, no zero column, rank $4$ over $\mathbb F_2$, trace $2$, an anti-diagonal sum congruent to $0$ modulo $3$, and exactly $5$ columns of odd sum. Put... | 69 | 069 | AIME Hard |
aimepp-hard-0017 | An urn contains $11$ red, $11$ blue, and $10$ green balls. Balls are drawn uniformly without replacement until the number drawn of some color first reaches its threshold: $5$ for red, $4$ for blue, and $2$ for green. The probability that the stopping color is green and the total number of draws is congruent to $3$ modu... | 205 | 205 | AIME Hard |
aimepp-hard-0018 | Triangle $ABC$ has side lengths $BC=21$, $CA=15$, and $AB=16$. An interior point $P$ has barycentric coordinates $(8:7:3)$, and $Q$ is the isotomic conjugate of $P$. The perpendiculars from $Q$ to $BC,CA,AB$ meet those lines at $D,E,F$, respectively. If $([DEF]/[ABC])^2=p/q$ in lowest terms, let $Z$ be the least nonneg... | 376 | 376 | AIME Hard |
aimepp-hard-0019 | For a positive integer $t$, let $J_2(t)=t^2\prod_{p\mid t}(1-p^{-2})$. Let $N=780$. Let $Z$ be the least nonnegative residue modulo $997$ of $\displaystyle\sum (d+1)J_2(N/d)$, where the sum is over squarefree divisors $d$ of $N$ such that $d\equiv 1\pmod{11}$ and $\Omega(d)\equiv 1\pmod2$. A Markov chain has transition... | 115 | 115 | AIME Hard |
aimepp-hard-0020 | The sequence $(a_n)$ is defined by $(a_0,a_1,\ldots,a_5)=(17, 16, -12, -18, 13, 10)$ and $a_{n+6}=8a_{n+5}-2a_{n+4}+7a_{n+3}-6a_{n+2}-5a_{n+1}+8a_n$ for $n\ge0$. Let $M$ be the $5\times5$ matrix whose $(i,j)$ entry is $a_{8798202908543+i+j}-5a_{8798202908543+i+j+1}$, where $0\le i,j\le4$. Let $Z$ be the least nonnegati... | 475 | 475 | AIME Hard |
aimepp-hard-0021 | The sequence $(a_n)$ is defined by $(a_0,a_1,\ldots,a_5)=(-18, -13, 14, 11, -13, -9)$ and $a_{n+6}=5a_{n+5}+7a_{n+4}+8a_{n+3}-8a_{n+2}+7a_{n+1}+3a_n$ for $n\ge0$. Let $M$ be the $5\times5$ matrix whose $(i,j)$ entry is $a_{2431472563692+i+j}-1a_{2431472563692+i+j+1}$, where $0\le i,j\le4$. Let $Z$ be the least nonnegat... | 967 | 967 | AIME Hard |
aimepp-hard-0022 | Let $N=38909$. Among all pairs of positive integers $(x,y)$ satisfying $x<y$, $x^2+y^2=N$, and $\gcd(x,y)=1$, retain only those for which $x+2y\equiv 1\pmod{13}$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\displaystyle\sum(2x+3y)$ over the retained pairs. Put $L=Z+50$. Let $S(n,k)$ be a Stirling number... | 65 | 065 | AIME Hard |
aimepp-hard-0023 | For $0\le i,j\le5$, define $m_{ij}=\binom{29+i+j}{23+2i-j}+3\binom{29+i+j}{23-i+2j}+3\binom{29+i+j}{23+i-j-2}$, where a binomial coefficient is $0$ when its lower index is outside its usual range. If $M=(m_{ij})_{0\le i,j\le5}$, let $Z$ be the least nonnegative residue of $\det M$ modulo $991$. A Markov chain has trans... | 149 | 149 | AIME Hard |
aimepp-hard-0024 | On an $8\times 8$ board, row $i$ contains exactly the first $b_i$ allowed squares, where $(b_1,\ldots,b_8)=(2, 3, 6, 8, 8, 8, 8, 8)$. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of ways to place $8$ nonattacking rooks, one in each row and column, so that exactly $0$ rooks lie on the main diagon... | 145 | 145 | AIME Hard |
aimepp-hard-0025 | Let $C_{267}=\frac1{268}\binom{534}{267}$ be a Catalan number. Write $C_{267}=5^e u$ with $5\nmid u$. Let $Z$ be the least nonnegative residue of $u+28e$ modulo $1000$. Put $L=Z+70$. Let $S(n,k)$ be a Stirling number of the second kind, and let $A(n,j)$ be the number of permutations of $\{1,\ldots,n\}$ with exactly $j$... | 900 | 900 | AIME Hard |
