= Solution
It is enough to assume that $h_k:\pi_k(X)\otimes\mathbb Q\to H_k(X;\mathbb Q)$ is surjective for every $k\ge2$; an integrally surjective <Hurewicz homomorphism> certainly has this property. First give the minimal-model argument suggested by the hint. Let $A=(\Lambda V,d)$ be a <simply connected> <Sullivan minimal model>, with $V$ in degrees at least two. Put $A^+=\bigoplus_{k>0}A^k$. Its <indecomposable quotient of an augmented algebra> is
$$
A^+/(A^+)^2\cong V.
$$
Minimality makes the induced differential on $V$ zero, so projection is a <chain map> $q:A^+\to(V,0)$. The Sullivan model identification says that the induced map
$$
q_*:\widetilde H^*(X;\mathbb Q)\longrightarrow V
$$
is dual to rational <Hurewicz homomorphism>. Under the usual finite-type duality convention, the assumed surjectivity makes $q_*$ injective.
A product of two positive <cohomology classes> has a decomposable cocycle representative, so its image under $q_*$ is zero. Injectivity therefore gives
$$
\boxed{\widetilde H^{>0}(X;\mathbb Q)\cdot
\widetilde H^{>0}(X;\mathbb Q)=0.}
$$
This deduction uses the hypothesis; zero <cup products> by themselves would not yet prove <formality of a topological space>.
Choose a homogeneous <cohomology> <basis> $h_\lambda$ and cocycle representatives $z_\lambda\in A$. Their indecomposable parts $q(z_\lambda)$ are linearly independent. Extend them to a graded <basis> of $V$ and replace the corresponding generators by the actual cocycles $z_\lambda$. This is a valid change of free algebra generators: each $z_\lambda$ differs from its indecomposable part by products of generators of strictly smaller degree, so degree induction constructs the inverse substitution. Call these closed generators $C$, and the remaining generators $N$.
Define a map of graded algebras
$$
\phi:A\longrightarrow H^*(X;\mathbb Q),
\qquad z_\lambda\longmapsto h_\lambda,\quad N\longmapsto0.
$$
Its target has zero differential. Every differential in $A$ is decomposable, and all products of positive target classes vanish, so $\phi$ is a <chain map>. If $z$ is any positive cocycle, its indecomposable part is in the span of the $q(z_\lambda)$, because these form the image of $q_*$. Write $q(z)=\sum_\lambda c_\lambda q(z_\lambda)$. Then $\phi(z)=\sum_\lambda c_\lambda h_\lambda$, while injectivity of $q_*$ gives $[z]=\sum_\lambda c_\lambda h_\lambda$. Thus $\phi$ induces the identity on <cohomology> and is a <quasi-isomorphism>. This proves that $X$ is a <formal space>.
The rational <wedge sum> assertion has a direct topological proof, which also avoids finite-dimensional duality restrictions in the preceding model argument. Choose a homogeneous <basis> $b_\lambda$ of $\widetilde H_*(X;\mathbb Q)$. By surjectivity, each <basis> class is the image under the <Hurewicz homomorphism> of an element of $\pi_{n_\lambda}(X)\otimes\mathbb Q$. In $X_{\mathbb Q}$ this element is represented by an actual sphere map $S^{n_\lambda}\to X_{\mathbb Q}$. Simple connectivity and $H_1=0$ ensure $n_\lambda\ge2$. The <wedge sum> of these maps gives
$$
F:\bigvee_\lambda S^{n_\lambda}\longrightarrow X_{\mathbb Q}
$$
and induces an isomorphism of <rational homology> by the choice of <basis>. The <rational Whitehead theorem> now makes it a <rational homotopy equivalence>, giving
$$
\boxed{X\simeq_{\mathbb Q}\bigvee_\lambda S^{n_\lambda}.}
$$
A <wedge sum> of spheres is a <formal space>. For each individual sphere, its augmented <minimal model> maps by a <quasi-isomorphism> to its sphere cochain algebra, and also to its <cohomology> algebra, by killing the extra odd generator in the even-sphere case. Take based simplicial models of the spheres. Their <wedge sum> is a simplicial pushout over the common basepoint, so the algebra of <rational polynomial differential forms> is the corresponding fibre product of augmented sphere algebras. Taking fibre products of these model maps gives a <quasi-isomorphism> zigzag to the square-zero <cohomology> algebra. This also works for infinitely many sphere summands: the positive parts are products of the individual complexes, and products of vector-space exact sequences over a <field> are exact. In the finite case its minimal resolution is the <minimal model of a wedge of simply connected spheres>. Thus the topological proof establishes both conclusions without imposing finite rank on the sphere <basis>. For the zero reduced-homology case the <wedge sum> is a point. Both the minimal-model construction and the explicit sphere map explain why the surjectivity hypothesis is decisive.
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