aimepp-hard-0026 | For $1\le n\le 72793$, set $a_n=\lfloor n\sqrt{12}\rfloor$ and $b_n=\lfloor n/\sqrt{12}\rfloor$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all $n$ such that $a_n\equiv 2\pmod{11}$, $b_n\equiv 4\pmod{6}$, and $n\equiv 1\pmod2$. Let $L=Z+52$ and $P=\displaystyle\prod_{j=1}^L(8j+7)$. Write $P=5^... | 546 | 546 | AIME Hard |
aimepp-hard-0027 | Let $p=3079$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$11$ digits is congruent to $5$ modulo $11$, and the sum of the base-$11$ digits of its least positive inverse modulo $p$ is congruent to $9$ modulo $10$. Let $Z$ be the least nonnegative residue mod... | 460 | 460 | AIME Hard |
aimepp-hard-0028 | Two walkers move simultaneously on the lattice, each taking one unit north or east step per second. The lower walker goes from $(0,0)$ to $(9,11)$, and the upper walker goes from $(0,2)$ to $(9,13)$. After every equal number of steps, their current vertices must be distinct. A turn is a change of direction by either wa... | 55 | 055 | AIME Hard |
aimepp-hard-0029 | Let $N=30240$. Consider nondecreasing $4$-tuples of integers $2\le a_1\le\cdots\le a_4$ whose product is $N$. Among those having exactly $2$ even entries, greatest common divisor $1$, and $a_1+\cdots+a_4\equiv 4\pmod{6}$, let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all possible values of $a_4$.... | 17 | 017 | AIME Hard |
aimepp-hard-0030 | A sphere $\Sigma$ is orthogonal to four spheres having, respectively, center $(-8,5,-7)$ and squared radius $\frac{221}{2}$; center $(-5,-1,4)$ and squared radius $\frac{133}{2}$; center $(-6,8,-6)$ and squared radius $\frac{177}{2}$; and center $(1,3,-5)$ and squared radius $\frac{11}{2}$. If the center of $\Sigma$ is... | 413 | 413 | AIME Hard |
aimepp-hard-0031 | In triangle $ABC$, the side lengths opposite $A,B,C$ are $26,32,12$, respectively. An interior point $P$ has barycentric coordinates $(5:2:2)$, and $Q$ is the isogonal conjugate of $P$. If $PQ^2=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. In the quotient ring $(\mathbb Z/1000\... | 9 | 009 | AIME Hard |
aimepp-hard-0032 | Let $r_1,r_2,r_3,r_4$ be the roots of $P(x)=x^{4} + 7 x^{3} - 3 x^{2} - 3 x + 7$. For $i<j$ define $u_{ij}=r_ir_j+4$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\displaystyle\sum_{i<j}\left(u_{ij}^{1282}-7u_{ij}^{1281}\right)$. Let $L=Z+47$ and $P=\displaystyle\prod_{j=1}^L(16j+9)$. Write $P=5^eU$ with ... | 945 | 945 | AIME Hard |
aimepp-hard-0033 | Write the base-$7$ expansion of $1/83$ as $0.\overline{d_1d_2\cdots d_h}_{7}$ with the shortest possible period, and read subscripts cyclically modulo $h$. For each $j$, let $B_j$ be the base-$7$ integer with digits $d_j,d_{j+1},\ldots,d_{j+6}$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all $... | 828 | 828 | AIME Hard |
aimepp-hard-0034 | Consider ordered pairs $(x,y)$ of least positive residues modulo $1296$ satisfying $xy\equiv 701\pmod{1296}$, $x+y\equiv 8\pmod{14}$, and $x+2y\equiv 4\pmod{5}$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\displaystyle\sum(x+3y)$ over all such pairs. Find the least nonnegative residue modulo $1000$ of t... | 124 | 124 | AIME Hard |
aimepp-hard-0035 | Let $N=329509$. For all coprime positive integer pairs $(x,y)$ with $x>y$ and $x^2-xy+y^2=N$, consider those satisfying $2x-y\equiv 7\pmod{9}$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of $3x+y$ over all such pairs. A Markov chain has transition matrix $\frac1{10}\begin{pmatrix}4 & 2 & 2 & 2\\2... | 887 | 887 | AIME Hard |
aimepp-hard-0036 | Let $C_{315}=\frac1{316}\binom{630}{315}$ be a Catalan number. Write $C_{315}=3^e u$ with $3\nmid u$. Let $Z$ be the least nonnegative residue of $u+81e$ modulo $1000$. Define $u_0=402$ and $u_1=422$, and for $k\ge 0$ define $u_{k+2}$ by $u_{k+2}\equiv 19u_{k+1}-17u_k+7(-1)^k\pmod{1000}$, with every $u_k$ chosen from $... | 192 | 192 | AIME Hard |
aimepp-hard-0037 | Let $p=2017$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$10$ digits is congruent to $7$ modulo $9$, and the sum of the base-$10$ digits of its least positive inverse modulo $p$ is congruent to $3$ modulo $9$. Let $Z$ be the least nonnegative residue modul... | 759 | 759 | AIME Hard |
aimepp-hard-0038 | A sphere $\Sigma$ is orthogonal to four spheres having, respectively, center $(7,5,4)$ and squared radius $\frac{521}{4}$; center $(6,3,7)$ and squared radius $\frac{617}{4}$; center $(-6,-3,-6)$ and squared radius $\frac{133}{4}$; and center $(-6,4,7)$ and squared radius $\frac{441}{4}$. If the center of $\Sigma$ is $... | 800 | 800 | AIME Hard |
aimepp-hard-0039 | Over the ring $(\mathbb Z/1000\mathbb Z)[x]$, let $c_0+c_1x+\cdots+c_6x^6$ be the remainder when $(x^2+x^3-2)^{110460296634}$ is divided by the monic polynomial $P(x)=x^{7} + x^{6} + 4 x^{5} - 4 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 2$. Let $Z$ be the least nonnegative residue of $1c_0+7c_1+4c_2+4c_3+9c_4+2c_5+7c_6$ modulo... | 864 | 864 | AIME Hard |
aimepp-hard-0040 | Over the ring $(\mathbb Z/1000\mathbb Z)[x]$, let $c_0+c_1x+\cdots+c_6x^6$ be the remainder when $(x^1+x^5+2)^{267703890501}$ is divided by the monic polynomial $P(x)=x^{7} + 5 x^{6} + 6 x^{5} - 6 x^{4} + 2 x^{3} - 5 x^{2} + x + 4$. Let $Z$ be the least nonnegative residue of $9c_0+2c_1+2c_2+4c_3+7c_4+1c_5+4c_6$ modulo... | 720 | 720 | AIME Hard |
aimepp-hard-0041 | Let $\theta_k=\frac{2\pi k}{9}$ for $1\le k\le 8$. If $\displaystyle\sum_{k=1}^{8}\left(\frac1{(8-2\cos\theta_k)^2}+\frac{4}{8-2\cos\theta_k}\right)=\frac pq$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $p=1889$, set $x_0=Z$, and for $k\ge0$ let $x_{k+1}$ be the least nonnegati... | 923 | 923 | AIME Hard |
aimepp-hard-0042 | A plane partition in an $3\times 5\times 4$ box is a $3\times 5$ array $(p_{ij})$ of integers from $0$ through $4$ that is weakly decreasing across every row and down every column. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of such arrays that have total sum congruent to $2$ modulo $5$, diagon... | 36 | 036 | AIME Hard |
aimepp-hard-0043 | A simple graph on $6$ labeled vertices is chosen uniformly from all connected such graphs. The probability that it has exactly $2$ vertices of odd degree and exactly $4$ triangles is $p/q$ in lowest terms. Let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Find the least nonnegative residue modulo $1000$ ... | 13 | 013 | AIME Hard |
aimepp-hard-0044 | The eight vertices of a cube are colored red, green, or blue, using the colors exactly $2, 5, 1$ times, respectively. Two colorings are considered the same when a rotation of the cube carries one to the other. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of equivalence classes that have exactly ... | 276 | 276 | AIME Hard |
aimepp-hard-0045 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of permutations $\pi$ of $\{1,2,\ldots,9\}$ that have exactly $3$ cycles in their disjoint-cycle decomposition, exactly $2$ excedances, exactly $1$ fixed points, and major index congruent to $6$ modulo $7$. Put $L=18+(Z\bmod 13)$ and, for $1\le j\le L... | 743 | 743 | AIME Hard |
aimepp-hard-0046 | Triangle $ABC$ has side lengths $BC=17$, $CA=12$, and $AB=26$. An interior point $P$ has barycentric coordinates $(6:8:7)$, and $Q$ is the isotomic conjugate of $P$. The perpendiculars from $Q$ to $BC,CA,AB$ meet those lines at $D,E,F$, respectively. If $([DEF]/[ABC])^2=p/q$ in lowest terms, let $Z$ be the least nonneg... | 687 | 687 | AIME Hard |
aimepp-hard-0047 | A partition of $\{1,2,\ldots,9\}$ has a crossing quadruple when $a<b<c<d$, the numbers $a,c$ lie in one block, and $b,d$ lie in a different block. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of set partitions that have exactly $6$ blocks, exactly $5$ singleton blocks, and a number of crossing q... | 508 | 508 | AIME Hard |
aimepp-hard-0048 | In triangle $ABC$, the side lengths opposite $A,B,C$ are $33,33,13$, respectively. An interior point $P$ has barycentric coordinates $(5:4:3)$, and $Q$ is the isogonal conjugate of $P$. If $PQ^2=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $A=\begin{pmatrix}3 & 4 & 1 & 4\\2... | 295 | 295 | AIME Hard |
aimepp-hard-0049 | For a positive integer $t$, let $J_2(t)=t^2\prod_{p\mid t}(1-p^{-2})$. Let $N=2012472$. Let $Z$ be the least nonnegative residue modulo $997$ of $\displaystyle\sum (d+1)J_2(N/d)$, where the sum is over squarefree divisors $d$ of $N$ such that $d\equiv 6\pmod{13}$ and $\Omega(d)\equiv 0\pmod2$. Let $A=\begin{pmatrix}0 &... | 334 | 334 | AIME Hard |
aimepp-hard-0050 | Let $a_n$ be defined by $\displaystyle\sum_{n\ge0}a_nx^n=\frac{(1+x^8)^{5}(1+x^7+x^{14})^{5}}{(1-x^2)^{3}(1-x^5)^{2}}$. Let $Z$ be the least nonnegative residue modulo $983$ of $\displaystyle\sum_{\substack{0\le n\le 116\n\equiv 6\pmod{9}}}a_n$. Let $n=7+(Z\bmod 4)$ and let $M=(m_{ij})_{0\le i,j<n}$, where $m_{ij}$ is ... | 720 | 720 | AIME Hard |
aimepp-hard-0051 | The graph $C_5\square P_15$ has $15$ cyclic rows of $5$ vertices; adjacent vertices in each row are joined, including the wraparound pair, and corresponding vertices in consecutive rows are joined. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of independent sets that contain exactly $5$ vertices... | 36 | 036 | AIME Hard |
aimepp-hard-0052 | A tetrahedron $ABCD$ has squared edge lengths $AB^2=36$, $AC^2=128$, $AD^2=54$, $BC^2=68$, $BD^2=102$, and $CD^2=230$. Let $V$ be its volume and $R$ its circumradius. Write $R^2=p/q$ in lowest terms. Let $Z$ be the least nonnegative residue modulo $1000$ of $288V^2+p+q$. Let $R$ be the least nonnegative residue modulo ... | 30 | 030 | AIME Hard |
aimepp-hard-0053 | Let $p=5843$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$5$ digits is congruent to $4$ modulo $10$, and the sum of the base-$5$ digits of its least positive inverse modulo $p$ is congruent to $5$ modulo $7$. Let $Z$ be the least nonnegative residue modulo... | 320 | 320 | AIME Hard |
aimepp-hard-0054 | A tetrahedron $ABCD$ has squared edge lengths $AB^2=49$, $AC^2=208$, $AD^2=116$, $BC^2=145$, $BD^2=109$, and $CD^2=68$. Let $V$ be its volume and $R$ its circumradius. Write $R^2=p/q$ in lowest terms. Let $Z$ be the least nonnegative residue modulo $1000$ of $288V^2+p+q$. Define $u_0=616$ and $u_1=471$, and for $k\ge 0... | 536 | 536 | AIME Hard |
aimepp-hard-0055 | Let $r_1,r_2,r_3,r_4$ be the complex roots, counted with multiplicity, of $P(x)=x^{4} + x^{3} + x^{2} - 3 x + 3$. For the six numbers $s_{ij}=r_i+r_j$ with $1\le i<j\le4$, set $T=\displaystyle\sum_{i<j}\left(\frac{1}{(11-s_{ij})^2}+4\frac{1}{11-s_{ij}}\right)$. If $T=\frac pq$ in lowest terms with $q>0$, let $Z$ be the... | 234 | 234 | AIME Hard |
aimepp-hard-0056 | For $0\le i,j\le5$, define $m_{ij}=\binom{37+i+j}{20+2i-j}+4\binom{37+i+j}{20-i+2j}+2\binom{37+i+j}{20+i-j-2}$, where a binomial coefficient is $0$ when its lower index is outside its usual range. If $M=(m_{ij})_{0\le i,j\le5}$, let $Z$ be the least nonnegative residue of $\det M$ modulo $991$. Write $Z=100d_2+10d_1+d_... | 0 | 000 | AIME Hard |
aimepp-hard-0057 | Let $p=1579$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$11$ digits is congruent to $6$ modulo $11$, and the sum of the base-$11$ digits of its least positive inverse modulo $p$ is congruent to $3$ modulo $6$. Let $Z$ be the least nonnegative residue modu... | 0 | 000 | AIME Hard |
aimepp-hard-0058 | The sequence $(a_n)$ is defined by $(a_0,a_1,\ldots,a_5)=(-3, 14, -17, -17, 3, 13)$ and $a_{n+6}=8a_{n+5}-5a_{n+4}+2a_{n+2}+7a_{n+1}+a_n$ for $n\ge0$. Let $M$ be the $5\times5$ matrix whose $(i,j)$ entry is $a_{3800044329232+i+j}-3a_{3800044329232+i+j+1}$, where $0\le i,j\le4$. Let $Z$ be the least nonnegative residue ... | 573 | 573 | AIME Hard |
aimepp-hard-0059 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of length-$42$ words $a_1a_2\cdots a_42$ over $\{0,1,2,3\}$ that contain none of the blocks $00$, $123$, or $232$, contain exactly $38$ occurrences of $3$, satisfy $\sum_{i=1}^42 i a_i\equiv 3\pmod7$, and have a number of indices with $a_{i+1}>a_i$ co... | 133 | 133 | AIME Hard |
aimepp-hard-0060 | Consider ordered pairs $(x,y)$ of least positive residues modulo $12960$ satisfying $xy\equiv 5729\pmod{12960}$, $x+y\equiv 6\pmod{8}$, and $x+2y\equiv 2\pmod{9}$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\displaystyle\sum(x+3y)$ over all such pairs. Put $L=18+(Z\bmod 13)$ and, for $1\le j\le L$, put ... | 534 | 534 | AIME Hard |
aimepp-hard-0061 | The sequence $(a_n)$ is defined by $(a_0,a_1,\ldots,a_5)=(16, -18, 18, -17, -16, 16)$ and $a_{n+6}=-8a_{n+5}+4a_{n+2}-7a_{n+1}-a_n$ for $n\ge0$. Let $M$ be the $5\times5$ matrix whose $(i,j)$ entry is $a_{917841055891+i+j}-4a_{917841055891+i+j+1}$, where $0\le i,j\le4$. Let $Z$ be the least nonnegative residue of $\det... | 910 | 910 | AIME Hard |
aimepp-hard-0062 | Let $N=1440$. Consider nondecreasing $4$-tuples of integers $2\le a_1\le\cdots\le a_4$ whose product is $N$. Among those having exactly $2$ even entries, greatest common divisor $1$, and $a_1+\cdots+a_4\equiv 7\pmod{9}$, let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all possible values of $a_4$. ... | 228 | 228 | AIME Hard |
aimepp-hard-0063 | Let $C_{243}=\frac1{244}\binom{486}{243}$ be a Catalan number. Write $C_{243}=11^e u$ with $11\nmid u$. Let $Z$ be the least nonnegative residue of $u+67e$ modulo $1000$. Let $M=3080$ and let $R$ be the least nonnegative residue modulo $M$ of $(10Z+239)^2+1(10Z+239)$. Let $\mathcal S$ be the set of residues $z\in\{0,1,... | 236 | 236 | AIME Hard |
aimepp-hard-0064 | Let $p=4597$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$8$ digits is congruent to $1$ modulo $5$, and the sum of the base-$8$ digits of its least positive inverse modulo $p$ is congruent to $0$ modulo $10$. Let $Z$ be the least nonnegative residue modulo... | 30 | 030 | AIME Hard |
aimepp-hard-0065 | In triangle $ABC$ with $BC=36$, $CA=36$, and $AB=31$, points $D,E,F$ lie on $BC,CA,AB$, respectively, and satisfy $BD:DC=2:4$, $CE:EA=1:5$, and $AF:FB=3:3$. The circles $(ADE)$, $(BEF)$, and $(CFD)$ have radical center $X$. If the power of $X$ with respect to the circumcircle of $ABC$ is $p_0/q_0$ in lowest terms with ... | 590 | 590 | AIME Hard |
aimepp-hard-0066 | For a positive integer $t$, let $J_2(t)=t^2\prod_{p\mid t}(1-p^{-2})$. Let $N=26583700$. Let $Z$ be the least nonnegative residue modulo $997$ of $\displaystyle\sum (d+1)J_2(N/d)$, where the sum is over squarefree divisors $d$ of $N$ such that $d\equiv 0\pmod{10}$ and $\Omega(d)\equiv 1\pmod2$. Write $Z=100d_2+10d_1+d_... | 320 | 320 | AIME Hard |
End of preview. Expand in Data Studio
AIME++ Sample
AIME++ is Ulam AI's exact-answer mathematical reasoning environment. It keeps one of the most useful properties of AIME-style evaluation—a compact, deterministic answer in the integer range 0–999—and extends it across four levels of mathematical depth, from competition-style problems to research-level challenges.
This repository contains a 157-problem, MIT-licensed sample of Ulam AI's much larger problem catalog. Every problem has a canonical integer answer and a zero-padded three-digit representation, making evaluation inexpensive, reproducible, and free of judge-model variance.
Ulam AI's broader database contains 100,000+ problems. The commercially available AIME++ collections include 24,700+ AIME-family problems spanning AIME and AIME Hard, 1,000+ AIME-Graduate problems, and 100+ AIME-Researcher problems. This repository is designed to let teams inspect the format and difficulty range before licensing a production-scale collection.
Name clarification: In this dataset, AIME means AI Mathematical Environment. “AIME-style” describes the
0–999answer format. This project is not affiliated with or endorsed by the Mathematical Association of America or its competitions, and the sample does not claim to contain official competition problems.
Why AIME++
- Verifier-friendly: exact-match rewards are deterministic and require no subjective rubric or model judge.
- One interface, four depths: the answer contract stays fixed while the mathematical demands increase.
- Evaluation-ready: stable IDs, explicit tiers, a versioned JSON Schema, and a reference scorer are included.
- Useful for capability profiling: results can be compared overall and per tier without conflating output-format changes with problem difficulty.
- Frictionless sample: the MIT license allows teams to test the data in their own training and evaluation stacks before discussing a larger license.
The dataset is intentionally answer-only: each record contains a problem and its authoritative golden answer in 0–999, not a worked derivation or chain-of-thought trace. For the AIME++ task, that final answer is the golden solution. A different normalized integer is incorrect.
Difficulty tiers
| Config | Tier | Description | Records |
|---|---|---|---|
aime |
AIME | Standard AIME-style mathematical problems | 34 |
aime-hard |
AIME Hard | Harder AIME-style mathematical problems | 98 |
aime-graduate |
AIME-Graduate | Graduate-level problems with AIME-style answers | 20 |
aime-researcher |
AIME-Researcher | Research-level problems with AIME-style answers | 5 |
all |
All four tiers | Default combined evaluation config | 157 |
All configurations expose a single test split. This is an evaluation sample, not a train/test partition.
From sample to full catalog
| Collection | This sample | Larger Ulam collection | Scope |
|---|---|---|---|
| AIME family | 132 | 24,700+ | Standard and harder AIME-style problems |
| AIME-Graduate | 20 | 1,000+ | Graduate-level mathematics with AIME-style answers |
| AIME-Researcher | 5 | 100+ | Research-level mathematics with AIME-style answers |
| Broader Ulam problem database | — | 100,000+ | Mathematical reasoning problems across Ulam collections |
The sample exposes only a small fraction of the commercial inventory while providing enough material to test parsing, training, RLVR rewards, evaluation code, and tier-level behavior.
Quick start
When the repository is published on Hugging Face, load the complete sample with:
from datasets import load_dataset
dataset = load_dataset("ulamai/AIME-Plus-Plus", "all", split="test")
print(dataset[0])
Load a single tier by replacing all with aime, aime-hard, aime-graduate, or aime-researcher.
The release files can also be loaded before publication:
from datasets import load_dataset
dataset = load_dataset(
"json",
data_files={"test": "data/*.jsonl"},
split="test",
)
Data schema
Each JSONL row has five fields:
| Field | Type | Description |
|---|---|---|
id |
string | Stable identifier, such as aimepp-aime-0001 |
problem |
string | English problem statement with LaTeX markup |
answer |
integer | Canonical answer in 0–999 |
answer_str |
string | The same answer zero-padded to exactly three digits |
tier |
string | One of the four human-readable difficulty tiers |
Example:
{
"id": "aimepp-aime-0001",
"problem": "Let $X,Y,Z>1$ and $W>1$ satisfy $\\log_X W=54$, $\\log_Y W=24$, and $\\log_{X^{3}Y^{2}Z^{2}}W=6$. Determine $\\log_Z W$.",
"answer": 72,
"answer_str": "072",
"tier": "AIME"
}
The normative record definition is schema/dataset.schema.json.
Evaluation
Create a JSONL file with one prediction per problem:
{"id":"aimepp-aime-0001","prediction":"072"}
{"id":"aimepp-aime-0002","prediction":314}
Then run:
python3 scripts/score.py predictions.jsonl
Use --config aime-hard (or another config name) to score only one tier. The default protocol accepts an integer or a string containing only a one-to-three-digit integer. 72 and "072" are equivalent. Missing, malformed, or out-of-range predictions are incorrect. The scorer reports overall and per-tier accuracy, always using the full selected gold set as the denominator.
For systems that emit worked reasoning, --allow-boxed also accepts the last \boxed{N} in a string. Report which parsing mode was used whenever publishing results.
Intended uses
- Exact-answer evaluation of mathematical reasoning systems
- Reinforcement learning with deterministic verifiable rewards
- Test-time-compute and inference-strategy comparisons
- Per-tier capability profiling and regression testing
- Technical inspection by prospective data or evaluation partners
Out-of-scope uses
- Treating this public-answer sample as a hidden or contamination-resistant benchmark
- Claiming broad mathematical, scientific, or safety capability from this sample alone
- Comparing scores produced with different prompts, tool policies, budgets, or answer parsers as if they were directly equivalent
Creation and rights
All problems in this sample were created internally by Ulam AI, and Ulam AI holds the rights to the dataset. The records are original Ulam AI material rather than official competition questions. The dataset does not contain personal data or user-contributed content.
Worked derivations and reasoning traces are not part of this dataset's product format. The final integer supplied with each problem is its golden solution and the normative target used by the scorer.
Quality and validation
The packaged files pass deterministic checks for schema conformance, stable and unique IDs, non-empty problems, answer range, answer-string consistency, duplicate problems, control characters, and balanced dollar-sign LaTeX delimiters.
Run the checks locally with no third-party dependencies:
make validate
See QUALITY_REPORT.md for measured results. The supplied answers are the authoritative ground truth for AIME++ exact-match evaluation.
Reproducible reporting
Every reported result should include:
- dataset version or immutable commit hash;
- config and record count;
- system and model version;
- prompt template;
- tool-access policy;
- sampling parameters and number of attempts;
- token or compute budget;
- strict or boxed answer parsing mode; and
- overall plus per-tier accuracy.
Once answers are distributed, those records should be treated as inspection, development, or training data rather than a private holdout. Ulam AI can create separately governed evaluation material for commercial partners.
License and commercial access
The sample is distributed under the permissive MIT License, allowing teams to inspect, evaluate, train on, modify, and redistribute the sample subject to the license terms.
The larger 24,700+ AIME-family, 1,000+ AIME-Graduate, and 100+ AIME-Researcher collections are available separately, as are custom difficulty mixes and private evaluation services. See docs/COMMERCIAL_ACCESS.md or visit ulam.ai.
Citation
Citation metadata is provided in CITATION.cff. Until a paper or technical report is published, cite the dataset by organization, title, version, and repository URL.
Version
This package is version 0.1.0. See CHANGELOG.md for release notes.
Copyright © 2026 Ulam AI.
- Downloads last month
- -